LMFDB - Embedded newform 1521.2.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,2,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, 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("1521.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1521.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:	$12.1452461474$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 507)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.35690 q^{2} +3.55496 q^{4} -3.69202 q^{5} -0.801938 q^{7} -3.66487 q^{8} +O(q^{10})$	$q-2.35690 q^{2} +3.55496 q^{4} -3.69202 q^{5} -0.801938 q^{7} -3.66487 q^{8} +8.70171 q^{10} +2.85086 q^{11} +1.89008 q^{14} +1.52781 q^{16} -2.93900 q^{17} -2.44504 q^{19} -13.1250 q^{20} -6.71917 q^{22} +7.78986 q^{23} +8.63102 q^{25} -2.85086 q^{28} -3.85086 q^{29} +2.34481 q^{31} +3.72886 q^{32} +6.92692 q^{34} +2.96077 q^{35} +7.44504 q^{37} +5.76271 q^{38} +13.5308 q^{40} +0.850855 q^{41} -1.61596 q^{43} +10.1347 q^{44} -18.3599 q^{46} -2.44504 q^{47} -6.35690 q^{49} -20.3424 q^{50} +9.96077 q^{53} -10.5254 q^{55} +2.93900 q^{56} +9.07606 q^{58} +5.38404 q^{59} -13.2567 q^{61} -5.52648 q^{62} -11.8442 q^{64} +14.3937 q^{67} -10.4480 q^{68} -6.97823 q^{70} -8.12498 q^{71} -11.8877 q^{73} -17.5472 q^{74} -8.69202 q^{76} -2.28621 q^{77} +5.40581 q^{79} -5.64071 q^{80} -2.00538 q^{82} -7.04892 q^{83} +10.8509 q^{85} +3.80864 q^{86} -10.4480 q^{88} -1.13169 q^{89} +27.6926 q^{92} +5.76271 q^{94} +9.02715 q^{95} -5.94438 q^{97} +14.9825 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 11 q^{4} - 6 q^{5} + 2 q^{7} - 12 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 11 * q^4 - 6 * q^5 + 2 * q^7 - 12 * q^8	$ 3 q - 3 q^{2} + 11 q^{4} - 6 q^{5} + 2 q^{7} - 12 q^{8} - q^{10} - 5 q^{11} + 5 q^{14} + 11 q^{16} + q^{17} - 7 q^{19} - 15 q^{20} - 9 q^{22} + 11 q^{25} + 5 q^{28} + 2 q^{29} - 16 q^{31} - 22 q^{32} - 8 q^{34} - 4 q^{35} + 22 q^{37} + 3 q^{40} - 11 q^{41} - 15 q^{43} - 16 q^{44} - 7 q^{46} - 7 q^{47} - 15 q^{49} + 3 q^{50} + 17 q^{53} + 3 q^{55} - q^{56} + 12 q^{58} + 6 q^{59} - 13 q^{61} + 2 q^{62} + 11 q^{67} + 13 q^{68} - 24 q^{70} + 6 q^{73} - 15 q^{74} - 21 q^{76} - 15 q^{77} + 3 q^{79} + 20 q^{80} - 3 q^{82} - 12 q^{83} + 19 q^{85} + 29 q^{86} + 13 q^{88} - q^{89} - 7 q^{92} + 21 q^{95} - 5 q^{97} + 29 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 11 * q^4 - 6 * q^5 + 2 * q^7 - 12 * q^8 - q^10 - 5 * q^11 + 5 * q^14 + 11 * q^16 + q^17 - 7 * q^19 - 15 * q^20 - 9 * q^22 + 11 * q^25 + 5 * q^28 + 2 * q^29 - 16 * q^31 - 22 * q^32 - 8 * q^34 - 4 * q^35 + 22 * q^37 + 3 * q^40 - 11 * q^41 - 15 * q^43 - 16 * q^44 - 7 * q^46 - 7 * q^47 - 15 * q^49 + 3 * q^50 + 17 * q^53 + 3 * q^55 - q^56 + 12 * q^58 + 6 * q^59 - 13 * q^61 + 2 * q^62 + 11 * q^67 + 13 * q^68 - 24 * q^70 + 6 * q^73 - 15 * q^74 - 21 * q^76 - 15 * q^77 + 3 * q^79 + 20 * q^80 - 3 * q^82 - 12 * q^83 + 19 * q^85 + 29 * q^86 + 13 * q^88 - q^89 - 7 * q^92 + 21 * q^95 - 5 * q^97 + 29 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.35690	−1.66658	−0.833289	−	0.552838i	$-0.813545\pi$
$2$	−2.35690	−1.66658	−0.833289	+	0.552838i	$0.813545\pi$
$3$	0	0
$3$	0	0
$4$	3.55496	1.77748
$5$	−3.69202	−1.65112	−0.825561	−	0.564313i	$-0.809141\pi$
$5$	−3.69202	−1.65112	−0.825561	+	0.564313i	$0.809141\pi$
$6$	0	0
$7$	−0.801938	−0.303104	−0.151552	−	0.988449i	$-0.548427\pi$
$7$	−0.801938	−0.303104	−0.151552	+	0.988449i	$0.548427\pi$
$8$	−3.66487	−1.29573
$9$	0	0
$10$	8.70171	2.75172
$11$	2.85086	0.859565	0.429783	−	0.902932i	$-0.358590\pi$
$11$	2.85086	0.859565	0.429783	+	0.902932i	$0.358590\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	1.89008	0.505146
$15$	0	0
$16$	1.52781	0.381953
$17$	−2.93900	−0.712812	−0.356406	−	0.934331i	$-0.615998\pi$
$17$	−2.93900	−0.712812	−0.356406	+	0.934331i	$0.615998\pi$
$18$	0	0
$19$	−2.44504	−0.560931	−0.280466	−	0.959864i	$-0.590489\pi$
$19$	−2.44504	−0.560931	−0.280466	+	0.959864i	$0.590489\pi$
$20$	−13.1250	−2.93484
$21$	0	0
$22$	−6.71917	−1.43253
$23$	7.78986	1.62430	0.812149	−	0.583451i	$-0.198298\pi$
$23$	7.78986	1.62430	0.812149	+	0.583451i	$0.198298\pi$
$24$	0	0
$25$	8.63102	1.72620
$26$	0	0
$27$	0	0
$28$	−2.85086	−0.538761
$29$	−3.85086	−0.715086	−0.357543	−	0.933897i	$-0.616385\pi$
$29$	−3.85086	−0.715086	−0.357543	+	0.933897i	$0.616385\pi$
$30$	0	0
$31$	2.34481	0.421141	0.210571	−	0.977579i	$-0.432468\pi$
$31$	2.34481	0.421141	0.210571	+	0.977579i	$0.432468\pi$
$32$	3.72886	0.659175
$33$	0	0
$34$	6.92692	1.18796
$35$	2.96077	0.500462
$36$	0	0
$37$	7.44504	1.22396	0.611979	−	0.790874i	$-0.290374\pi$
$37$	7.44504	1.22396	0.611979	+	0.790874i	$0.290374\pi$
$38$	5.76271	0.934835
$39$	0	0
$40$	13.5308	2.13941
$41$	0.850855	0.132881	0.0664406	−	0.997790i	$-0.478836\pi$
$41$	0.850855	0.132881	0.0664406	+	0.997790i	$0.478836\pi$
$42$	0	0
$43$	−1.61596	−0.246431	−0.123216	−	0.992380i	$-0.539321\pi$
$43$	−1.61596	−0.246431	−0.123216	+	0.992380i	$0.539321\pi$
$44$	10.1347	1.52786
$45$	0	0
$46$	−18.3599	−2.70702
$47$	−2.44504	−0.356646	−0.178323	−	0.983972i	$-0.557067\pi$
$47$	−2.44504	−0.356646	−0.178323	+	0.983972i	$0.557067\pi$
$48$	0	0
$49$	−6.35690	−0.908128
$50$	−20.3424	−2.87685
$51$	0	0
$52$	0	0
$53$	9.96077	1.36822	0.684109	−	0.729380i	$-0.260191\pi$
$53$	9.96077	1.36822	0.684109	+	0.729380i	$0.260191\pi$
$54$	0	0
$55$	−10.5254	−1.41925
$56$	2.93900	0.392741
$57$	0	0
$58$	9.07606	1.19175
$59$	5.38404	0.700943	0.350471	−	0.936573i	$-0.386021\pi$
$59$	5.38404	0.700943	0.350471	+	0.936573i	$0.386021\pi$
$60$	0	0
$61$	−13.2567	−1.69734	−0.848671	−	0.528921i	$-0.822597\pi$
$61$	−13.2567	−1.69734	−0.848671	+	0.528921i	$0.822597\pi$
$62$	−5.52648	−0.701864
$63$	0	0
$64$	−11.8442	−1.48052
$65$	0	0
$66$	0	0
$67$	14.3937	1.75847	0.879237	−	0.476384i	$-0.158053\pi$
$67$	14.3937	1.75847	0.879237	+	0.476384i	$0.158053\pi$
$68$	−10.4480	−1.26701
$69$	0	0
$70$	−6.97823	−0.834058
$71$	−8.12498	−0.964258	−0.482129	−	0.876100i	$-0.660136\pi$
$71$	−8.12498	−0.964258	−0.482129	+	0.876100i	$0.660136\pi$
$72$	0	0
$73$	−11.8877	−1.39135	−0.695674	−	0.718357i	$-0.744894\pi$
$73$	−11.8877	−1.39135	−0.695674	+	0.718357i	$0.744894\pi$
$74$	−17.5472	−2.03982
$75$	0	0
$76$	−8.69202	−0.997043
$77$	−2.28621	−0.260538
$78$	0	0
$79$	5.40581	0.608202	0.304101	−	0.952640i	$-0.401644\pi$
$79$	5.40581	0.608202	0.304101	+	0.952640i	$0.401644\pi$
$80$	−5.64071	−0.630651
$81$	0	0
$82$	−2.00538	−0.221457
$83$	−7.04892	−0.773719	−0.386860	−	0.922139i	$-0.626440\pi$
$83$	−7.04892	−0.773719	−0.386860	+	0.922139i	$0.626440\pi$
$84$	0	0
$85$	10.8509	1.17694
$86$	3.80864	0.410696
$87$	0	0
$88$	−10.4480	−1.11376
$89$	−1.13169	−0.119959	−0.0599793	−	0.998200i	$-0.519103\pi$
$89$	−1.13169	−0.119959	−0.0599793	+	0.998200i	$0.519103\pi$
$90$	0	0
$91$	0	0
$92$	27.6926	2.88715
$93$	0	0
$94$	5.76271	0.594378
$95$	9.02715	0.926166
$96$	0	0
$97$	−5.94438	−0.603560	−0.301780	−	0.953378i	$-0.597581\pi$
$97$	−5.94438	−0.603560	−0.301780	+	0.953378i	$0.597581\pi$
$98$	14.9825	1.51347
$99$	0	0
$100$	30.6829	3.06829
$101$	−4.62565	−0.460269	−0.230134	−	0.973159i	$-0.573917\pi$
$101$	−4.62565	−0.460269	−0.230134	+	0.973159i	$0.573917\pi$
$102$	0	0
$103$	1.20775	0.119003	0.0595016	−	0.998228i	$-0.481049\pi$
$103$	1.20775	0.119003	0.0595016	+	0.998228i	$0.481049\pi$
$104$	0	0
$105$	0	0
$106$	−23.4765	−2.28024
$107$	9.52111	0.920440	0.460220	−	0.887805i	$-0.347771\pi$
$107$	9.52111	0.920440	0.460220	+	0.887805i	$0.347771\pi$
$108$	0	0
$109$	−1.78448	−0.170922	−0.0854611	−	0.996342i	$-0.527236\pi$
$109$	−1.78448	−0.170922	−0.0854611	+	0.996342i	$0.527236\pi$
$110$	24.8073	2.36528
$111$	0	0
$112$	−1.22521	−0.115771
$113$	−4.95108	−0.465759	−0.232879	−	0.972506i	$-0.574815\pi$
$113$	−4.95108	−0.465759	−0.232879	+	0.972506i	$0.574815\pi$
$114$	0	0
$115$	−28.7603	−2.68191
$116$	−13.6896	−1.27105
$117$	0	0
$118$	−12.6896	−1.16817
$119$	2.35690	0.216056
$120$	0	0
$121$	−2.87263	−0.261148
$122$	31.2446	2.82875
$123$	0	0
$124$	8.33572	0.748569
$125$	−13.4058	−1.19905
$126$	0	0
$127$	5.67025	0.503153	0.251577	−	0.967837i	$-0.419051\pi$
$127$	5.67025	0.503153	0.251577	+	0.967837i	$0.419051\pi$
$128$	20.4577	1.80822
$129$	0	0
$130$	0	0
$131$	−18.2228	−1.59213	−0.796067	−	0.605208i	$-0.793090\pi$
$131$	−18.2228	−1.59213	−0.796067	+	0.605208i	$0.793090\pi$
$132$	0	0
$133$	1.96077	0.170020
$134$	−33.9245	−2.93063
$135$	0	0
$136$	10.7711	0.923612
$137$	−9.45042	−0.807404	−0.403702	−	0.914891i	$-0.632277\pi$
$137$	−9.45042	−0.807404	−0.403702	+	0.914891i	$0.632277\pi$
$138$	0	0
$139$	4.01507	0.340553	0.170277	−	0.985396i	$-0.445534\pi$
$139$	4.01507	0.340553	0.170277	+	0.985396i	$0.445534\pi$
$140$	10.5254	0.889560
$141$	0	0
$142$	19.1497	1.60701
$143$	0	0
$144$	0	0
$145$	14.2174	1.18069
$146$	28.0180	2.31879
$147$	0	0
$148$	26.4668	2.17556
$149$	−19.4058	−1.58979	−0.794893	−	0.606750i	$-0.792473\pi$
$149$	−19.4058	−1.58979	−0.794893	+	0.606750i	$0.792473\pi$
$150$	0	0
$151$	−12.3623	−1.00603	−0.503014	−	0.864278i	$-0.667775\pi$
$151$	−12.3623	−1.00603	−0.503014	+	0.864278i	$0.667775\pi$
$152$	8.96077	0.726815
$153$	0	0
$154$	5.38835	0.434206
$155$	−8.65710	−0.695355
$156$	0	0
$157$	−18.6775	−1.49063	−0.745315	−	0.666712i	$-0.767701\pi$
$157$	−18.6775	−1.49063	−0.745315	+	0.666712i	$0.767701\pi$
$158$	−12.7409	−1.01361
$159$	0	0
$160$	−13.7670	−1.08838
$161$	−6.24698	−0.492331
$162$	0	0
$163$	12.3394	0.966499	0.483250	−	0.875483i	$-0.339456\pi$
$163$	12.3394	0.966499	0.483250	+	0.875483i	$0.339456\pi$
$164$	3.02475	0.236194
$165$	0	0
$166$	16.6136	1.28946
$167$	−11.4940	−0.889429	−0.444715	−	0.895672i	$-0.646695\pi$
$167$	−11.4940	−0.889429	−0.444715	+	0.895672i	$0.646695\pi$
$168$	0	0
$169$	0	0
$170$	−25.5743	−1.96146
$171$	0	0
$172$	−5.74466	−0.438026
$173$	−12.1142	−0.921028	−0.460514	−	0.887653i	$-0.652335\pi$
$173$	−12.1142	−0.921028	−0.460514	+	0.887653i	$0.652335\pi$
$174$	0	0
$175$	−6.92154	−0.523219
$176$	4.35557	0.328313
$177$	0	0
$178$	2.66727	0.199920
$179$	0.538565	0.0402542	0.0201271	−	0.999797i	$-0.493593\pi$
$179$	0.538565	0.0402542	0.0201271	+	0.999797i	$0.493593\pi$
$180$	0	0
$181$	−23.2838	−1.73067	−0.865336	−	0.501192i	$-0.832895\pi$
$181$	−23.2838	−1.73067	−0.865336	+	0.501192i	$0.832895\pi$
$182$	0	0
$183$	0	0
$184$	−28.5488	−2.10465
$185$	−27.4873	−2.02090
$186$	0	0
$187$	−8.37867	−0.612709
$188$	−8.69202	−0.633931
$189$	0	0
$190$	−21.2760	−1.54353
$191$	−16.7657	−1.21312	−0.606561	−	0.795037i	$-0.707452\pi$
$191$	−16.7657	−1.21312	−0.606561	+	0.795037i	$0.707452\pi$
$192$	0	0
$193$	25.7439	1.85309	0.926544	−	0.376186i	$-0.122765\pi$
$193$	25.7439	1.85309	0.926544	+	0.376186i	$0.122765\pi$
$194$	14.0103	1.00588
$195$	0	0
$196$	−22.5985	−1.61418
$197$	−21.4209	−1.52617	−0.763087	−	0.646296i	$-0.776317\pi$
$197$	−21.4209	−1.52617	−0.763087	+	0.646296i	$0.776317\pi$
$198$	0	0
$199$	3.52781	0.250080	0.125040	−	0.992152i	$-0.460094\pi$
$199$	3.52781	0.250080	0.125040	+	0.992152i	$0.460094\pi$
$200$	−31.6316	−2.23669
$201$	0	0
$202$	10.9022	0.767074
$203$	3.08815	0.216745
$204$	0	0
$205$	−3.14138	−0.219403
$206$	−2.84654	−0.198328
$207$	0	0
$208$	0	0
$209$	−6.97046	−0.482157
$210$	0	0
$211$	1.21552	0.0836799	0.0418399	−	0.999124i	$-0.486678\pi$
$211$	1.21552	0.0836799	0.0418399	+	0.999124i	$0.486678\pi$
$212$	35.4101	2.43198
$213$	0	0
$214$	−22.4403	−1.53398
$215$	5.96615	0.406888
$216$	0	0
$217$	−1.88040	−0.127650
$218$	4.20583	0.284855
$219$	0	0
$220$	−37.4174	−2.52268
$221$	0	0
$222$	0	0
$223$	−17.3884	−1.16441	−0.582205	−	0.813042i	$-0.697810\pi$
$223$	−17.3884	−1.16441	−0.582205	+	0.813042i	$0.697810\pi$
$224$	−2.99031	−0.199799
$225$	0	0
$226$	11.6692	0.776223
$227$	−17.4155	−1.15591	−0.577954	−	0.816070i	$-0.696149\pi$
$227$	−17.4155	−1.15591	−0.577954	+	0.816070i	$0.696149\pi$
$228$	0	0
$229$	18.7603	1.23972	0.619858	−	0.784714i	$-0.287190\pi$
$229$	18.7603	1.23972	0.619858	+	0.784714i	$0.287190\pi$
$230$	67.7851	4.46962
$231$	0	0
$232$	14.1129	0.926557
$233$	−3.95108	−0.258844	−0.129422	−	0.991590i	$-0.541312\pi$
$233$	−3.95108	−0.258844	−0.129422	+	0.991590i	$0.541312\pi$
$234$	0	0
$235$	9.02715	0.588866
$236$	19.1400	1.24591
$237$	0	0
$238$	−5.55496	−0.360074
$239$	−0.818331	−0.0529334	−0.0264667	−	0.999650i	$-0.508426\pi$
$239$	−0.818331	−0.0529334	−0.0264667	+	0.999650i	$0.508426\pi$
$240$	0	0
$241$	6.03252	0.388589	0.194295	−	0.980943i	$-0.437758\pi$
$241$	6.03252	0.388589	0.194295	+	0.980943i	$0.437758\pi$
$242$	6.77048	0.435223
$243$	0	0
$244$	−47.1269	−3.01699
$245$	23.4698	1.49943
$246$	0	0
$247$	0	0
$248$	−8.59345	−0.545685
$249$	0	0
$250$	31.5961	1.99831
$251$	26.8799	1.69665	0.848323	−	0.529479i	$-0.177613\pi$
$251$	26.8799	1.69665	0.848323	+	0.529479i	$0.177613\pi$
$252$	0	0
$253$	22.2078	1.39619
$254$	−13.3642	−0.838544
$255$	0	0
$256$	−24.5284	−1.53303
$257$	9.05323	0.564725	0.282362	−	0.959308i	$-0.408882\pi$
$257$	9.05323	0.564725	0.282362	+	0.959308i	$0.408882\pi$
$258$	0	0
$259$	−5.97046	−0.370986
$260$	0	0
$261$	0	0
$262$	42.9493	2.65342
$263$	−23.1511	−1.42756	−0.713778	−	0.700372i	$-0.753017\pi$
$263$	−23.1511	−1.42756	−0.713778	+	0.700372i	$0.753017\pi$
$264$	0	0
$265$	−36.7754	−2.25909
$266$	−4.62133	−0.283352
$267$	0	0
$268$	51.1691	3.12565
$269$	2.42088	0.147604	0.0738018	−	0.997273i	$-0.476487\pi$
$269$	2.42088	0.147604	0.0738018	+	0.997273i	$0.476487\pi$
$270$	0	0
$271$	−21.4450	−1.30269	−0.651347	−	0.758780i	$-0.725796\pi$
$271$	−21.4450	−1.30269	−0.651347	+	0.758780i	$0.725796\pi$
$272$	−4.49024	−0.272261
$273$	0	0
$274$	22.2737	1.34560
$275$	24.6058	1.48379
$276$	0	0
$277$	−14.8073	−0.889685	−0.444843	−	0.895609i	$-0.646740\pi$
$277$	−14.8073	−0.889685	−0.444843	+	0.895609i	$0.646740\pi$
$278$	−9.46309	−0.567558
$279$	0	0
$280$	−10.8509	−0.648463
$281$	−14.5036	−0.865215	−0.432608	−	0.901582i	$-0.642406\pi$
$281$	−14.5036	−0.865215	−0.432608	+	0.901582i	$0.642406\pi$
$282$	0	0
$283$	25.6722	1.52605	0.763026	−	0.646368i	$-0.223713\pi$
$283$	25.6722	1.52605	0.763026	+	0.646368i	$0.223713\pi$
$284$	−28.8840	−1.71395
$285$	0	0
$286$	0	0
$287$	−0.682333	−0.0402768
$288$	0	0
$289$	−8.36227	−0.491898
$290$	−33.5090	−1.96772
$291$	0	0
$292$	−42.2602	−2.47309
$293$	26.5230	1.54949	0.774746	−	0.632273i	$-0.217878\pi$
$293$	26.5230	1.54949	0.774746	+	0.632273i	$0.217878\pi$
$294$	0	0
$295$	−19.8780	−1.15734
$296$	−27.2851	−1.58592
$297$	0	0
$298$	45.7375	2.64950
$299$	0	0
$300$	0	0
$301$	1.29590	0.0746943
$302$	29.1366	1.67662
$303$	0	0
$304$	−3.73556	−0.214249
$305$	48.9439	2.80252
$306$	0	0
$307$	−8.24698	−0.470680	−0.235340	−	0.971913i	$-0.575620\pi$
$307$	−8.24698	−0.470680	−0.235340	+	0.971913i	$0.575620\pi$
$308$	−8.12737	−0.463100
$309$	0	0
$310$	20.4039	1.15886
$311$	14.4179	0.817564	0.408782	−	0.912632i	$-0.365954\pi$
$311$	14.4179	0.817564	0.408782	+	0.912632i	$0.365954\pi$
$312$	0	0
$313$	14.2338	0.804544	0.402272	−	0.915520i	$-0.368221\pi$
$313$	14.2338	0.804544	0.402272	+	0.915520i	$0.368221\pi$
$314$	44.0210	2.48425
$315$	0	0
$316$	19.2174	1.08107
$317$	6.84415	0.384406	0.192203	−	0.981355i	$-0.438437\pi$
$317$	6.84415	0.384406	0.192203	+	0.981355i	$0.438437\pi$
$318$	0	0
$319$	−10.9782	−0.614663
$320$	43.7289	2.44452
$321$	0	0
$322$	14.7235	0.820507
$323$	7.18598	0.399839
$324$	0	0
$325$	0	0
$326$	−29.0828	−1.61075
$327$	0	0
$328$	−3.11828	−0.172178
$329$	1.96077	0.108101
$330$	0	0
$331$	9.44265	0.519015	0.259507	−	0.965741i	$-0.416440\pi$
$331$	9.44265	0.519015	0.259507	+	0.965741i	$0.416440\pi$
$332$	−25.0586	−1.37527
$333$	0	0
$334$	27.0901	1.48230
$335$	−53.1420	−2.90346
$336$	0	0
$337$	2.64310	0.143979	0.0719895	−	0.997405i	$-0.477065\pi$
$337$	2.64310	0.143979	0.0719895	+	0.997405i	$0.477065\pi$
$338$	0	0
$339$	0	0
$340$	38.5743	2.09199
$341$	6.68473	0.361998
$342$	0	0
$343$	10.7114	0.578361
$344$	5.92228	0.319308
$345$	0	0
$346$	28.5520	1.53496
$347$	10.1588	0.545355	0.272677	−	0.962106i	$-0.412091\pi$
$347$	10.1588	0.545355	0.272677	+	0.962106i	$0.412091\pi$
$348$	0	0
$349$	10.4397	0.558822	0.279411	−	0.960172i	$-0.409861\pi$
$349$	10.4397	0.558822	0.279411	+	0.960172i	$0.409861\pi$
$350$	16.3134	0.871986
$351$	0	0
$352$	10.6304	0.566604
$353$	−18.2911	−0.973538	−0.486769	−	0.873531i	$-0.661825\pi$
$353$	−18.2911	−0.973538	−0.486769	+	0.873531i	$0.661825\pi$
$354$	0	0
$355$	29.9976	1.59211
$356$	−4.02310	−0.213224
$357$	0	0
$358$	−1.26934	−0.0670867
$359$	15.2731	0.806081	0.403041	−	0.915182i	$-0.367953\pi$
$359$	15.2731	0.806081	0.403041	+	0.915182i	$0.367953\pi$
$360$	0	0
$361$	−13.0218	−0.685356
$362$	54.8775	2.88430
$363$	0	0
$364$	0	0
$365$	43.8896	2.29729
$366$	0	0
$367$	−22.2717	−1.16258	−0.581288	−	0.813698i	$-0.697451\pi$
$367$	−22.2717	−1.16258	−0.581288	+	0.813698i	$0.697451\pi$
$368$	11.9014	0.620405
$369$	0	0
$370$	64.7846	3.36799
$371$	−7.98792	−0.414712
$372$	0	0
$373$	4.12631	0.213652	0.106826	−	0.994278i	$-0.465931\pi$
$373$	4.12631	0.213652	0.106826	+	0.994278i	$0.465931\pi$
$374$	19.7476	1.02113
$375$	0	0
$376$	8.96077	0.462116
$377$	0	0
$378$	0	0
$379$	−10.7071	−0.549986	−0.274993	−	0.961446i	$-0.588676\pi$
$379$	−10.7071	−0.549986	−0.274993	+	0.961446i	$0.588676\pi$
$380$	32.0911	1.64624
$381$	0	0
$382$	39.5150	2.02176
$383$	−6.52648	−0.333488	−0.166744	−	0.986000i	$-0.553325\pi$
$383$	−6.52648	−0.333488	−0.166744	+	0.986000i	$0.553325\pi$
$384$	0	0
$385$	8.44073	0.430179
$386$	−60.6757	−3.08831
$387$	0	0
$388$	−21.1320	−1.07282
$389$	11.7922	0.597891	0.298945	−	0.954270i	$-0.403365\pi$
$389$	11.7922	0.597891	0.298945	+	0.954270i	$0.403365\pi$
$390$	0	0
$391$	−22.8944	−1.15782
$392$	23.2972	1.17669
$393$	0	0
$394$	50.4868	2.54349
$395$	−19.9584	−1.00422
$396$	0	0
$397$	−12.5429	−0.629509	−0.314754	−	0.949173i	$-0.601922\pi$
$397$	−12.5429	−0.629509	−0.314754	+	0.949173i	$0.601922\pi$
$398$	−8.31468	−0.416777
$399$	0	0
$400$	13.1866	0.659329
$401$	−17.8702	−0.892397	−0.446198	−	0.894934i	$-0.647222\pi$
$401$	−17.8702	−0.892397	−0.446198	+	0.894934i	$0.647222\pi$
$402$	0	0
$403$	0	0
$404$	−16.4440	−0.818118
$405$	0	0
$406$	−7.27844	−0.361223
$407$	21.2247	1.05207
$408$	0	0
$409$	−27.9119	−1.38015	−0.690076	−	0.723737i	$-0.742423\pi$
$409$	−27.9119	−1.38015	−0.690076	+	0.723737i	$0.742423\pi$
$410$	7.40389	0.365652
$411$	0	0
$412$	4.29350	0.211526
$413$	−4.31767	−0.212459
$414$	0	0
$415$	26.0248	1.27750
$416$	0	0
$417$	0	0
$418$	16.4286	0.803551
$419$	16.4034	0.801360	0.400680	−	0.916218i	$-0.368774\pi$
$419$	16.4034	0.801360	0.400680	+	0.916218i	$0.368774\pi$
$420$	0	0
$421$	−3.03684	−0.148006	−0.0740032	−	0.997258i	$-0.523577\pi$
$421$	−3.03684	−0.148006	−0.0740032	+	0.997258i	$0.523577\pi$
$422$	−2.86486	−0.139459
$423$	0	0
$424$	−36.5050	−1.77284
$425$	−25.3666	−1.23046
$426$	0	0
$427$	10.6310	0.514471
$428$	33.8471	1.63606
$429$	0	0
$430$	−14.0616	−0.678110
$431$	3.33811	0.160791	0.0803955	−	0.996763i	$-0.474382\pi$
$431$	3.33811	0.160791	0.0803955	+	0.996763i	$0.474382\pi$
$432$	0	0
$433$	11.9028	0.572010	0.286005	−	0.958228i	$-0.407673\pi$
$433$	11.9028	0.572010	0.286005	+	0.958228i	$0.407673\pi$
$434$	4.43190	0.212738
$435$	0	0
$436$	−6.34375	−0.303810
$437$	−19.0465	−0.911119
$438$	0	0
$439$	3.71810	0.177455	0.0887277	−	0.996056i	$-0.471720\pi$
$439$	3.71810	0.177455	0.0887277	+	0.996056i	$0.471720\pi$
$440$	38.5743	1.83896
$441$	0	0
$442$	0	0
$443$	−1.45712	−0.0692300	−0.0346150	−	0.999401i	$-0.511021\pi$
$443$	−1.45712	−0.0692300	−0.0346150	+	0.999401i	$0.511021\pi$
$444$	0	0
$445$	4.17821	0.198066
$446$	40.9825	1.94058
$447$	0	0
$448$	9.49827	0.448751
$449$	12.1274	0.572326	0.286163	−	0.958181i	$-0.407620\pi$
$449$	12.1274	0.572326	0.286163	+	0.958181i	$0.407620\pi$
$450$	0	0
$451$	2.42566	0.114220
$452$	−17.6009	−0.827876
$453$	0	0
$454$	41.0465	1.92641
$455$	0	0
$456$	0	0
$457$	−3.44803	−0.161292	−0.0806459	−	0.996743i	$-0.525698\pi$
$457$	−3.44803	−0.161292	−0.0806459	+	0.996743i	$0.525698\pi$
$458$	−44.2161	−2.06608
$459$	0	0
$460$	−102.242	−4.76704
$461$	6.75600	0.314658	0.157329	−	0.987546i	$-0.449712\pi$
$461$	6.75600	0.314658	0.157329	+	0.987546i	$0.449712\pi$
$462$	0	0
$463$	−7.45175	−0.346312	−0.173156	−	0.984894i	$-0.555396\pi$
$463$	−7.45175	−0.346312	−0.173156	+	0.984894i	$0.555396\pi$
$464$	−5.88338	−0.273129
$465$	0	0
$466$	9.31229	0.431384
$467$	−32.6098	−1.50900	−0.754502	−	0.656298i	$-0.772121\pi$
$467$	−32.6098	−1.50900	−0.754502	+	0.656298i	$0.772121\pi$
$468$	0	0
$469$	−11.5429	−0.533001
$470$	−21.2760	−0.981391
$471$	0	0
$472$	−19.7318	−0.908232
$473$	−4.60686	−0.211824
$474$	0	0
$475$	−21.1032	−0.968282
$476$	8.37867	0.384036
$477$	0	0
$478$	1.92872	0.0882177
$479$	−2.82908	−0.129264	−0.0646321	−	0.997909i	$-0.520587\pi$
$479$	−2.82908	−0.129264	−0.0646321	+	0.997909i	$0.520587\pi$
$480$	0	0
$481$	0	0
$482$	−14.2180	−0.647614
$483$	0	0
$484$	−10.2121	−0.464185
$485$	21.9468	0.996552
$486$	0	0
$487$	−41.2935	−1.87119	−0.935594	−	0.353079i	$-0.885135\pi$
$487$	−41.2935	−1.87119	−0.935594	+	0.353079i	$0.885135\pi$
$488$	48.5840	2.19930
$489$	0	0
$490$	−55.3159	−2.49892
$491$	34.6698	1.56463	0.782313	−	0.622886i	$-0.214040\pi$
$491$	34.6698	1.56463	0.782313	+	0.622886i	$0.214040\pi$
$492$	0	0
$493$	11.3177	0.509722
$494$	0	0
$495$	0	0
$496$	3.58243	0.160856
$497$	6.51573	0.292270
$498$	0	0
$499$	17.9409	0.803146	0.401573	−	0.915827i	$-0.368464\pi$
$499$	17.9409	0.803146	0.401573	+	0.915827i	$0.368464\pi$
$500$	−47.6571	−2.13129
$501$	0	0
$502$	−63.3532	−2.82759
$503$	26.1812	1.16736	0.583681	−	0.811983i	$-0.301612\pi$
$503$	26.1812	1.16736	0.583681	+	0.811983i	$0.301612\pi$
$504$	0	0
$505$	17.0780	0.759960
$506$	−52.3414	−2.32686
$507$	0	0
$508$	20.1575	0.894345
$509$	5.50604	0.244051	0.122025	−	0.992527i	$-0.461061\pi$
$509$	5.50604	0.244051	0.122025	+	0.992527i	$0.461061\pi$
$510$	0	0
$511$	9.53319	0.421723
$512$	16.8955	0.746681
$513$	0	0
$514$	−21.3375	−0.941158
$515$	−4.45904	−0.196489
$516$	0	0
$517$	−6.97046	−0.306560
$518$	14.0718	0.618277
$519$	0	0
$520$	0	0
$521$	26.7211	1.17067	0.585336	−	0.810791i	$-0.300963\pi$
$521$	26.7211	1.17067	0.585336	+	0.810791i	$0.300963\pi$
$522$	0	0
$523$	36.5230	1.59704	0.798520	−	0.601968i	$-0.205617\pi$
$523$	36.5230	1.59704	0.798520	+	0.601968i	$0.205617\pi$
$524$	−64.7813	−2.82999
$525$	0	0
$526$	54.5646	2.37913
$527$	−6.89141	−0.300195
$528$	0	0
$529$	37.6819	1.63834
$530$	86.6757	3.76495
$531$	0	0
$532$	6.97046	0.302208
$533$	0	0
$534$	0	0
$535$	−35.1521	−1.51976
$536$	−52.7512	−2.27851
$537$	0	0
$538$	−5.70576	−0.245993
$539$	−18.1226	−0.780595
$540$	0	0
$541$	18.4655	0.793893	0.396947	−	0.917842i	$-0.370070\pi$
$541$	18.4655	0.793893	0.396947	+	0.917842i	$0.370070\pi$
$542$	50.5437	2.17104
$543$	0	0
$544$	−10.9591	−0.469868
$545$	6.58834	0.282213
$546$	0	0
$547$	39.8471	1.70374	0.851870	−	0.523753i	$-0.175468\pi$
$547$	39.8471	1.70374	0.851870	+	0.523753i	$0.175468\pi$
$548$	−33.5958	−1.43514
$549$	0	0
$550$	−57.9933	−2.47284
$551$	9.41550	0.401114
$552$	0	0
$553$	−4.33513	−0.184348
$554$	34.8993	1.48273
$555$	0	0
$556$	14.2734	0.605327
$557$	−9.20477	−0.390018	−0.195009	−	0.980801i	$-0.562474\pi$
$557$	−9.20477	−0.390018	−0.195009	+	0.980801i	$0.562474\pi$
$558$	0	0
$559$	0	0
$560$	4.52350	0.191153
$561$	0	0
$562$	34.1836	1.44195
$563$	0.975246	0.0411017	0.0205509	−	0.999789i	$-0.493458\pi$
$563$	0.975246	0.0411017	0.0205509	+	0.999789i	$0.493458\pi$
$564$	0	0
$565$	18.2795	0.769024
$566$	−60.5066	−2.54328
$567$	0	0
$568$	29.7770	1.24942
$569$	16.8944	0.708250	0.354125	−	0.935198i	$-0.384779\pi$
$569$	16.8944	0.708250	0.354125	+	0.935198i	$0.384779\pi$
$570$	0	0
$571$	−44.3226	−1.85484	−0.927421	−	0.374019i	$-0.877979\pi$
$571$	−44.3226	−1.85484	−0.927421	+	0.374019i	$0.877979\pi$
$572$	0	0
$573$	0	0
$574$	1.60819	0.0671244
$575$	67.2344	2.80387
$576$	0	0
$577$	3.56704	0.148498	0.0742489	−	0.997240i	$-0.476344\pi$
$577$	3.56704	0.148498	0.0742489	+	0.997240i	$0.476344\pi$
$578$	19.7090	0.819787
$579$	0	0
$580$	50.5424	2.09866
$581$	5.65279	0.234517
$582$	0	0
$583$	28.3967	1.17607
$584$	43.5669	1.80281
$585$	0	0
$586$	−62.5120	−2.58235
$587$	16.1172	0.665229	0.332614	−	0.943063i	$-0.392069\pi$
$587$	16.1172	0.665229	0.332614	+	0.943063i	$0.392069\pi$
$588$	0	0
$589$	−5.73317	−0.236231
$590$	46.8504	1.92880
$591$	0	0
$592$	11.3746	0.467494
$593$	42.8611	1.76010	0.880048	−	0.474885i	$-0.157510\pi$
$593$	42.8611	1.76010	0.880048	+	0.474885i	$0.157510\pi$
$594$	0	0
$595$	−8.70171	−0.356735
$596$	−68.9869	−2.82581
$597$	0	0
$598$	0	0
$599$	−40.9420	−1.67284	−0.836422	−	0.548086i	$-0.815357\pi$
$599$	−40.9420	−1.67284	−0.836422	+	0.548086i	$0.815357\pi$
$600$	0	0
$601$	1.18705	0.0484206	0.0242103	−	0.999707i	$-0.492293\pi$
$601$	1.18705	0.0484206	0.0242103	+	0.999707i	$0.492293\pi$
$602$	−3.05429	−0.124484
$603$	0	0
$604$	−43.9474	−1.78819
$605$	10.6058	0.431187
$606$	0	0
$607$	19.9922	0.811460	0.405730	−	0.913993i	$-0.367017\pi$
$607$	19.9922	0.811460	0.405730	+	0.913993i	$0.367017\pi$
$608$	−9.11721	−0.369752
$609$	0	0
$610$	−115.356	−4.67062
$611$	0	0
$612$	0	0
$613$	−33.3618	−1.34747	−0.673735	−	0.738973i	$-0.735311\pi$
$613$	−33.3618	−1.34747	−0.673735	+	0.738973i	$0.735311\pi$
$614$	19.4373	0.784424
$615$	0	0
$616$	8.37867	0.337586
$617$	−11.6233	−0.467935	−0.233967	−	0.972244i	$-0.575171\pi$
$617$	−11.6233	−0.467935	−0.233967	+	0.972244i	$0.575171\pi$
$618$	0	0
$619$	−16.5381	−0.664722	−0.332361	−	0.943152i	$-0.607845\pi$
$619$	−16.5381	−0.664722	−0.332361	+	0.943152i	$0.607845\pi$
$620$	−30.7756	−1.23598
$621$	0	0
$622$	−33.9815	−1.36253
$623$	0.907542	0.0363599
$624$	0	0
$625$	6.33944	0.253577
$626$	−33.5477	−1.34083
$627$	0	0
$628$	−66.3979	−2.64956
$629$	−21.8810	−0.872452
$630$	0	0
$631$	−36.4416	−1.45072	−0.725358	−	0.688372i	$-0.758326\pi$
$631$	−36.4416	−1.45072	−0.725358	+	0.688372i	$0.758326\pi$
$632$	−19.8116	−0.788064
$633$	0	0
$634$	−16.1309	−0.640642
$635$	−20.9347	−0.830768
$636$	0	0
$637$	0	0
$638$	25.8745	1.02438
$639$	0	0
$640$	−75.5303	−2.98560
$641$	−27.2067	−1.07460	−0.537300	−	0.843391i	$-0.680556\pi$
$641$	−27.2067	−1.07460	−0.537300	+	0.843391i	$0.680556\pi$
$642$	0	0
$643$	−5.06962	−0.199926	−0.0999632	−	0.994991i	$-0.531873\pi$
$643$	−5.06962	−0.199926	−0.0999632	+	0.994991i	$0.531873\pi$
$644$	−22.2078	−0.875108
$645$	0	0
$646$	−16.9366	−0.666362
$647$	−19.3207	−0.759573	−0.379787	−	0.925074i	$-0.624003\pi$
$647$	−19.3207	−0.759573	−0.379787	+	0.925074i	$0.624003\pi$
$648$	0	0
$649$	15.3491	0.602506
$650$	0	0
$651$	0	0
$652$	43.8662	1.71793
$653$	−35.2355	−1.37887	−0.689436	−	0.724347i	$-0.742141\pi$
$653$	−35.2355	−1.37887	−0.689436	+	0.724347i	$0.742141\pi$
$654$	0	0
$655$	67.2790	2.62881
$656$	1.29995	0.0507544
$657$	0	0
$658$	−4.62133	−0.180158
$659$	4.36168	0.169907	0.0849535	−	0.996385i	$-0.472926\pi$
$659$	4.36168	0.169907	0.0849535	+	0.996385i	$0.472926\pi$
$660$	0	0
$661$	−15.4709	−0.601747	−0.300873	−	0.953664i	$-0.597278\pi$
$661$	−15.4709	−0.601747	−0.300873	+	0.953664i	$0.597278\pi$
$662$	−22.2553	−0.864978
$663$	0	0
$664$	25.8334	1.00253
$665$	−7.23921	−0.280725
$666$	0	0
$667$	−29.9976	−1.16151
$668$	−40.8605	−1.58094
$669$	0	0
$670$	125.250	4.83883
$671$	−37.7928	−1.45898
$672$	0	0
$673$	−11.7409	−0.452580	−0.226290	−	0.974060i	$-0.572660\pi$
$673$	−11.7409	−0.452580	−0.226290	+	0.974060i	$0.572660\pi$
$674$	−6.22952	−0.239952
$675$	0	0
$676$	0	0
$677$	−3.44504	−0.132404	−0.0662019	−	0.997806i	$-0.521088\pi$
$677$	−3.44504	−0.132404	−0.0662019	+	0.997806i	$0.521088\pi$
$678$	0	0
$679$	4.76702	0.182941
$680$	−39.7670	−1.52500
$681$	0	0
$682$	−15.7552	−0.603298
$683$	−20.4058	−0.780807	−0.390403	−	0.920644i	$-0.627664\pi$
$683$	−20.4058	−0.780807	−0.390403	+	0.920644i	$0.627664\pi$
$684$	0	0
$685$	34.8911	1.33312
$686$	−25.2457	−0.963883
$687$	0	0
$688$	−2.46888	−0.0941251
$689$	0	0
$690$	0	0
$691$	27.8039	1.05771	0.528854	−	0.848713i	$-0.322622\pi$
$691$	27.8039	1.05771	0.528854	+	0.848713i	$0.322622\pi$
$692$	−43.0656	−1.63711
$693$	0	0
$694$	−23.9433	−0.908876
$695$	−14.8237	−0.562295
$696$	0	0
$697$	−2.50066	−0.0947194
$698$	−24.6052	−0.931321
$699$	0	0
$700$	−24.6058	−0.930012
$701$	−11.9715	−0.452158	−0.226079	−	0.974109i	$-0.572591\pi$
$701$	−11.9715	−0.452158	−0.226079	+	0.974109i	$0.572591\pi$
$702$	0	0
$703$	−18.2034	−0.686556
$704$	−33.7660	−1.27260
$705$	0	0
$706$	43.1102	1.62248
$707$	3.70948	0.139509
$708$	0	0
$709$	32.2664	1.21179	0.605894	−	0.795545i	$-0.292815\pi$
$709$	32.2664	1.21179	0.605894	+	0.795545i	$0.292815\pi$
$710$	−70.7012	−2.65337
$711$	0	0
$712$	4.14749	0.155434
$713$	18.2658	0.684058
$714$	0	0
$715$	0	0
$716$	1.91457	0.0715510
$717$	0	0
$718$	−35.9970	−1.34340
$719$	−12.1086	−0.451574	−0.225787	−	0.974177i	$-0.572495\pi$
$719$	−12.1086	−0.451574	−0.225787	+	0.974177i	$0.572495\pi$
$720$	0	0
$721$	−0.968541	−0.0360704
$722$	30.6910	1.14220
$723$	0	0
$724$	−82.7730	−3.07623
$725$	−33.2368	−1.23438
$726$	0	0
$727$	−16.6200	−0.616402	−0.308201	−	0.951321i	$-0.599727\pi$
$727$	−16.6200	−0.616402	−0.308201	+	0.951321i	$0.599727\pi$
$728$	0	0
$729$	0	0
$730$	−103.443	−3.82861
$731$	4.74930	0.175659
$732$	0	0
$733$	17.7912	0.657132	0.328566	−	0.944481i	$-0.393435\pi$
$733$	17.7912	0.657132	0.328566	+	0.944481i	$0.393435\pi$
$734$	52.4922	1.93752
$735$	0	0
$736$	29.0473	1.07070
$737$	41.0344	1.51152
$738$	0	0
$739$	−27.3618	−1.00652	−0.503260	−	0.864135i	$-0.667866\pi$
$739$	−27.3618	−1.00652	−0.503260	+	0.864135i	$0.667866\pi$
$740$	−97.7160	−3.59211
$741$	0	0
$742$	18.8267	0.691150
$743$	−8.38596	−0.307651	−0.153826	−	0.988098i	$-0.549159\pi$
$743$	−8.38596	−0.307651	−0.153826	+	0.988098i	$0.549159\pi$
$744$	0	0
$745$	71.6467	2.62493
$746$	−9.72528	−0.356068
$747$	0	0
$748$	−29.7858	−1.08908
$749$	−7.63533	−0.278989
$750$	0	0
$751$	−38.7778	−1.41502	−0.707511	−	0.706703i	$-0.750182\pi$
$751$	−38.7778	−1.41502	−0.707511	+	0.706703i	$0.750182\pi$
$752$	−3.73556	−0.136222
$753$	0	0
$754$	0	0
$755$	45.6418	1.66107
$756$	0	0
$757$	12.9729	0.471506	0.235753	−	0.971813i	$-0.424244\pi$
$757$	12.9729	0.471506	0.235753	+	0.971813i	$0.424244\pi$
$758$	25.2355	0.916594
$759$	0	0
$760$	−33.0834	−1.20006
$761$	5.15585	0.186899	0.0934497	−	0.995624i	$-0.470211\pi$
$761$	5.15585	0.186899	0.0934497	+	0.995624i	$0.470211\pi$
$762$	0	0
$763$	1.43104	0.0518072
$764$	−59.6013	−2.15630
$765$	0	0
$766$	15.3822	0.555783
$767$	0	0
$768$	0	0
$769$	−35.5013	−1.28021	−0.640104	−	0.768288i	$-0.721109\pi$
$769$	−35.5013	−1.28021	−0.640104	+	0.768288i	$0.721109\pi$
$770$	−19.8939	−0.716927
$771$	0	0
$772$	91.5186	3.29383
$773$	6.15585	0.221411	0.110705	−	0.993853i	$-0.464689\pi$
$773$	6.15585	0.221411	0.110705	+	0.993853i	$0.464689\pi$
$774$	0	0
$775$	20.2381	0.726976
$776$	21.7854	0.782050
$777$	0	0
$778$	−27.7931	−0.996431
$779$	−2.08038	−0.0745372
$780$	0	0
$781$	−23.1631	−0.828843
$782$	53.9597	1.92960
$783$	0	0
$784$	−9.71214	−0.346862
$785$	68.9579	2.46121
$786$	0	0
$787$	14.2107	0.506558	0.253279	−	0.967393i	$-0.418491\pi$
$787$	14.2107	0.506558	0.253279	+	0.967393i	$0.418491\pi$
$788$	−76.1503	−2.71274
$789$	0	0
$790$	47.0398	1.67360
$791$	3.97046	0.141173
$792$	0	0
$793$	0	0
$794$	29.5623	1.04913
$795$	0	0
$796$	12.5412	0.444512
$797$	−11.9022	−0.421596	−0.210798	−	0.977530i	$-0.567606\pi$
$797$	−11.9022	−0.421596	−0.210798	+	0.977530i	$0.567606\pi$
$798$	0	0
$799$	7.18598	0.254222
$800$	32.1839	1.13787
$801$	0	0
$802$	42.1183	1.48725
$803$	−33.8901	−1.19596
$804$	0	0
$805$	23.0640	0.812899
$806$	0	0
$807$	0	0
$808$	16.9524	0.596384
$809$	−1.61596	−0.0568140	−0.0284070	−	0.999596i	$-0.509043\pi$
$809$	−1.61596	−0.0568140	−0.0284070	+	0.999596i	$0.509043\pi$
$810$	0	0
$811$	−51.9657	−1.82476	−0.912381	−	0.409342i	$-0.865758\pi$
$811$	−51.9657	−1.82476	−0.912381	+	0.409342i	$0.865758\pi$
$812$	10.9782	0.385260
$813$	0	0
$814$	−50.0245	−1.75336
$815$	−45.5575	−1.59581
$816$	0	0
$817$	3.95108	0.138231
$818$	65.7853	2.30013
$819$	0	0
$820$	−11.1675	−0.389985
$821$	−54.8327	−1.91367	−0.956837	−	0.290627i	$-0.906136\pi$
$821$	−54.8327	−1.91367	−0.956837	+	0.290627i	$0.906136\pi$
$822$	0	0
$823$	−46.8514	−1.63314	−0.816569	−	0.577247i	$-0.804127\pi$
$823$	−46.8514	−1.63314	−0.816569	+	0.577247i	$0.804127\pi$
$824$	−4.42626	−0.154196
$825$	0	0
$826$	10.1763	0.354078
$827$	−21.2021	−0.737270	−0.368635	−	0.929574i	$-0.620175\pi$
$827$	−21.2021	−0.737270	−0.368635	+	0.929574i	$0.620175\pi$
$828$	0	0
$829$	18.2972	0.635489	0.317744	−	0.948176i	$-0.397075\pi$
$829$	18.2972	0.635489	0.317744	+	0.948176i	$0.397075\pi$
$830$	−61.3376	−2.12906
$831$	0	0
$832$	0	0
$833$	18.6829	0.647325
$834$	0	0
$835$	42.4359	1.46856
$836$	−24.7797	−0.857024
$837$	0	0
$838$	−38.6612	−1.33553
$839$	−21.1414	−0.729881	−0.364941	−	0.931031i	$-0.618911\pi$
$839$	−21.1414	−0.729881	−0.364941	+	0.931031i	$0.618911\pi$
$840$	0	0
$841$	−14.1709	−0.488652
$842$	7.15751	0.246664
$843$	0	0
$844$	4.32113	0.148739
$845$	0	0
$846$	0	0
$847$	2.30367	0.0791549
$848$	15.2182	0.522594
$849$	0	0
$850$	59.7864	2.05066
$851$	57.9958	1.98807
$852$	0	0
$853$	−7.13036	−0.244139	−0.122069	−	0.992522i	$-0.538953\pi$
$853$	−7.13036	−0.244139	−0.122069	+	0.992522i	$0.538953\pi$
$854$	−25.0562	−0.857406
$855$	0	0
$856$	−34.8937	−1.19264
$857$	44.7741	1.52945	0.764726	−	0.644355i	$-0.222874\pi$
$857$	44.7741	1.52945	0.764726	+	0.644355i	$0.222874\pi$
$858$	0	0
$859$	57.3782	1.95772	0.978859	−	0.204535i	$-0.0655681\pi$
$859$	57.3782	1.95772	0.978859	+	0.204535i	$0.0655681\pi$
$860$	21.2094	0.723235
$861$	0	0
$862$	−7.86758	−0.267971
$863$	6.67563	0.227241	0.113621	−	0.993524i	$-0.463755\pi$
$863$	6.67563	0.227241	0.113621	+	0.993524i	$0.463755\pi$
$864$	0	0
$865$	44.7260	1.52073
$866$	−28.0536	−0.953299
$867$	0	0
$868$	−6.68473	−0.226894
$869$	15.4112	0.522789
$870$	0	0
$871$	0	0
$872$	6.53989	0.221469
$873$	0	0
$874$	44.8907	1.51845
$875$	10.7506	0.363438
$876$	0	0
$877$	25.2983	0.854263	0.427131	−	0.904190i	$-0.359524\pi$
$877$	25.2983	0.854263	0.427131	+	0.904190i	$0.359524\pi$
$878$	−8.76318	−0.295743
$879$	0	0
$880$	−16.0809	−0.542085
$881$	35.3787	1.19194	0.595969	−	0.803008i	$-0.296768\pi$
$881$	35.3787	1.19194	0.595969	+	0.803008i	$0.296768\pi$
$882$	0	0
$883$	−11.5851	−0.389869	−0.194935	−	0.980816i	$-0.562449\pi$
$883$	−11.5851	−0.389869	−0.194935	+	0.980816i	$0.562449\pi$
$884$	0	0
$885$	0	0
$886$	3.43429	0.115377
$887$	27.2892	0.916281	0.458141	−	0.888880i	$-0.348516\pi$
$887$	27.2892	0.916281	0.458141	+	0.888880i	$0.348516\pi$
$888$	0	0
$889$	−4.54719	−0.152508
$890$	−9.84761	−0.330093
$891$	0	0
$892$	−61.8149	−2.06972
$893$	5.97823	0.200054
$894$	0	0
$895$	−1.98839	−0.0664646
$896$	−16.4058	−0.548080
$897$	0	0
$898$	−28.5830	−0.953826
$899$	−9.02954	−0.301152
$900$	0	0
$901$	−29.2747	−0.975282
$902$	−5.71704	−0.190357
$903$	0	0
$904$	18.1451	0.603497
$905$	85.9643	2.85755
$906$	0	0
$907$	−30.4219	−1.01014	−0.505072	−	0.863077i	$-0.668534\pi$
$907$	−30.4219	−1.01014	−0.505072	+	0.863077i	$0.668534\pi$
$908$	−61.9114	−2.05460
$909$	0	0
$910$	0	0
$911$	−53.5719	−1.77492	−0.887459	−	0.460887i	$-0.847531\pi$
$911$	−53.5719	−1.77492	−0.887459	+	0.460887i	$0.847531\pi$
$912$	0	0
$913$	−20.0954	−0.665062
$914$	8.12664	0.268805
$915$	0	0
$916$	66.6921	2.20357
$917$	14.6136	0.482582
$918$	0	0
$919$	−36.1672	−1.19305	−0.596523	−	0.802596i	$-0.703451\pi$
$919$	−36.1672	−1.19305	−0.596523	+	0.802596i	$0.703451\pi$
$920$	105.403	3.47503
$921$	0	0
$922$	−15.9232	−0.524403
$923$	0	0
$924$	0	0
$925$	64.2583	2.11280
$926$	17.5630	0.577156
$927$	0	0
$928$	−14.3593	−0.471367
$929$	13.3478	0.437927	0.218964	−	0.975733i	$-0.429732\pi$
$929$	13.3478	0.437927	0.218964	+	0.975733i	$0.429732\pi$
$930$	0	0
$931$	15.5429	0.509397
$932$	−14.0459	−0.460090
$933$	0	0
$934$	76.8580	2.51487
$935$	30.9342	1.01166
$936$	0	0
$937$	38.6872	1.26386	0.631928	−	0.775027i	$-0.282264\pi$
$937$	38.6872	1.26386	0.631928	+	0.775027i	$0.282264\pi$
$938$	27.2054	0.888286
$939$	0	0
$940$	32.0911	1.04670
$941$	−35.7275	−1.16468	−0.582342	−	0.812944i	$-0.697864\pi$
$941$	−35.7275	−1.16468	−0.582342	+	0.812944i	$0.697864\pi$
$942$	0	0
$943$	6.62804	0.215839
$944$	8.22580	0.267727
$945$	0	0
$946$	10.8579	0.353020
$947$	17.8436	0.579838	0.289919	−	0.957051i	$-0.406372\pi$
$947$	17.8436	0.579838	0.289919	+	0.957051i	$0.406372\pi$
$948$	0	0
$949$	0	0
$950$	49.7381	1.61372
$951$	0	0
$952$	−8.63773	−0.279950
$953$	−20.1691	−0.653342	−0.326671	−	0.945138i	$-0.605927\pi$
$953$	−20.1691	−0.653342	−0.326671	+	0.945138i	$0.605927\pi$
$954$	0	0
$955$	61.8993	2.00301
$956$	−2.90913	−0.0940881
$957$	0	0
$958$	6.66786	0.215429
$959$	7.57865	0.244727
$960$	0	0
$961$	−25.5018	−0.822640
$962$	0	0
$963$	0	0
$964$	21.4454	0.690709
$965$	−95.0471	−3.05967
$966$	0	0
$967$	−32.7894	−1.05444	−0.527218	−	0.849730i	$-0.676765\pi$
$967$	−32.7894	−1.05444	−0.527218	+	0.849730i	$0.676765\pi$
$968$	10.5278	0.338377
$969$	0	0
$970$	−51.7263	−1.66083
$971$	26.9845	0.865973	0.432986	−	0.901401i	$-0.357460\pi$
$971$	26.9845	0.865973	0.432986	+	0.901401i	$0.357460\pi$
$972$	0	0
$973$	−3.21983	−0.103223
$974$	97.3245	3.11848
$975$	0	0
$976$	−20.2537	−0.648305
$977$	16.9571	0.542504	0.271252	−	0.962508i	$-0.412562\pi$
$977$	16.9571	0.542504	0.271252	+	0.962508i	$0.412562\pi$
$978$	0	0
$979$	−3.22627	−0.103112
$980$	83.4341	2.66521
$981$	0	0
$982$	−81.7131	−2.60757
$983$	32.6631	1.04179	0.520895	−	0.853621i	$-0.325598\pi$
$983$	32.6631	1.04179	0.520895	+	0.853621i	$0.325598\pi$
$984$	0	0
$985$	79.0863	2.51990
$986$	−26.6746	−0.849491
$987$	0	0
$988$	0	0
$989$	−12.5881	−0.400277
$990$	0	0
$991$	7.64310	0.242791	0.121396	−	0.992604i	$-0.461263\pi$
$991$	7.64310	0.242791	0.121396	+	0.992604i	$0.461263\pi$
$992$	8.74348	0.277606
$993$	0	0
$994$	−15.3569	−0.487091
$995$	−13.0248	−0.412912
$996$	0	0
$997$	36.1256	1.14411	0.572054	−	0.820216i	$-0.306147\pi$
$997$	36.1256	1.14411	0.572054	+	0.820216i	$0.306147\pi$
$998$	−42.2849	−1.33850
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.2.a.n.1.2		3
3.2	odd	2		507.2.a.l.1.2	yes	3
12.11	even	2		8112.2.a.cp.1.3		3
13.5	odd	4		1521.2.b.k.1351.5		6
13.8	odd	4		1521.2.b.k.1351.2		6
13.12	even	2		1521.2.a.s.1.2		3
39.2	even	12		507.2.j.i.316.5		12
39.5	even	4		507.2.b.f.337.2		6
39.8	even	4		507.2.b.f.337.5		6
39.11	even	12		507.2.j.i.316.2		12
39.17	odd	6		507.2.e.l.484.2		6
39.20	even	12		507.2.j.i.361.5		12
39.23	odd	6		507.2.e.l.22.2		6
39.29	odd	6		507.2.e.i.22.2		6
39.32	even	12		507.2.j.i.361.2		12
39.35	odd	6		507.2.e.i.484.2		6
39.38	odd	2		507.2.a.i.1.2	✓	3
156.155	even	2		8112.2.a.cg.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.a.i.1.2	✓	3	39.38	odd	2
507.2.a.l.1.2	yes	3	3.2	odd	2
507.2.b.f.337.2		6	39.5	even	4
507.2.b.f.337.5		6	39.8	even	4
507.2.e.i.22.2		6	39.29	odd	6
507.2.e.i.484.2		6	39.35	odd	6
507.2.e.l.22.2		6	39.23	odd	6
507.2.e.l.484.2		6	39.17	odd	6
507.2.j.i.316.2		12	39.11	even	12
507.2.j.i.316.5		12	39.2	even	12
507.2.j.i.361.2		12	39.32	even	12
507.2.j.i.361.5		12	39.20	even	12
1521.2.a.n.1.2		3	1.1	even	1	trivial
1521.2.a.s.1.2		3	13.12	even	2
1521.2.b.k.1351.2		6	13.8	odd	4
1521.2.b.k.1351.5		6	13.5	odd	4
8112.2.a.cg.1.1		3	156.155	even	2
8112.2.a.cp.1.3		3	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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q-2.35690 q^{2} +3.55496 q^{4} -3.69202 q^{5} -0.801938 q^{7} -3.66487 q^{8} +O(q^{10})\)	\(q-2.35690 q^{2} +3.55496 q^{4} -3.69202 q^{5} -0.801938 q^{7} -3.66487 q^{8} +8.70171 q^{10} +2.85086 q^{11} +1.89008 q^{14} +1.52781 q^{16} -2.93900 q^{17} -2.44504 q^{19} -13.1250 q^{20} -6.71917 q^{22} +7.78986 q^{23} +8.63102 q^{25} -2.85086 q^{28} -3.85086 q^{29} +2.34481 q^{31} +3.72886 q^{32} +6.92692 q^{34} +2.96077 q^{35} +7.44504 q^{37} +5.76271 q^{38} +13.5308 q^{40} +0.850855 q^{41} -1.61596 q^{43} +10.1347 q^{44} -18.3599 q^{46} -2.44504 q^{47} -6.35690 q^{49} -20.3424 q^{50} +9.96077 q^{53} -10.5254 q^{55} +2.93900 q^{56} +9.07606 q^{58} +5.38404 q^{59} -13.2567 q^{61} -5.52648 q^{62} -11.8442 q^{64} +14.3937 q^{67} -10.4480 q^{68} -6.97823 q^{70} -8.12498 q^{71} -11.8877 q^{73} -17.5472 q^{74} -8.69202 q^{76} -2.28621 q^{77} +5.40581 q^{79} -5.64071 q^{80} -2.00538 q^{82} -7.04892 q^{83} +10.8509 q^{85} +3.80864 q^{86} -10.4480 q^{88} -1.13169 q^{89} +27.6926 q^{92} +5.76271 q^{94} +9.02715 q^{95} -5.94438 q^{97} +14.9825 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 11 q^{4} - 6 q^{5} + 2 q^{7} - 12 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 11 * q^4 - 6 * q^5 + 2 * q^7 - 12 * q^8	\( 3 q - 3 q^{2} + 11 q^{4} - 6 q^{5} + 2 q^{7} - 12 q^{8} - q^{10} - 5 q^{11} + 5 q^{14} + 11 q^{16} + q^{17} - 7 q^{19} - 15 q^{20} - 9 q^{22} + 11 q^{25} + 5 q^{28} + 2 q^{29} - 16 q^{31} - 22 q^{32} - 8 q^{34} - 4 q^{35} + 22 q^{37} + 3 q^{40} - 11 q^{41} - 15 q^{43} - 16 q^{44} - 7 q^{46} - 7 q^{47} - 15 q^{49} + 3 q^{50} + 17 q^{53} + 3 q^{55} - q^{56} + 12 q^{58} + 6 q^{59} - 13 q^{61} + 2 q^{62} + 11 q^{67} + 13 q^{68} - 24 q^{70} + 6 q^{73} - 15 q^{74} - 21 q^{76} - 15 q^{77} + 3 q^{79} + 20 q^{80} - 3 q^{82} - 12 q^{83} + 19 q^{85} + 29 q^{86} + 13 q^{88} - q^{89} - 7 q^{92} + 21 q^{95} - 5 q^{97} + 29 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 11 * q^4 - 6 * q^5 + 2 * q^7 - 12 * q^8 - q^10 - 5 * q^11 + 5 * q^14 + 11 * q^16 + q^17 - 7 * q^19 - 15 * q^20 - 9 * q^22 + 11 * q^25 + 5 * q^28 + 2 * q^29 - 16 * q^31 - 22 * q^32 - 8 * q^34 - 4 * q^35 + 22 * q^37 + 3 * q^40 - 11 * q^41 - 15 * q^43 - 16 * q^44 - 7 * q^46 - 7 * q^47 - 15 * q^49 + 3 * q^50 + 17 * q^53 + 3 * q^55 - q^56 + 12 * q^58 + 6 * q^59 - 13 * q^61 + 2 * q^62 + 11 * q^67 + 13 * q^68 - 24 * q^70 + 6 * q^73 - 15 * q^74 - 21 * q^76 - 15 * q^77 + 3 * q^79 + 20 * q^80 - 3 * q^82 - 12 * q^83 + 19 * q^85 + 29 * q^86 + 13 * q^88 - q^89 - 7 * q^92 + 21 * q^95 - 5 * q^97 + 29 * q^98

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