LMFDB - Embedded newform 1287.2.a.q.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1287,2,Mod(1,1287)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1287, 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("1287.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1287 = 3^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1287.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:	$10.2767467401$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.194616205.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 10x^{4} - 2x^{3} + 24x^{2} + 7x - 12 $ x^6 - 10x^4 - 2x^3 + 24x^2 + 7x - 12
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 143)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.70899$ of defining polynomial
Character	$\chi$	$=$	1287.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.70899 q^{2} +5.33863 q^{4} -0.610508 q^{5} +2.05182 q^{7} +9.04431 q^{8} +O(q^{10})$	\(q+2.70899 q^{2} +5.33863 q^{4} -0.610508 q^{5} +2.05182 q^{7} +9.04431 q^{8} -1.65386 q^{10} +1.00000 q^{11} +1.00000 q^{13} +5.55836 q^{14} +13.8237 q^{16} -5.41798 q^{17} -0.783578 q^{19} -3.25927 q^{20} +2.70899 q^{22} -8.01493 q^{23} -4.62728 q^{25} +2.70899 q^{26} +10.9539 q^{28} +10.3311 q^{29} -1.04997 q^{31} +19.3596 q^{32} -14.6773 q^{34} -1.25265 q^{35} +4.64877 q^{37} -2.12271 q^{38} -5.52162 q^{40} -8.72907 q^{41} -0.339519 q^{43} +5.33863 q^{44} -21.7124 q^{46} +7.07184 q^{47} -2.79004 q^{49} -12.5353 q^{50} +5.33863 q^{52} +5.79004 q^{53} -0.610508 q^{55} +18.5573 q^{56} +27.9869 q^{58} +3.97342 q^{59} -1.69212 q^{61} -2.84437 q^{62} +24.7976 q^{64} -0.610508 q^{65} -6.06675 q^{67} -28.9246 q^{68} -3.39342 q^{70} +13.7860 q^{71} -1.85542 q^{73} +12.5935 q^{74} -4.18323 q^{76} +2.05182 q^{77} -8.33111 q^{79} -8.43947 q^{80} -23.6470 q^{82} -6.00516 q^{83} +3.30772 q^{85} -0.919754 q^{86} +9.04431 q^{88} -12.6991 q^{89} +2.05182 q^{91} -42.7887 q^{92} +19.1575 q^{94} +0.478381 q^{95} +6.70383 q^{97} -7.55818 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 8 q^{4} - q^{5} + 4 q^{7} + 6 q^{8}+O(q^{10}) $ 6 * q + 8 * q^4 - q^5 + 4 * q^7 + 6 * q^8	$ 6 q + 8 q^{4} - q^{5} + 4 q^{7} + 6 q^{8} + 6 q^{10} + 6 q^{11} + 6 q^{13} + 12 q^{14} + 8 q^{16} - 10 q^{19} - 4 q^{20} - 11 q^{23} + 23 q^{25} + 9 q^{28} - 2 q^{29} - 9 q^{31} + 17 q^{32} - 40 q^{34} + 24 q^{35} + 15 q^{37} + 9 q^{38} + 16 q^{40} + 4 q^{41} - 2 q^{43} + 8 q^{44} - 6 q^{46} - 6 q^{47} + 20 q^{49} + 4 q^{50} + 8 q^{52} - 2 q^{53} - q^{55} + 39 q^{56} + 18 q^{58} - 11 q^{59} + 16 q^{61} - 16 q^{62} + 36 q^{64} - q^{65} + 9 q^{67} - 12 q^{68} + 32 q^{70} + 15 q^{71} + 32 q^{73} - 22 q^{74} - 26 q^{76} + 4 q^{77} + 14 q^{79} - 56 q^{80} - 24 q^{82} + 26 q^{83} - 12 q^{85} + 10 q^{86} + 6 q^{88} + 23 q^{89} + 4 q^{91} - 83 q^{92} + 46 q^{94} + 52 q^{95} + 27 q^{97} + 3 q^{98}+O(q^{100}) $ 6 * q + 8 * q^4 - q^5 + 4 * q^7 + 6 * q^8 + 6 * q^10 + 6 * q^11 + 6 * q^13 + 12 * q^14 + 8 * q^16 - 10 * q^19 - 4 * q^20 - 11 * q^23 + 23 * q^25 + 9 * q^28 - 2 * q^29 - 9 * q^31 + 17 * q^32 - 40 * q^34 + 24 * q^35 + 15 * q^37 + 9 * q^38 + 16 * q^40 + 4 * q^41 - 2 * q^43 + 8 * q^44 - 6 * q^46 - 6 * q^47 + 20 * q^49 + 4 * q^50 + 8 * q^52 - 2 * q^53 - q^55 + 39 * q^56 + 18 * q^58 - 11 * q^59 + 16 * q^61 - 16 * q^62 + 36 * q^64 - q^65 + 9 * q^67 - 12 * q^68 + 32 * q^70 + 15 * q^71 + 32 * q^73 - 22 * q^74 - 26 * q^76 + 4 * q^77 + 14 * q^79 - 56 * q^80 - 24 * q^82 + 26 * q^83 - 12 * q^85 + 10 * q^86 + 6 * q^88 + 23 * q^89 + 4 * q^91 - 83 * q^92 + 46 * q^94 + 52 * q^95 + 27 * q^97 + 3 * 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 $

$2$	2.70899	1.91555	0.957773	−	0.287527i	$-0.0928330\pi$
$2$	2.70899	1.91555	0.957773	+	0.287527i	$0.0928330\pi$
$3$	0	0
$3$	0	0
$4$	5.33863	2.66931
$5$	−0.610508	−0.273027	−0.136514	−	0.990638i	$-0.543590\pi$
$5$	−0.610508	−0.273027	−0.136514	+	0.990638i	$0.543590\pi$
$6$	0	0
$7$	2.05182	0.775515	0.387757	−	0.921761i	$-0.373250\pi$
$7$	2.05182	0.775515	0.387757	+	0.921761i	$0.373250\pi$
$8$	9.04431	3.19765
$9$	0	0
$10$	−1.65386	−0.522996
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	5.55836	1.48553
$15$	0	0
$16$	13.8237	3.45592
$17$	−5.41798	−1.31405	−0.657027	−	0.753867i	$-0.728186\pi$
$17$	−5.41798	−1.31405	−0.657027	+	0.753867i	$0.728186\pi$
$18$	0	0
$19$	−0.783578	−0.179765	−0.0898826	−	0.995952i	$-0.528649\pi$
$19$	−0.783578	−0.179765	−0.0898826	+	0.995952i	$0.528649\pi$
$20$	−3.25927	−0.728796
$21$	0	0
$22$	2.70899	0.577559
$23$	−8.01493	−1.67123	−0.835614	−	0.549317i	$-0.814888\pi$
$23$	−8.01493	−1.67123	−0.835614	+	0.549317i	$0.814888\pi$
$24$	0	0
$25$	−4.62728	−0.925456
$26$	2.70899	0.531277
$27$	0	0
$28$	10.9539	2.07009
$29$	10.3311	1.91844	0.959220	−	0.282661i	$-0.0912173\pi$
$29$	10.3311	1.91844	0.959220	+	0.282661i	$0.0912173\pi$
$30$	0	0
$31$	−1.04997	−0.188581	−0.0942904	−	0.995545i	$-0.530058\pi$
$31$	−1.04997	−0.188581	−0.0942904	+	0.995545i	$0.530058\pi$
$32$	19.3596	3.42233
$33$	0	0
$34$	−14.6773	−2.51713
$35$	−1.25265	−0.211737
$36$	0	0
$37$	4.64877	0.764252	0.382126	−	0.924110i	$-0.375192\pi$
$37$	4.64877	0.764252	0.382126	+	0.924110i	$0.375192\pi$
$38$	−2.12271	−0.344348
$39$	0	0
$40$	−5.52162	−0.873045
$41$	−8.72907	−1.36325	−0.681626	−	0.731701i	$-0.738727\pi$
$41$	−8.72907	−1.36325	−0.681626	+	0.731701i	$0.738727\pi$
$42$	0	0
$43$	−0.339519	−0.0517762	−0.0258881	−	0.999665i	$-0.508241\pi$
$43$	−0.339519	−0.0517762	−0.0258881	+	0.999665i	$0.508241\pi$
$44$	5.33863	0.804828
$45$	0	0
$46$	−21.7124	−3.20131
$47$	7.07184	1.03153	0.515767	−	0.856729i	$-0.327507\pi$
$47$	7.07184	1.03153	0.515767	+	0.856729i	$0.327507\pi$
$48$	0	0
$49$	−2.79004	−0.398577
$50$	−12.5353	−1.77275
$51$	0	0
$52$	5.33863	0.740334
$53$	5.79004	0.795323	0.397661	−	0.917532i	$-0.369822\pi$
$53$	5.79004	0.795323	0.397661	+	0.917532i	$0.369822\pi$
$54$	0	0
$55$	−0.610508	−0.0823209
$56$	18.5573	2.47982
$57$	0	0
$58$	27.9869	3.67486
$59$	3.97342	0.517295	0.258648	−	0.965972i	$-0.416723\pi$
$59$	3.97342	0.517295	0.258648	+	0.965972i	$0.416723\pi$
$60$	0	0
$61$	−1.69212	−0.216653	−0.108327	−	0.994115i	$-0.534549\pi$
$61$	−1.69212	−0.216653	−0.108327	+	0.994115i	$0.534549\pi$
$62$	−2.84437	−0.361235
$63$	0	0
$64$	24.7976	3.09970
$65$	−0.610508	−0.0757242
$66$	0	0
$67$	−6.06675	−0.741171	−0.370585	−	0.928798i	$-0.620843\pi$
$67$	−6.06675	−0.741171	−0.370585	+	0.928798i	$0.620843\pi$
$68$	−28.9246	−3.50762
$69$	0	0
$70$	−3.39342	−0.405591
$71$	13.7860	1.63610	0.818048	−	0.575150i	$-0.195056\pi$
$71$	13.7860	1.63610	0.818048	+	0.575150i	$0.195056\pi$
$72$	0	0
$73$	−1.85542	−0.217160	−0.108580	−	0.994088i	$-0.534630\pi$
$73$	−1.85542	−0.217160	−0.108580	+	0.994088i	$0.534630\pi$
$74$	12.5935	1.46396
$75$	0	0
$76$	−4.18323	−0.479849
$77$	2.05182	0.233827
$78$	0	0
$79$	−8.33111	−0.937323	−0.468662	−	0.883378i	$-0.655264\pi$
$79$	−8.33111	−0.937323	−0.468662	+	0.883378i	$0.655264\pi$
$80$	−8.43947	−0.943561
$81$	0	0
$82$	−23.6470	−2.61137
$83$	−6.00516	−0.659151	−0.329576	−	0.944129i	$-0.606906\pi$
$83$	−6.00516	−0.659151	−0.329576	+	0.944129i	$0.606906\pi$
$84$	0	0
$85$	3.30772	0.358773
$86$	−0.919754	−0.0991796
$87$	0	0
$88$	9.04431	0.964126
$89$	−12.6991	−1.34610	−0.673052	−	0.739595i	$-0.735017\pi$
$89$	−12.6991	−1.34610	−0.673052	+	0.739595i	$0.735017\pi$
$90$	0	0
$91$	2.05182	0.215089
$92$	−42.7887	−4.46103
$93$	0	0
$94$	19.1575	1.97595
$95$	0.478381	0.0490808
$96$	0	0
$97$	6.70383	0.680671	0.340336	−	0.940304i	$-0.389459\pi$
$97$	6.70383	0.680671	0.340336	+	0.940304i	$0.389459\pi$
$98$	−7.55818	−0.763491
$99$	0	0
$100$	−24.7033	−2.47033
$101$	−0.401208	−0.0399216	−0.0199608	−	0.999801i	$-0.506354\pi$
$101$	−0.401208	−0.0399216	−0.0199608	+	0.999801i	$0.506354\pi$
$102$	0	0
$103$	−11.2236	−1.10589	−0.552947	−	0.833216i	$-0.686497\pi$
$103$	−11.2236	−1.10589	−0.552947	+	0.833216i	$0.686497\pi$
$104$	9.04431	0.886867
$105$	0	0
$106$	15.6851	1.52348
$107$	5.49515	0.531236	0.265618	−	0.964078i	$-0.414424\pi$
$107$	5.49515	0.531236	0.265618	+	0.964078i	$0.414424\pi$
$108$	0	0
$109$	−7.42257	−0.710954	−0.355477	−	0.934685i	$-0.615681\pi$
$109$	−7.42257	−0.710954	−0.355477	+	0.934685i	$0.615681\pi$
$110$	−1.65386	−0.157689
$111$	0	0
$112$	28.3637	2.68012
$113$	−14.0239	−1.31926	−0.659628	−	0.751592i	$-0.729286\pi$
$113$	−14.0239	−1.31926	−0.659628	+	0.751592i	$0.729286\pi$
$114$	0	0
$115$	4.89317	0.456291
$116$	55.1540	5.12092
$117$	0	0
$118$	10.7640	0.990903
$119$	−11.1167	−1.01907
$120$	0	0
$121$	1.00000	0.0909091
$122$	−4.58393	−0.415009
$123$	0	0
$124$	−5.60541	−0.503381
$125$	5.87753	0.525702
$126$	0	0
$127$	1.49515	0.132673	0.0663367	−	0.997797i	$-0.478869\pi$
$127$	1.49515	0.132673	0.0663367	+	0.997797i	$0.478869\pi$
$128$	28.4573	2.51529
$129$	0	0
$130$	−1.65386	−0.145053
$131$	−9.31625	−0.813964	−0.406982	−	0.913436i	$-0.633419\pi$
$131$	−9.31625	−0.813964	−0.406982	+	0.913436i	$0.633419\pi$
$132$	0	0
$133$	−1.60776	−0.139411
$134$	−16.4348	−1.41975
$135$	0	0
$136$	−49.0019	−4.20188
$137$	−9.58902	−0.819245	−0.409623	−	0.912255i	$-0.634340\pi$
$137$	−9.58902	−0.819245	−0.409623	+	0.912255i	$0.634340\pi$
$138$	0	0
$139$	−11.5624	−0.980714	−0.490357	−	0.871522i	$-0.663134\pi$
$139$	−11.5624	−0.980714	−0.490357	+	0.871522i	$0.663134\pi$
$140$	−6.68744	−0.565192
$141$	0	0
$142$	37.3461	3.13402
$143$	1.00000	0.0836242
$144$	0	0
$145$	−6.30723	−0.523787
$146$	−5.02631	−0.415980
$147$	0	0
$148$	24.8180	2.04003
$149$	−8.83271	−0.723604	−0.361802	−	0.932255i	$-0.617838\pi$
$149$	−8.83271	−0.723604	−0.361802	+	0.932255i	$0.617838\pi$
$150$	0	0
$151$	2.51855	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$151$	2.51855	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$152$	−7.08692	−0.574825
$153$	0	0
$154$	5.55836	0.447905
$155$	0.641017	0.0514877
$156$	0	0
$157$	13.3946	1.06901	0.534505	−	0.845166i	$-0.320498\pi$
$157$	13.3946	1.06901	0.534505	+	0.845166i	$0.320498\pi$
$158$	−22.5689	−1.79549
$159$	0	0
$160$	−11.8192	−0.934389
$161$	−16.4452	−1.29606
$162$	0	0
$163$	13.2372	1.03681	0.518407	−	0.855134i	$-0.326525\pi$
$163$	13.2372	1.03681	0.518407	+	0.855134i	$0.326525\pi$
$164$	−46.6013	−3.63895
$165$	0	0
$166$	−16.2679	−1.26263
$167$	20.0381	1.55060	0.775299	−	0.631595i	$-0.217599\pi$
$167$	20.0381	1.55060	0.775299	+	0.631595i	$0.217599\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	8.96058	0.687245
$171$	0	0
$172$	−1.81257	−0.138207
$173$	−22.6688	−1.72348	−0.861740	−	0.507350i	$-0.830625\pi$
$173$	−22.6688	−1.72348	−0.861740	+	0.507350i	$0.830625\pi$
$174$	0	0
$175$	−9.49435	−0.717705
$176$	13.8237	1.04200
$177$	0	0
$178$	−34.4018	−2.57852
$179$	6.94824	0.519336	0.259668	−	0.965698i	$-0.416387\pi$
$179$	6.94824	0.519336	0.259668	+	0.965698i	$0.416387\pi$
$180$	0	0
$181$	10.2352	0.760778	0.380389	−	0.924827i	$-0.375790\pi$
$181$	10.2352	0.760778	0.380389	+	0.924827i	$0.375790\pi$
$182$	5.55836	0.412013
$183$	0	0
$184$	−72.4894	−5.34399
$185$	−2.83811	−0.208662
$186$	0	0
$187$	−5.41798	−0.396202
$188$	37.7539	2.75349
$189$	0	0
$190$	1.29593	0.0940165
$191$	−10.6527	−0.770802	−0.385401	−	0.922749i	$-0.625937\pi$
$191$	−10.6527	−0.770802	−0.385401	+	0.922749i	$0.625937\pi$
$192$	0	0
$193$	8.02018	0.577305	0.288653	−	0.957434i	$-0.406793\pi$
$193$	8.02018	0.577305	0.288653	+	0.957434i	$0.406793\pi$
$194$	18.1606	1.30386
$195$	0	0
$196$	−14.8950	−1.06393
$197$	19.5561	1.39331	0.696656	−	0.717405i	$-0.254670\pi$
$197$	19.5561	1.39331	0.696656	+	0.717405i	$0.254670\pi$
$198$	0	0
$199$	6.49263	0.460250	0.230125	−	0.973161i	$-0.426086\pi$
$199$	6.49263	0.460250	0.230125	+	0.973161i	$0.426086\pi$
$200$	−41.8505	−2.95928
$201$	0	0
$202$	−1.08687	−0.0764717
$203$	21.1976	1.48778
$204$	0	0
$205$	5.32917	0.372205
$206$	−30.4046	−2.11839
$207$	0	0
$208$	13.8237	0.958500
$209$	−0.783578	−0.0542012
$210$	0	0
$211$	−2.58136	−0.177708	−0.0888542	−	0.996045i	$-0.528321\pi$
$211$	−2.58136	−0.177708	−0.0888542	+	0.996045i	$0.528321\pi$
$212$	30.9108	2.12297
$213$	0	0
$214$	14.8863	1.01761
$215$	0.207279	0.0141363
$216$	0	0
$217$	−2.15436	−0.146247
$218$	−20.1077	−1.36186
$219$	0	0
$220$	−3.25927	−0.219740
$221$	−5.41798	−0.364453
$222$	0	0
$223$	10.3213	0.691168	0.345584	−	0.938388i	$-0.387681\pi$
$223$	10.3213	0.691168	0.345584	+	0.938388i	$0.387681\pi$
$224$	39.7224	2.65407
$225$	0	0
$226$	−37.9906	−2.52710
$227$	12.7098	0.843578	0.421789	−	0.906694i	$-0.361402\pi$
$227$	12.7098	0.843578	0.421789	+	0.906694i	$0.361402\pi$
$228$	0	0
$229$	9.15552	0.605014	0.302507	−	0.953147i	$-0.402176\pi$
$229$	9.15552	0.605014	0.302507	+	0.953147i	$0.402176\pi$
$230$	13.2556	0.874046
$231$	0	0
$232$	93.4377	6.13449
$233$	11.6241	0.761522	0.380761	−	0.924673i	$-0.375662\pi$
$233$	11.6241	0.761522	0.380761	+	0.924673i	$0.375662\pi$
$234$	0	0
$235$	−4.31741	−0.281637
$236$	21.2126	1.38082
$237$	0	0
$238$	−30.1151	−1.95207
$239$	5.65845	0.366015	0.183007	−	0.983112i	$-0.441417\pi$
$239$	5.65845	0.366015	0.183007	+	0.983112i	$0.441417\pi$
$240$	0	0
$241$	26.2974	1.69396	0.846981	−	0.531624i	$-0.178418\pi$
$241$	26.2974	1.69396	0.846981	+	0.531624i	$0.178418\pi$
$242$	2.70899	0.174140
$243$	0	0
$244$	−9.03358	−0.578316
$245$	1.70334	0.108822
$246$	0	0
$247$	−0.783578	−0.0498579
$248$	−9.49628	−0.603014
$249$	0	0
$250$	15.9222	1.00701
$251$	−16.7642	−1.05815	−0.529073	−	0.848576i	$-0.677460\pi$
$251$	−16.7642	−1.05815	−0.529073	+	0.848576i	$0.677460\pi$
$252$	0	0
$253$	−8.01493	−0.503894
$254$	4.05035	0.254142
$255$	0	0
$256$	27.4952	1.71845
$257$	−12.5968	−0.785769	−0.392884	−	0.919588i	$-0.628523\pi$
$257$	−12.5968	−0.785769	−0.392884	+	0.919588i	$0.628523\pi$
$258$	0	0
$259$	9.53843	0.592689
$260$	−3.25927	−0.202132
$261$	0	0
$262$	−25.2376	−1.55919
$263$	21.3883	1.31886	0.659428	−	0.751768i	$-0.270799\pi$
$263$	21.3883	1.31886	0.659428	+	0.751768i	$0.270799\pi$
$264$	0	0
$265$	−3.53486	−0.217145
$266$	−4.35541	−0.267047
$267$	0	0
$268$	−32.3881	−1.97842
$269$	5.97465	0.364281	0.182140	−	0.983273i	$-0.441697\pi$
$269$	5.97465	0.364281	0.182140	+	0.983273i	$0.441697\pi$
$270$	0	0
$271$	−13.7062	−0.832595	−0.416297	−	0.909228i	$-0.636672\pi$
$271$	−13.7062	−0.832595	−0.416297	+	0.909228i	$0.636672\pi$
$272$	−74.8964	−4.54126
$273$	0	0
$274$	−25.9766	−1.56930
$275$	−4.62728	−0.279035
$276$	0	0
$277$	12.7275	0.764720	0.382360	−	0.924013i	$-0.375111\pi$
$277$	12.7275	0.764720	0.382360	+	0.924013i	$0.375111\pi$
$278$	−31.3225	−1.87860
$279$	0	0
$280$	−11.3294	−0.677059
$281$	3.95536	0.235957	0.117979	−	0.993016i	$-0.462359\pi$
$281$	3.95536	0.235957	0.117979	+	0.993016i	$0.462359\pi$
$282$	0	0
$283$	19.7407	1.17346	0.586731	−	0.809782i	$-0.300415\pi$
$283$	19.7407	1.17346	0.586731	+	0.809782i	$0.300415\pi$
$284$	73.5982	4.36725
$285$	0	0
$286$	2.70899	0.160186
$287$	−17.9105	−1.05722
$288$	0	0
$289$	12.3545	0.726736
$290$	−17.0862	−1.00334
$291$	0	0
$292$	−9.90538	−0.579669
$293$	−2.96058	−0.172959	−0.0864794	−	0.996254i	$-0.527562\pi$
$293$	−2.96058	−0.172959	−0.0864794	+	0.996254i	$0.527562\pi$
$294$	0	0
$295$	−2.42580	−0.141236
$296$	42.0449	2.44381
$297$	0	0
$298$	−23.9277	−1.38610
$299$	−8.01493	−0.463515
$300$	0	0
$301$	−0.696632	−0.0401532
$302$	6.82272	0.392603
$303$	0	0
$304$	−10.8319	−0.621254
$305$	1.03305	0.0591523
$306$	0	0
$307$	−13.8263	−0.789107	−0.394553	−	0.918873i	$-0.629101\pi$
$307$	−13.8263	−0.789107	−0.394553	+	0.918873i	$0.629101\pi$
$308$	10.9539	0.624156
$309$	0	0
$310$	1.73651	0.0986270
$311$	−13.8004	−0.782546	−0.391273	−	0.920275i	$-0.627965\pi$
$311$	−13.8004	−0.782546	−0.391273	+	0.920275i	$0.627965\pi$
$312$	0	0
$313$	24.2773	1.37223	0.686116	−	0.727492i	$-0.259314\pi$
$313$	24.2773	1.37223	0.686116	+	0.727492i	$0.259314\pi$
$314$	36.2860	2.04774
$315$	0	0
$316$	−44.4767	−2.50201
$317$	28.9758	1.62744	0.813722	−	0.581254i	$-0.197438\pi$
$317$	28.9758	1.62744	0.813722	+	0.581254i	$0.197438\pi$
$318$	0	0
$319$	10.3311	0.578431
$320$	−15.1391	−0.846303
$321$	0	0
$322$	−44.5498	−2.48267
$323$	4.24541	0.236221
$324$	0	0
$325$	−4.62728	−0.256675
$326$	35.8594	1.98607
$327$	0	0
$328$	−78.9484	−4.35920
$329$	14.5101	0.799970
$330$	0	0
$331$	6.19751	0.340646	0.170323	−	0.985388i	$-0.445519\pi$
$331$	6.19751	0.340646	0.170323	+	0.985388i	$0.445519\pi$
$332$	−32.0593	−1.75948
$333$	0	0
$334$	54.2831	2.97024
$335$	3.70380	0.202360
$336$	0	0
$337$	15.1490	0.825219	0.412610	−	0.910908i	$-0.364617\pi$
$337$	15.1490	0.825219	0.412610	+	0.910908i	$0.364617\pi$
$338$	2.70899	0.147350
$339$	0	0
$340$	17.6587	0.957676
$341$	−1.04997	−0.0568592
$342$	0	0
$343$	−20.0874	−1.08462
$344$	−3.07072	−0.165562
$345$	0	0
$346$	−61.4097	−3.30140
$347$	−9.24425	−0.496257	−0.248129	−	0.968727i	$-0.579816\pi$
$347$	−9.24425	−0.496257	−0.248129	+	0.968727i	$0.579816\pi$
$348$	0	0
$349$	−13.0440	−0.698231	−0.349116	−	0.937080i	$-0.613518\pi$
$349$	−13.0440	−0.698231	−0.349116	+	0.937080i	$0.613518\pi$
$350$	−25.7201	−1.37480
$351$	0	0
$352$	19.3596	1.03187
$353$	3.94100	0.209758	0.104879	−	0.994485i	$-0.466554\pi$
$353$	3.94100	0.209758	0.104879	+	0.994485i	$0.466554\pi$
$354$	0	0
$355$	−8.41645	−0.446699
$356$	−67.7959	−3.59317
$357$	0	0
$358$	18.8227	0.994812
$359$	−27.4761	−1.45013	−0.725066	−	0.688679i	$-0.758191\pi$
$359$	−27.4761	−1.45013	−0.725066	+	0.688679i	$0.758191\pi$
$360$	0	0
$361$	−18.3860	−0.967684
$362$	27.7271	1.45730
$363$	0	0
$364$	10.9539	0.574140
$365$	1.13275	0.0592907
$366$	0	0
$367$	−15.9136	−0.830685	−0.415342	−	0.909665i	$-0.636338\pi$
$367$	−15.9136	−0.830685	−0.415342	+	0.909665i	$0.636338\pi$
$368$	−110.796	−5.77563
$369$	0	0
$370$	−7.68841	−0.399701
$371$	11.8801	0.616785
$372$	0	0
$373$	15.6083	0.808168	0.404084	−	0.914722i	$-0.367590\pi$
$373$	15.6083	0.808168	0.404084	+	0.914722i	$0.367590\pi$
$374$	−14.6773	−0.758943
$375$	0	0
$376$	63.9599	3.29848
$377$	10.3311	0.532079
$378$	0	0
$379$	5.13858	0.263951	0.131976	−	0.991253i	$-0.457868\pi$
$379$	5.13858	0.263951	0.131976	+	0.991253i	$0.457868\pi$
$380$	2.55389	0.131012
$381$	0	0
$382$	−28.8581	−1.47651
$383$	−1.76321	−0.0900960	−0.0450480	−	0.998985i	$-0.514344\pi$
$383$	−1.76321	−0.0900960	−0.0450480	+	0.998985i	$0.514344\pi$
$384$	0	0
$385$	−1.25265	−0.0638411
$386$	21.7266	1.10585
$387$	0	0
$388$	35.7893	1.81692
$389$	31.9410	1.61947	0.809737	−	0.586793i	$-0.199610\pi$
$389$	31.9410	1.61947	0.809737	+	0.586793i	$0.199610\pi$
$390$	0	0
$391$	43.4247	2.19608
$392$	−25.2339	−1.27451
$393$	0	0
$394$	52.9772	2.66895
$395$	5.08621	0.255915
$396$	0	0
$397$	−37.8377	−1.89902	−0.949510	−	0.313737i	$-0.898419\pi$
$397$	−37.8377	−1.89902	−0.949510	+	0.313737i	$0.898419\pi$
$398$	17.5885	0.881630
$399$	0	0
$400$	−63.9660	−3.19830
$401$	−13.5352	−0.675915	−0.337958	−	0.941161i	$-0.609736\pi$
$401$	−13.5352	−0.675915	−0.337958	+	0.941161i	$0.609736\pi$
$402$	0	0
$403$	−1.04997	−0.0523029
$404$	−2.14190	−0.106563
$405$	0	0
$406$	57.4240	2.84991
$407$	4.64877	0.230431
$408$	0	0
$409$	8.92572	0.441348	0.220674	−	0.975348i	$-0.429174\pi$
$409$	8.92572	0.441348	0.220674	+	0.975348i	$0.429174\pi$
$410$	14.4367	0.712976
$411$	0	0
$412$	−59.9186	−2.95198
$413$	8.15274	0.401170
$414$	0	0
$415$	3.66619	0.179966
$416$	19.3596	0.949183
$417$	0	0
$418$	−2.12271	−0.103825
$419$	−7.22361	−0.352896	−0.176448	−	0.984310i	$-0.556461\pi$
$419$	−7.22361	−0.352896	−0.176448	+	0.984310i	$0.556461\pi$
$420$	0	0
$421$	8.38361	0.408592	0.204296	−	0.978909i	$-0.434509\pi$
$421$	8.38361	0.408592	0.204296	+	0.978909i	$0.434509\pi$
$422$	−6.99288	−0.340408
$423$	0	0
$424$	52.3669	2.54316
$425$	25.0705	1.21610
$426$	0	0
$427$	−3.47192	−0.168018
$428$	29.3366	1.41804
$429$	0	0
$430$	0.561517	0.0270788
$431$	−36.7242	−1.76894	−0.884472	−	0.466593i	$-0.845481\pi$
$431$	−36.7242	−1.76894	−0.884472	+	0.466593i	$0.845481\pi$
$432$	0	0
$433$	34.2922	1.64798	0.823989	−	0.566606i	$-0.191744\pi$
$433$	34.2922	1.64798	0.823989	+	0.566606i	$0.191744\pi$
$434$	−5.83613	−0.280143
$435$	0	0
$436$	−39.6263	−1.89776
$437$	6.28032	0.300428
$438$	0	0
$439$	25.6977	1.22649	0.613243	−	0.789895i	$-0.289865\pi$
$439$	25.6977	1.22649	0.613243	+	0.789895i	$0.289865\pi$
$440$	−5.52162	−0.263233
$441$	0	0
$442$	−14.6773	−0.698126
$443$	0.797604	0.0378953	0.0189476	−	0.999820i	$-0.493968\pi$
$443$	0.797604	0.0378953	0.0189476	+	0.999820i	$0.493968\pi$
$444$	0	0
$445$	7.75291	0.367523
$446$	27.9604	1.32396
$447$	0	0
$448$	50.8802	2.40386
$449$	37.2845	1.75956	0.879782	−	0.475377i	$-0.157688\pi$
$449$	37.2845	1.75956	0.879782	+	0.475377i	$0.157688\pi$
$450$	0	0
$451$	−8.72907	−0.411036
$452$	−74.8683	−3.52151
$453$	0	0
$454$	34.4307	1.61591
$455$	−1.25265	−0.0587252
$456$	0	0
$457$	20.3780	0.953244	0.476622	−	0.879108i	$-0.341861\pi$
$457$	20.3780	0.953244	0.476622	+	0.879108i	$0.341861\pi$
$458$	24.8022	1.15893
$459$	0	0
$460$	26.1228	1.21798
$461$	8.77841	0.408851	0.204426	−	0.978882i	$-0.434467\pi$
$461$	8.77841	0.408851	0.204426	+	0.978882i	$0.434467\pi$
$462$	0	0
$463$	8.43856	0.392173	0.196087	−	0.980587i	$-0.437177\pi$
$463$	8.43856	0.392173	0.196087	+	0.980587i	$0.437177\pi$
$464$	142.814	6.62997
$465$	0	0
$466$	31.4897	1.45873
$467$	−35.2257	−1.63005	−0.815026	−	0.579424i	$-0.803278\pi$
$467$	−35.2257	−1.63005	−0.815026	+	0.579424i	$0.803278\pi$
$468$	0	0
$469$	−12.4479	−0.574789
$470$	−11.6958	−0.539488
$471$	0	0
$472$	35.9368	1.65413
$473$	−0.339519	−0.0156111
$474$	0	0
$475$	3.62584	0.166365
$476$	−59.3480	−2.72021
$477$	0	0
$478$	15.3287	0.701118
$479$	−13.6396	−0.623210	−0.311605	−	0.950212i	$-0.600867\pi$
$479$	−13.6396	−0.623210	−0.311605	+	0.950212i	$0.600867\pi$
$480$	0	0
$481$	4.64877	0.211965
$482$	71.2393	3.24486
$483$	0	0
$484$	5.33863	0.242665
$485$	−4.09274	−0.185842
$486$	0	0
$487$	−11.9531	−0.541647	−0.270823	−	0.962629i	$-0.587296\pi$
$487$	−11.9531	−0.541647	−0.270823	+	0.962629i	$0.587296\pi$
$488$	−15.3040	−0.692781
$489$	0	0
$490$	4.61433	0.208454
$491$	7.83580	0.353625	0.176812	−	0.984245i	$-0.443421\pi$
$491$	7.83580	0.353625	0.176812	+	0.984245i	$0.443421\pi$
$492$	0	0
$493$	−55.9738	−2.52093
$494$	−2.12271	−0.0955050
$495$	0	0
$496$	−14.5145	−0.651720
$497$	28.2864	1.26882
$498$	0	0
$499$	−2.33224	−0.104405	−0.0522027	−	0.998637i	$-0.516624\pi$
$499$	−2.33224	−0.104405	−0.0522027	+	0.998637i	$0.516624\pi$
$500$	31.3779	1.40326
$501$	0	0
$502$	−45.4140	−2.02693
$503$	24.3797	1.08704	0.543518	−	0.839397i	$-0.317092\pi$
$503$	24.3797	1.08704	0.543518	+	0.839397i	$0.317092\pi$
$504$	0	0
$505$	0.244940	0.0108997
$506$	−21.7124	−0.965232
$507$	0	0
$508$	7.98206	0.354147
$509$	19.8121	0.878158	0.439079	−	0.898448i	$-0.355305\pi$
$509$	19.8121	0.878158	0.439079	+	0.898448i	$0.355305\pi$
$510$	0	0
$511$	−3.80698	−0.168411
$512$	17.5697	0.776480
$513$	0	0
$514$	−34.1247	−1.50518
$515$	6.85210	0.301940
$516$	0	0
$517$	7.07184	0.311019
$518$	25.8395	1.13532
$519$	0	0
$520$	−5.52162	−0.242139
$521$	31.8533	1.39552	0.697759	−	0.716333i	$-0.254181\pi$
$521$	31.8533	1.39552	0.697759	+	0.716333i	$0.254181\pi$
$522$	0	0
$523$	6.10993	0.267169	0.133584	−	0.991037i	$-0.457351\pi$
$523$	6.10993	0.267169	0.133584	+	0.991037i	$0.457351\pi$
$524$	−49.7360	−2.17273
$525$	0	0
$526$	57.9406	2.52633
$527$	5.68873	0.247805
$528$	0	0
$529$	41.2390	1.79300
$530$	−9.57591	−0.415951
$531$	0	0
$532$	−8.58324	−0.372130
$533$	−8.72907	−0.378098
$534$	0	0
$535$	−3.35483	−0.145042
$536$	−54.8695	−2.37000
$537$	0	0
$538$	16.1853	0.697796
$539$	−2.79004	−0.120175
$540$	0	0
$541$	5.53888	0.238135	0.119067	−	0.992886i	$-0.462010\pi$
$541$	5.53888	0.238135	0.119067	+	0.992886i	$0.462010\pi$
$542$	−37.1301	−1.59487
$543$	0	0
$544$	−104.890	−4.49712
$545$	4.53154	0.194110
$546$	0	0
$547$	−9.53586	−0.407724	−0.203862	−	0.979000i	$-0.565349\pi$
$547$	−9.53586	−0.407724	−0.203862	+	0.979000i	$0.565349\pi$
$548$	−51.1922	−2.18682
$549$	0	0
$550$	−12.5353	−0.534505
$551$	−8.09523	−0.344869
$552$	0	0
$553$	−17.0939	−0.726908
$554$	34.4786	1.46486
$555$	0	0
$556$	−61.7276	−2.61783
$557$	−24.1005	−1.02117	−0.510586	−	0.859827i	$-0.670571\pi$
$557$	−24.1005	−1.02117	−0.510586	+	0.859827i	$0.670571\pi$
$558$	0	0
$559$	−0.339519	−0.0143601
$560$	−17.3163	−0.731746
$561$	0	0
$562$	10.7150	0.451987
$563$	40.1512	1.69217	0.846085	−	0.533047i	$-0.178953\pi$
$563$	40.1512	1.69217	0.846085	+	0.533047i	$0.178953\pi$
$564$	0	0
$565$	8.56170	0.360193
$566$	53.4773	2.24782
$567$	0	0
$568$	124.685	5.23165
$569$	21.7983	0.913833	0.456917	−	0.889510i	$-0.348954\pi$
$569$	21.7983	0.913833	0.456917	+	0.889510i	$0.348954\pi$
$570$	0	0
$571$	−34.9698	−1.46344	−0.731720	−	0.681605i	$-0.761282\pi$
$571$	−34.9698	−1.46344	−0.731720	+	0.681605i	$0.761282\pi$
$572$	5.33863	0.223219
$573$	0	0
$574$	−48.5193	−2.02516
$575$	37.0873	1.54665
$576$	0	0
$577$	0.708854	0.0295100	0.0147550	−	0.999891i	$-0.495303\pi$
$577$	0.708854	0.0295100	0.0147550	+	0.999891i	$0.495303\pi$
$578$	33.4682	1.39210
$579$	0	0
$580$	−33.6719	−1.39815
$581$	−12.3215	−0.511182
$582$	0	0
$583$	5.79004	0.239799
$584$	−16.7810	−0.694401
$585$	0	0
$586$	−8.02018	−0.331310
$587$	−11.9493	−0.493200	−0.246600	−	0.969117i	$-0.579313\pi$
$587$	−11.9493	−0.493200	−0.246600	+	0.969117i	$0.579313\pi$
$588$	0	0
$589$	0.822736	0.0339002
$590$	−6.57148	−0.270544
$591$	0	0
$592$	64.2630	2.64120
$593$	−41.7347	−1.71384	−0.856920	−	0.515449i	$-0.827625\pi$
$593$	−41.7347	−1.71384	−0.856920	+	0.515449i	$0.827625\pi$
$594$	0	0
$595$	6.78684	0.278233
$596$	−47.1546	−1.93153
$597$	0	0
$598$	−21.7124	−0.887884
$599$	−3.80539	−0.155484	−0.0777421	−	0.996974i	$-0.524771\pi$
$599$	−3.80539	−0.155484	−0.0777421	+	0.996974i	$0.524771\pi$
$600$	0	0
$601$	−18.8023	−0.766963	−0.383482	−	0.923548i	$-0.625275\pi$
$601$	−18.8023	−0.766963	−0.383482	+	0.923548i	$0.625275\pi$
$602$	−1.88717	−0.0769153
$603$	0	0
$604$	13.4456	0.547093
$605$	−0.610508	−0.0248207
$606$	0	0
$607$	30.8116	1.25060	0.625301	−	0.780383i	$-0.284976\pi$
$607$	30.8116	1.25060	0.625301	+	0.780383i	$0.284976\pi$
$608$	−15.1698	−0.615215
$609$	0	0
$610$	2.79852	0.113309
$611$	7.07184	0.286096
$612$	0	0
$613$	−24.2392	−0.979010	−0.489505	−	0.872000i	$-0.662823\pi$
$613$	−24.2392	−0.979010	−0.489505	+	0.872000i	$0.662823\pi$
$614$	−37.4552	−1.51157
$615$	0	0
$616$	18.5573	0.747694
$617$	−26.4076	−1.06313	−0.531566	−	0.847017i	$-0.678396\pi$
$617$	−26.4076	−1.06313	−0.531566	+	0.847017i	$0.678396\pi$
$618$	0	0
$619$	−32.2887	−1.29779	−0.648897	−	0.760877i	$-0.724769\pi$
$619$	−32.2887	−1.29779	−0.648897	+	0.760877i	$0.724769\pi$
$620$	3.42215	0.137437
$621$	0	0
$622$	−37.3850	−1.49900
$623$	−26.0563	−1.04392
$624$	0	0
$625$	19.5481	0.781925
$626$	65.7669	2.62857
$627$	0	0
$628$	71.5090	2.85352
$629$	−25.1869	−1.00427
$630$	0	0
$631$	−7.00848	−0.279003	−0.139502	−	0.990222i	$-0.544550\pi$
$631$	−7.00848	−0.279003	−0.139502	+	0.990222i	$0.544550\pi$
$632$	−75.3491	−2.99723
$633$	0	0
$634$	78.4952	3.11744
$635$	−0.912803	−0.0362235
$636$	0	0
$637$	−2.79004	−0.110545
$638$	27.9869	1.10801
$639$	0	0
$640$	−17.3734	−0.686743
$641$	−14.9537	−0.590636	−0.295318	−	0.955399i	$-0.595426\pi$
$641$	−14.9537	−0.590636	−0.295318	+	0.955399i	$0.595426\pi$
$642$	0	0
$643$	−38.1198	−1.50330	−0.751649	−	0.659563i	$-0.770741\pi$
$643$	−38.1198	−1.50330	−0.751649	+	0.659563i	$0.770741\pi$
$644$	−87.7947	−3.45960
$645$	0	0
$646$	11.5008	0.452492
$647$	−22.7794	−0.895551	−0.447776	−	0.894146i	$-0.647784\pi$
$647$	−22.7794	−0.895551	−0.447776	+	0.894146i	$0.647784\pi$
$648$	0	0
$649$	3.97342	0.155970
$650$	−12.5353	−0.491673
$651$	0	0
$652$	70.6683	2.76758
$653$	−18.0459	−0.706190	−0.353095	−	0.935587i	$-0.614871\pi$
$653$	−18.0459	−0.706190	−0.353095	+	0.935587i	$0.614871\pi$
$654$	0	0
$655$	5.68764	0.222235
$656$	−120.668	−4.71129
$657$	0	0
$658$	39.3078	1.53238
$659$	3.03751	0.118325	0.0591623	−	0.998248i	$-0.481157\pi$
$659$	3.03751	0.118325	0.0591623	+	0.998248i	$0.481157\pi$
$660$	0	0
$661$	−32.6389	−1.26951	−0.634753	−	0.772715i	$-0.718898\pi$
$661$	−32.6389	−1.26951	−0.634753	+	0.772715i	$0.718898\pi$
$662$	16.7890	0.652523
$663$	0	0
$664$	−54.3125	−2.10773
$665$	0.981551	0.0380629
$666$	0	0
$667$	−82.8031	−3.20615
$668$	106.976	4.13903
$669$	0	0
$670$	10.0335	0.387630
$671$	−1.69212	−0.0653235
$672$	0	0
$673$	−8.19080	−0.315732	−0.157866	−	0.987461i	$-0.550461\pi$
$673$	−8.19080	−0.315732	−0.157866	+	0.987461i	$0.550461\pi$
$674$	41.0385	1.58074
$675$	0	0
$676$	5.33863	0.205332
$677$	20.0647	0.771150	0.385575	−	0.922676i	$-0.374003\pi$
$677$	20.0647	0.771150	0.385575	+	0.922676i	$0.374003\pi$
$678$	0	0
$679$	13.7551	0.527871
$680$	29.9160	1.14723
$681$	0	0
$682$	−2.84437	−0.108916
$683$	35.5118	1.35882	0.679410	−	0.733759i	$-0.262236\pi$
$683$	35.5118	1.35882	0.679410	+	0.733759i	$0.262236\pi$
$684$	0	0
$685$	5.85417	0.223676
$686$	−54.4165	−2.07763
$687$	0	0
$688$	−4.69340	−0.178934
$689$	5.79004	0.220583
$690$	0	0
$691$	−24.4626	−0.930600	−0.465300	−	0.885153i	$-0.654054\pi$
$691$	−24.4626	−0.930600	−0.465300	+	0.885153i	$0.654054\pi$
$692$	−121.021	−4.60051
$693$	0	0
$694$	−25.0426	−0.950603
$695$	7.05896	0.267762
$696$	0	0
$697$	47.2939	1.79139
$698$	−35.3362	−1.33749
$699$	0	0
$700$	−50.6868	−1.91578
$701$	1.51602	0.0572592	0.0286296	−	0.999590i	$-0.490886\pi$
$701$	1.51602	0.0572592	0.0286296	+	0.999590i	$0.490886\pi$
$702$	0	0
$703$	−3.64267	−0.137386
$704$	24.7976	0.934595
$705$	0	0
$706$	10.6761	0.401801
$707$	−0.823206	−0.0309598
$708$	0	0
$709$	30.5302	1.14659	0.573293	−	0.819350i	$-0.305666\pi$
$709$	30.5302	1.14659	0.573293	+	0.819350i	$0.305666\pi$
$710$	−22.8001	−0.855672
$711$	0	0
$712$	−114.855	−4.30436
$713$	8.41546	0.315161
$714$	0	0
$715$	−0.610508	−0.0228317
$716$	37.0941	1.38627
$717$	0	0
$718$	−74.4325	−2.77779
$719$	33.8706	1.26316	0.631580	−	0.775311i	$-0.282407\pi$
$719$	33.8706	1.26316	0.631580	+	0.775311i	$0.282407\pi$
$720$	0	0
$721$	−23.0288	−0.857638
$722$	−49.8075	−1.85364
$723$	0	0
$724$	54.6420	2.03075
$725$	−47.8050	−1.77543
$726$	0	0
$727$	−3.68023	−0.136492	−0.0682460	−	0.997669i	$-0.521740\pi$
$727$	−3.68023	−0.136492	−0.0682460	+	0.997669i	$0.521740\pi$
$728$	18.5573	0.687779
$729$	0	0
$730$	3.06860	0.113574
$731$	1.83951	0.0680367
$732$	0	0
$733$	−9.01936	−0.333138	−0.166569	−	0.986030i	$-0.553269\pi$
$733$	−9.01936	−0.333138	−0.166569	+	0.986030i	$0.553269\pi$
$734$	−43.1099	−1.59121
$735$	0	0
$736$	−155.166	−5.71949
$737$	−6.06675	−0.223471
$738$	0	0
$739$	44.1854	1.62539	0.812694	−	0.582691i	$-0.198000\pi$
$739$	44.1854	1.62539	0.812694	+	0.582691i	$0.198000\pi$
$740$	−15.1516	−0.556984
$741$	0	0
$742$	32.1831	1.18148
$743$	11.3104	0.414940	0.207470	−	0.978241i	$-0.433477\pi$
$743$	11.3104	0.414940	0.207470	+	0.978241i	$0.433477\pi$
$744$	0	0
$745$	5.39244	0.197564
$746$	42.2828	1.54808
$747$	0	0
$748$	−28.9246	−1.05759
$749$	11.2751	0.411982
$750$	0	0
$751$	3.94751	0.144047	0.0720233	−	0.997403i	$-0.477054\pi$
$751$	3.94751	0.144047	0.0720233	+	0.997403i	$0.477054\pi$
$752$	97.7589	3.56490
$753$	0	0
$754$	27.9869	1.01922
$755$	−1.53759	−0.0559587
$756$	0	0
$757$	3.39667	0.123454	0.0617271	−	0.998093i	$-0.480339\pi$
$757$	3.39667	0.123454	0.0617271	+	0.998093i	$0.480339\pi$
$758$	13.9204	0.505611
$759$	0	0
$760$	4.32662	0.156943
$761$	37.1075	1.34515	0.672573	−	0.740031i	$-0.265189\pi$
$761$	37.1075	1.34515	0.672573	+	0.740031i	$0.265189\pi$
$762$	0	0
$763$	−15.2298	−0.551355
$764$	−56.8708	−2.05751
$765$	0	0
$766$	−4.77653	−0.172583
$767$	3.97342	0.143472
$768$	0	0
$769$	35.2642	1.27166	0.635830	−	0.771829i	$-0.280658\pi$
$769$	35.2642	1.27166	0.635830	+	0.771829i	$0.280658\pi$
$770$	−3.39342	−0.122290
$771$	0	0
$772$	42.8168	1.54101
$773$	23.4879	0.844800	0.422400	−	0.906410i	$-0.361188\pi$
$773$	23.4879	0.844800	0.422400	+	0.906410i	$0.361188\pi$
$774$	0	0
$775$	4.85852	0.174523
$776$	60.6315	2.17654
$777$	0	0
$778$	86.5279	3.10217
$779$	6.83991	0.245065
$780$	0	0
$781$	13.7860	0.493301
$782$	117.637	4.20669
$783$	0	0
$784$	−38.5686	−1.37745
$785$	−8.17754	−0.291869
$786$	0	0
$787$	9.32837	0.332521	0.166260	−	0.986082i	$-0.446831\pi$
$787$	9.32837	0.332521	0.166260	+	0.986082i	$0.446831\pi$
$788$	104.403	3.71919
$789$	0	0
$790$	13.7785	0.490217
$791$	−28.7745	−1.02310
$792$	0	0
$793$	−1.69212	−0.0600888
$794$	−102.502	−3.63766
$795$	0	0
$796$	34.6617	1.22855
$797$	13.4757	0.477332	0.238666	−	0.971102i	$-0.423290\pi$
$797$	13.4757	0.477332	0.238666	+	0.971102i	$0.423290\pi$
$798$	0	0
$799$	−38.3151	−1.35549
$800$	−89.5823	−3.16721
$801$	0	0
$802$	−36.6667	−1.29475
$803$	−1.85542	−0.0654763
$804$	0	0
$805$	10.0399	0.353860
$806$	−2.84437	−0.100189
$807$	0	0
$808$	−3.62864	−0.127655
$809$	17.8552	0.627754	0.313877	−	0.949464i	$-0.398372\pi$
$809$	17.8552	0.627754	0.313877	+	0.949464i	$0.398372\pi$
$810$	0	0
$811$	25.1094	0.881711	0.440855	−	0.897578i	$-0.354675\pi$
$811$	25.1094	0.881711	0.440855	+	0.897578i	$0.354675\pi$
$812$	113.166	3.97135
$813$	0	0
$814$	12.5935	0.441401
$815$	−8.08139	−0.283079
$816$	0	0
$817$	0.266040	0.00930755
$818$	24.1797	0.845423
$819$	0	0
$820$	28.4504	0.993532
$821$	−31.3013	−1.09242	−0.546212	−	0.837647i	$-0.683931\pi$
$821$	−31.3013	−1.09242	−0.546212	+	0.837647i	$0.683931\pi$
$822$	0	0
$823$	38.2894	1.33468	0.667342	−	0.744752i	$-0.267432\pi$
$823$	38.2894	1.33468	0.667342	+	0.744752i	$0.267432\pi$
$824$	−101.510	−3.53626
$825$	0	0
$826$	22.0857	0.768460
$827$	−11.1904	−0.389129	−0.194565	−	0.980890i	$-0.562329\pi$
$827$	−11.1904	−0.389129	−0.194565	+	0.980890i	$0.562329\pi$
$828$	0	0
$829$	−22.2137	−0.771513	−0.385756	−	0.922601i	$-0.626059\pi$
$829$	−22.2137	−0.771513	−0.385756	+	0.922601i	$0.626059\pi$
$830$	9.93168	0.344734
$831$	0	0
$832$	24.7976	0.859702
$833$	15.1164	0.523751
$834$	0	0
$835$	−12.2334	−0.423356
$836$	−4.18323	−0.144680
$837$	0	0
$838$	−19.5687	−0.675989
$839$	55.7220	1.92374	0.961868	−	0.273513i	$-0.0881858\pi$
$839$	55.7220	1.92374	0.961868	+	0.273513i	$0.0881858\pi$
$840$	0	0
$841$	77.7319	2.68041
$842$	22.7111	0.782677
$843$	0	0
$844$	−13.7809	−0.474359
$845$	−0.610508	−0.0210021
$846$	0	0
$847$	2.05182	0.0705014
$848$	80.0396	2.74857
$849$	0	0
$850$	67.9158	2.32949
$851$	−37.2595	−1.27724
$852$	0	0
$853$	22.8770	0.783294	0.391647	−	0.920115i	$-0.371905\pi$
$853$	22.8770	0.783294	0.391647	+	0.920115i	$0.371905\pi$
$854$	−9.40540	−0.321846
$855$	0	0
$856$	49.6998	1.69871
$857$	16.3979	0.560143	0.280072	−	0.959979i	$-0.409642\pi$
$857$	16.3979	0.560143	0.280072	+	0.959979i	$0.409642\pi$
$858$	0	0
$859$	−51.7295	−1.76499	−0.882493	−	0.470325i	$-0.844137\pi$
$859$	−51.7295	−1.76499	−0.882493	+	0.470325i	$0.844137\pi$
$860$	1.10659	0.0377343
$861$	0	0
$862$	−99.4856	−3.38849
$863$	−15.3977	−0.524142	−0.262071	−	0.965049i	$-0.584406\pi$
$863$	−15.3977	−0.524142	−0.262071	+	0.965049i	$0.584406\pi$
$864$	0	0
$865$	13.8395	0.470557
$866$	92.8972	3.15678
$867$	0	0
$868$	−11.5013	−0.390380
$869$	−8.33111	−0.282614
$870$	0	0
$871$	−6.06675	−0.205564
$872$	−67.1320	−2.27338
$873$	0	0
$874$	17.0133	0.575484
$875$	12.0596	0.407690
$876$	0	0
$877$	41.7580	1.41007	0.705035	−	0.709173i	$-0.250931\pi$
$877$	41.7580	1.41007	0.705035	+	0.709173i	$0.250931\pi$
$878$	69.6149	2.34939
$879$	0	0
$880$	−8.43947	−0.284494
$881$	−18.9438	−0.638234	−0.319117	−	0.947715i	$-0.603386\pi$
$881$	−18.9438	−0.638234	−0.319117	+	0.947715i	$0.603386\pi$
$882$	0	0
$883$	−2.22790	−0.0749747	−0.0374874	−	0.999297i	$-0.511935\pi$
$883$	−2.22790	−0.0749747	−0.0374874	+	0.999297i	$0.511935\pi$
$884$	−28.9246	−0.972839
$885$	0	0
$886$	2.16070	0.0725901
$887$	47.7029	1.60171	0.800853	−	0.598861i	$-0.204380\pi$
$887$	47.7029	1.60171	0.800853	+	0.598861i	$0.204380\pi$
$888$	0	0
$889$	3.06778	0.102890
$890$	21.0026	0.704007
$891$	0	0
$892$	55.1018	1.84494
$893$	−5.54134	−0.185434
$894$	0	0
$895$	−4.24196	−0.141793
$896$	58.3892	1.95064
$897$	0	0
$898$	101.003	3.37052
$899$	−10.8474	−0.361781
$900$	0	0
$901$	−31.3703	−1.04510
$902$	−23.6470	−0.787358
$903$	0	0
$904$	−126.836	−4.21852
$905$	−6.24868	−0.207713
$906$	0	0
$907$	−20.5287	−0.681645	−0.340822	−	0.940128i	$-0.610706\pi$
$907$	−20.5287	−0.681645	−0.340822	+	0.940128i	$0.610706\pi$
$908$	67.8528	2.25177
$909$	0	0
$910$	−3.39342	−0.112491
$911$	−41.6188	−1.37889	−0.689446	−	0.724337i	$-0.742146\pi$
$911$	−41.6188	−1.37889	−0.689446	+	0.724337i	$0.742146\pi$
$912$	0	0
$913$	−6.00516	−0.198742
$914$	55.2038	1.82598
$915$	0	0
$916$	48.8779	1.61497
$917$	−19.1153	−0.631242
$918$	0	0
$919$	−7.43379	−0.245218	−0.122609	−	0.992455i	$-0.539126\pi$
$919$	−7.43379	−0.245218	−0.122609	+	0.992455i	$0.539126\pi$
$920$	44.2554	1.45906
$921$	0	0
$922$	23.7806	0.783173
$923$	13.7860	0.453771
$924$	0	0
$925$	−21.5111	−0.707282
$926$	22.8600	0.751225
$927$	0	0
$928$	200.006	6.56553
$929$	−9.56393	−0.313782	−0.156891	−	0.987616i	$-0.550147\pi$
$929$	−9.56393	−0.313782	−0.156891	+	0.987616i	$0.550147\pi$
$930$	0	0
$931$	2.18621	0.0716502
$932$	62.0569	2.03274
$933$	0	0
$934$	−95.4262	−3.12244
$935$	3.30772	0.108174
$936$	0	0
$937$	−10.7352	−0.350705	−0.175352	−	0.984506i	$-0.556106\pi$
$937$	−10.7352	−0.350705	−0.175352	+	0.984506i	$0.556106\pi$
$938$	−33.7211	−1.10103
$939$	0	0
$940$	−23.0491	−0.751777
$941$	−16.9637	−0.553002	−0.276501	−	0.961014i	$-0.589175\pi$
$941$	−16.9637	−0.553002	−0.276501	+	0.961014i	$0.589175\pi$
$942$	0	0
$943$	69.9629	2.27830
$944$	54.9273	1.78773
$945$	0	0
$946$	−0.919754	−0.0299038
$947$	−51.6780	−1.67931	−0.839654	−	0.543121i	$-0.817242\pi$
$947$	−51.6780	−1.67931	−0.839654	+	0.543121i	$0.817242\pi$
$948$	0	0
$949$	−1.85542	−0.0602294
$950$	9.82235	0.318679
$951$	0	0
$952$	−100.543	−3.25862
$953$	−47.9985	−1.55482	−0.777411	−	0.628992i	$-0.783468\pi$
$953$	−47.9985	−1.55482	−0.777411	+	0.628992i	$0.783468\pi$
$954$	0	0
$955$	6.50356	0.210450
$956$	30.2084	0.977008
$957$	0	0
$958$	−36.9496	−1.19379
$959$	−19.6749	−0.635337
$960$	0	0
$961$	−29.8976	−0.964437
$962$	12.5935	0.406029
$963$	0	0
$964$	140.392	4.52171
$965$	−4.89638	−0.157620
$966$	0	0
$967$	−27.6426	−0.888927	−0.444464	−	0.895797i	$-0.646606\pi$
$967$	−27.6426	−0.888927	−0.444464	+	0.895797i	$0.646606\pi$
$968$	9.04431	0.290695
$969$	0	0
$970$	−11.0872	−0.355988
$971$	−45.9185	−1.47359	−0.736797	−	0.676114i	$-0.763662\pi$
$971$	−45.9185	−1.47359	−0.736797	+	0.676114i	$0.763662\pi$
$972$	0	0
$973$	−23.7240	−0.760558
$974$	−32.3808	−1.03755
$975$	0	0
$976$	−23.3913	−0.748737
$977$	7.38783	0.236358	0.118179	−	0.992992i	$-0.462294\pi$
$977$	7.38783	0.236358	0.118179	+	0.992992i	$0.462294\pi$
$978$	0	0
$979$	−12.6991	−0.405866
$980$	9.09349	0.290481
$981$	0	0
$982$	21.2271	0.677384
$983$	−35.7636	−1.14068	−0.570340	−	0.821409i	$-0.693189\pi$
$983$	−35.7636	−1.14068	−0.570340	+	0.821409i	$0.693189\pi$
$984$	0	0
$985$	−11.9391	−0.380412
$986$	−151.632	−4.82896
$987$	0	0
$988$	−4.18323	−0.133086
$989$	2.72122	0.0865298
$990$	0	0
$991$	−11.3451	−0.360388	−0.180194	−	0.983631i	$-0.557673\pi$
$991$	−11.3451	−0.360388	−0.180194	+	0.983631i	$0.557673\pi$
$992$	−20.3271	−0.645385
$993$	0	0
$994$	76.6275	2.43048
$995$	−3.96380	−0.125661
$996$	0	0
$997$	−0.528077	−0.0167244	−0.00836218	−	0.999965i	$-0.502662\pi$
$997$	−0.528077	−0.0167244	−0.00836218	+	0.999965i	$0.502662\pi$
$998$	−6.31802	−0.199993
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1287.2.a.q.1.6		6
3.2	odd	2		143.2.a.c.1.1	✓	6
12.11	even	2		2288.2.a.z.1.3		6
15.14	odd	2		3575.2.a.p.1.6		6
21.20	even	2		7007.2.a.r.1.1		6
24.5	odd	2		9152.2.a.cm.1.3		6
24.11	even	2		9152.2.a.cs.1.4		6
33.32	even	2		1573.2.a.m.1.6		6
39.38	odd	2		1859.2.a.m.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
143.2.a.c.1.1	✓	6	3.2	odd	2
1287.2.a.q.1.6		6	1.1	even	1	trivial
1573.2.a.m.1.6		6	33.32	even	2
1859.2.a.m.1.6		6	39.38	odd	2
2288.2.a.z.1.3		6	12.11	even	2
3575.2.a.p.1.6		6	15.14	odd	2
7007.2.a.r.1.1		6	21.20	even	2
9152.2.a.cm.1.3		6	24.5	odd	2
9152.2.a.cs.1.4		6	24.11	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:	\( N \)	\(=\)	\( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1287.a (trivial)

\(f(q)\)	\(=\)	\(q+2.70899 q^{2} +5.33863 q^{4} -0.610508 q^{5} +2.05182 q^{7} +9.04431 q^{8} +O(q^{10})\)	\(q+2.70899 q^{2} +5.33863 q^{4} -0.610508 q^{5} +2.05182 q^{7} +9.04431 q^{8} -1.65386 q^{10} +1.00000 q^{11} +1.00000 q^{13} +5.55836 q^{14} +13.8237 q^{16} -5.41798 q^{17} -0.783578 q^{19} -3.25927 q^{20} +2.70899 q^{22} -8.01493 q^{23} -4.62728 q^{25} +2.70899 q^{26} +10.9539 q^{28} +10.3311 q^{29} -1.04997 q^{31} +19.3596 q^{32} -14.6773 q^{34} -1.25265 q^{35} +4.64877 q^{37} -2.12271 q^{38} -5.52162 q^{40} -8.72907 q^{41} -0.339519 q^{43} +5.33863 q^{44} -21.7124 q^{46} +7.07184 q^{47} -2.79004 q^{49} -12.5353 q^{50} +5.33863 q^{52} +5.79004 q^{53} -0.610508 q^{55} +18.5573 q^{56} +27.9869 q^{58} +3.97342 q^{59} -1.69212 q^{61} -2.84437 q^{62} +24.7976 q^{64} -0.610508 q^{65} -6.06675 q^{67} -28.9246 q^{68} -3.39342 q^{70} +13.7860 q^{71} -1.85542 q^{73} +12.5935 q^{74} -4.18323 q^{76} +2.05182 q^{77} -8.33111 q^{79} -8.43947 q^{80} -23.6470 q^{82} -6.00516 q^{83} +3.30772 q^{85} -0.919754 q^{86} +9.04431 q^{88} -12.6991 q^{89} +2.05182 q^{91} -42.7887 q^{92} +19.1575 q^{94} +0.478381 q^{95} +6.70383 q^{97} -7.55818 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 8 q^{4} - q^{5} + 4 q^{7} + 6 q^{8}+O(q^{10}) \) 6 * q + 8 * q^4 - q^5 + 4 * q^7 + 6 * q^8	\( 6 q + 8 q^{4} - q^{5} + 4 q^{7} + 6 q^{8} + 6 q^{10} + 6 q^{11} + 6 q^{13} + 12 q^{14} + 8 q^{16} - 10 q^{19} - 4 q^{20} - 11 q^{23} + 23 q^{25} + 9 q^{28} - 2 q^{29} - 9 q^{31} + 17 q^{32} - 40 q^{34} + 24 q^{35} + 15 q^{37} + 9 q^{38} + 16 q^{40} + 4 q^{41} - 2 q^{43} + 8 q^{44} - 6 q^{46} - 6 q^{47} + 20 q^{49} + 4 q^{50} + 8 q^{52} - 2 q^{53} - q^{55} + 39 q^{56} + 18 q^{58} - 11 q^{59} + 16 q^{61} - 16 q^{62} + 36 q^{64} - q^{65} + 9 q^{67} - 12 q^{68} + 32 q^{70} + 15 q^{71} + 32 q^{73} - 22 q^{74} - 26 q^{76} + 4 q^{77} + 14 q^{79} - 56 q^{80} - 24 q^{82} + 26 q^{83} - 12 q^{85} + 10 q^{86} + 6 q^{88} + 23 q^{89} + 4 q^{91} - 83 q^{92} + 46 q^{94} + 52 q^{95} + 27 q^{97} + 3 q^{98}+O(q^{100}) \) 6 * q + 8 * q^4 - q^5 + 4 * q^7 + 6 * q^8 + 6 * q^10 + 6 * q^11 + 6 * q^13 + 12 * q^14 + 8 * q^16 - 10 * q^19 - 4 * q^20 - 11 * q^23 + 23 * q^25 + 9 * q^28 - 2 * q^29 - 9 * q^31 + 17 * q^32 - 40 * q^34 + 24 * q^35 + 15 * q^37 + 9 * q^38 + 16 * q^40 + 4 * q^41 - 2 * q^43 + 8 * q^44 - 6 * q^46 - 6 * q^47 + 20 * q^49 + 4 * q^50 + 8 * q^52 - 2 * q^53 - q^55 + 39 * q^56 + 18 * q^58 - 11 * q^59 + 16 * q^61 - 16 * q^62 + 36 * q^64 - q^65 + 9 * q^67 - 12 * q^68 + 32 * q^70 + 15 * q^71 + 32 * q^73 - 22 * q^74 - 26 * q^76 + 4 * q^77 + 14 * q^79 - 56 * q^80 - 24 * q^82 + 26 * q^83 - 12 * q^85 + 10 * q^86 + 6 * q^88 + 23 * q^89 + 4 * q^91 - 83 * q^92 + 46 * q^94 + 52 * q^95 + 27 * q^97 + 3 * q^98

Introduction

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