LMFDB - Embedded newform 1755.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1755,2,Mod(1,1755)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1755, 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("1755.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1755 = 3^{3} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1755.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:	$14.0137455547$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	1755.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.732051 q^{2} -1.46410 q^{4} +1.00000 q^{5} +3.73205 q^{7} +2.53590 q^{8} +O(q^{10})$	\(q-0.732051 q^{2} -1.46410 q^{4} +1.00000 q^{5} +3.73205 q^{7} +2.53590 q^{8} -0.732051 q^{10} +1.46410 q^{11} -1.00000 q^{13} -2.73205 q^{14} +1.07180 q^{16} +4.00000 q^{17} +6.00000 q^{19} -1.46410 q^{20} -1.07180 q^{22} -3.00000 q^{23} +1.00000 q^{25} +0.732051 q^{26} -5.46410 q^{28} -4.19615 q^{29} +0.732051 q^{31} -5.85641 q^{32} -2.92820 q^{34} +3.73205 q^{35} -1.19615 q^{37} -4.39230 q^{38} +2.53590 q^{40} +1.73205 q^{41} +6.19615 q^{43} -2.14359 q^{44} +2.19615 q^{46} -5.26795 q^{47} +6.92820 q^{49} -0.732051 q^{50} +1.46410 q^{52} -1.53590 q^{53} +1.46410 q^{55} +9.46410 q^{56} +3.07180 q^{58} -7.73205 q^{59} +2.92820 q^{61} -0.535898 q^{62} +2.14359 q^{64} -1.00000 q^{65} +7.19615 q^{67} -5.85641 q^{68} -2.73205 q^{70} +2.66025 q^{71} -11.1962 q^{73} +0.875644 q^{74} -8.78461 q^{76} +5.46410 q^{77} +9.00000 q^{79} +1.07180 q^{80} -1.26795 q^{82} -8.92820 q^{83} +4.00000 q^{85} -4.53590 q^{86} +3.71281 q^{88} -1.19615 q^{89} -3.73205 q^{91} +4.39230 q^{92} +3.85641 q^{94} +6.00000 q^{95} +14.3923 q^{97} -5.07180 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 4 q^{4} + 2 q^{5} + 4 q^{7} + 12 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 4 * q^4 + 2 * q^5 + 4 * q^7 + 12 * q^8	$ 2 q + 2 q^{2} + 4 q^{4} + 2 q^{5} + 4 q^{7} + 12 q^{8} + 2 q^{10} - 4 q^{11} - 2 q^{13} - 2 q^{14} + 16 q^{16} + 8 q^{17} + 12 q^{19} + 4 q^{20} - 16 q^{22} - 6 q^{23} + 2 q^{25} - 2 q^{26} - 4 q^{28} + 2 q^{29} - 2 q^{31} + 16 q^{32} + 8 q^{34} + 4 q^{35} + 8 q^{37} + 12 q^{38} + 12 q^{40} + 2 q^{43} - 32 q^{44} - 6 q^{46} - 14 q^{47} + 2 q^{50} - 4 q^{52} - 10 q^{53} - 4 q^{55} + 12 q^{56} + 20 q^{58} - 12 q^{59} - 8 q^{61} - 8 q^{62} + 32 q^{64} - 2 q^{65} + 4 q^{67} + 16 q^{68} - 2 q^{70} - 12 q^{71} - 12 q^{73} + 26 q^{74} + 24 q^{76} + 4 q^{77} + 18 q^{79} + 16 q^{80} - 6 q^{82} - 4 q^{83} + 8 q^{85} - 16 q^{86} - 48 q^{88} + 8 q^{89} - 4 q^{91} - 12 q^{92} - 20 q^{94} + 12 q^{95} + 8 q^{97} - 24 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 4 * q^4 + 2 * q^5 + 4 * q^7 + 12 * q^8 + 2 * q^10 - 4 * q^11 - 2 * q^13 - 2 * q^14 + 16 * q^16 + 8 * q^17 + 12 * q^19 + 4 * q^20 - 16 * q^22 - 6 * q^23 + 2 * q^25 - 2 * q^26 - 4 * q^28 + 2 * q^29 - 2 * q^31 + 16 * q^32 + 8 * q^34 + 4 * q^35 + 8 * q^37 + 12 * q^38 + 12 * q^40 + 2 * q^43 - 32 * q^44 - 6 * q^46 - 14 * q^47 + 2 * q^50 - 4 * q^52 - 10 * q^53 - 4 * q^55 + 12 * q^56 + 20 * q^58 - 12 * q^59 - 8 * q^61 - 8 * q^62 + 32 * q^64 - 2 * q^65 + 4 * q^67 + 16 * q^68 - 2 * q^70 - 12 * q^71 - 12 * q^73 + 26 * q^74 + 24 * q^76 + 4 * q^77 + 18 * q^79 + 16 * q^80 - 6 * q^82 - 4 * q^83 + 8 * q^85 - 16 * q^86 - 48 * q^88 + 8 * q^89 - 4 * q^91 - 12 * q^92 - 20 * q^94 + 12 * q^95 + 8 * q^97 - 24 * 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 $

$2$	−0.732051	−0.517638	−0.258819	−	0.965926i	$-0.583333\pi$
$2$	−0.732051	−0.517638	−0.258819	+	0.965926i	$0.583333\pi$
$3$	0	0
$3$	0	0
$4$	−1.46410	−0.732051
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.73205	1.41058	0.705291	−	0.708918i	$-0.250816\pi$
$7$	3.73205	1.41058	0.705291	+	0.708918i	$0.250816\pi$
$8$	2.53590	0.896575
$9$	0	0
$10$	−0.732051	−0.231495
$11$	1.46410	0.441443	0.220722	−	0.975337i	$-0.429159\pi$
$11$	1.46410	0.441443	0.220722	+	0.975337i	$0.429159\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−2.73205	−0.730171
$15$	0	0
$16$	1.07180	0.267949
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	−1.46410	−0.327383
$21$	0	0
$22$	−1.07180	−0.228508
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.732051	0.143567
$27$	0	0
$28$	−5.46410	−1.03262
$29$	−4.19615	−0.779206	−0.389603	−	0.920983i	$-0.627388\pi$
$29$	−4.19615	−0.779206	−0.389603	+	0.920983i	$0.627388\pi$
$30$	0	0
$31$	0.732051	0.131480	0.0657401	−	0.997837i	$-0.479059\pi$
$31$	0.732051	0.131480	0.0657401	+	0.997837i	$0.479059\pi$
$32$	−5.85641	−1.03528
$33$	0	0
$34$	−2.92820	−0.502183
$35$	3.73205	0.630832
$36$	0	0
$37$	−1.19615	−0.196646	−0.0983231	−	0.995155i	$-0.531348\pi$
$37$	−1.19615	−0.196646	−0.0983231	+	0.995155i	$0.531348\pi$
$38$	−4.39230	−0.712526
$39$	0	0
$40$	2.53590	0.400961
$41$	1.73205	0.270501	0.135250	−	0.990811i	$-0.456816\pi$
$41$	1.73205	0.270501	0.135250	+	0.990811i	$0.456816\pi$
$42$	0	0
$43$	6.19615	0.944904	0.472452	−	0.881356i	$-0.343369\pi$
$43$	6.19615	0.944904	0.472452	+	0.881356i	$0.343369\pi$
$44$	−2.14359	−0.323159
$45$	0	0
$46$	2.19615	0.323805
$47$	−5.26795	−0.768409	−0.384205	−	0.923248i	$-0.625524\pi$
$47$	−5.26795	−0.768409	−0.384205	+	0.923248i	$0.625524\pi$
$48$	0	0
$49$	6.92820	0.989743
$50$	−0.732051	−0.103528
$51$	0	0
$52$	1.46410	0.203034
$53$	−1.53590	−0.210972	−0.105486	−	0.994421i	$-0.533640\pi$
$53$	−1.53590	−0.210972	−0.105486	+	0.994421i	$0.533640\pi$
$54$	0	0
$55$	1.46410	0.197419
$56$	9.46410	1.26469
$57$	0	0
$58$	3.07180	0.403347
$59$	−7.73205	−1.00663	−0.503314	−	0.864104i	$-0.667886\pi$
$59$	−7.73205	−1.00663	−0.503314	+	0.864104i	$0.667886\pi$
$60$	0	0
$61$	2.92820	0.374918	0.187459	−	0.982272i	$-0.439975\pi$
$61$	2.92820	0.374918	0.187459	+	0.982272i	$0.439975\pi$
$62$	−0.535898	−0.0680592
$63$	0	0
$64$	2.14359	0.267949
$65$	−1.00000	−0.124035
$66$	0	0
$67$	7.19615	0.879150	0.439575	−	0.898206i	$-0.355129\pi$
$67$	7.19615	0.879150	0.439575	+	0.898206i	$0.355129\pi$
$68$	−5.85641	−0.710194
$69$	0	0
$70$	−2.73205	−0.326543
$71$	2.66025	0.315714	0.157857	−	0.987462i	$-0.449541\pi$
$71$	2.66025	0.315714	0.157857	+	0.987462i	$0.449541\pi$
$72$	0	0
$73$	−11.1962	−1.31041	−0.655205	−	0.755451i	$-0.727418\pi$
$73$	−11.1962	−1.31041	−0.655205	+	0.755451i	$0.727418\pi$
$74$	0.875644	0.101792
$75$	0	0
$76$	−8.78461	−1.00766
$77$	5.46410	0.622692
$78$	0	0
$79$	9.00000	1.01258	0.506290	−	0.862364i	$-0.331017\pi$
$79$	9.00000	1.01258	0.506290	+	0.862364i	$0.331017\pi$
$80$	1.07180	0.119831
$81$	0	0
$82$	−1.26795	−0.140022
$83$	−8.92820	−0.979998	−0.489999	−	0.871723i	$-0.663003\pi$
$83$	−8.92820	−0.979998	−0.489999	+	0.871723i	$0.663003\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	−4.53590	−0.489119
$87$	0	0
$88$	3.71281	0.395787
$89$	−1.19615	−0.126792	−0.0633960	−	0.997988i	$-0.520193\pi$
$89$	−1.19615	−0.126792	−0.0633960	+	0.997988i	$0.520193\pi$
$90$	0	0
$91$	−3.73205	−0.391225
$92$	4.39230	0.457929
$93$	0	0
$94$	3.85641	0.397758
$95$	6.00000	0.615587
$96$	0	0
$97$	14.3923	1.46132	0.730659	−	0.682743i	$-0.239213\pi$
$97$	14.3923	1.46132	0.730659	+	0.682743i	$0.239213\pi$
$98$	−5.07180	−0.512329
$99$	0	0
$100$	−1.46410	−0.146410
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	0	0
$103$	11.1244	1.09612	0.548058	−	0.836441i	$-0.315367\pi$
$103$	11.1244	1.09612	0.548058	+	0.836441i	$0.315367\pi$
$104$	−2.53590	−0.248665
$105$	0	0
$106$	1.12436	0.109207
$107$	6.39230	0.617967	0.308984	−	0.951067i	$-0.400011\pi$
$107$	6.39230	0.617967	0.308984	+	0.951067i	$0.400011\pi$
$108$	0	0
$109$	4.92820	0.472036	0.236018	−	0.971749i	$-0.424158\pi$
$109$	4.92820	0.472036	0.236018	+	0.971749i	$0.424158\pi$
$110$	−1.07180	−0.102192
$111$	0	0
$112$	4.00000	0.377964
$113$	−20.8564	−1.96201	−0.981003	−	0.193993i	$-0.937856\pi$
$113$	−20.8564	−1.96201	−0.981003	+	0.193993i	$0.937856\pi$
$114$	0	0
$115$	−3.00000	−0.279751
$116$	6.14359	0.570418
$117$	0	0
$118$	5.66025	0.521069
$119$	14.9282	1.36847
$120$	0	0
$121$	−8.85641	−0.805128
$122$	−2.14359	−0.194072
$123$	0	0
$124$	−1.07180	−0.0962502
$125$	1.00000	0.0894427
$126$	0	0
$127$	22.0000	1.95218	0.976092	−	0.217357i	$-0.0697436\pi$
$127$	22.0000	1.95218	0.976092	+	0.217357i	$0.0697436\pi$
$128$	10.1436	0.896575
$129$	0	0
$130$	0.732051	0.0642051
$131$	3.66025	0.319798	0.159899	−	0.987133i	$-0.448883\pi$
$131$	3.66025	0.319798	0.159899	+	0.987133i	$0.448883\pi$
$132$	0	0
$133$	22.3923	1.94166
$134$	−5.26795	−0.455081
$135$	0	0
$136$	10.1436	0.869806
$137$	19.3205	1.65066	0.825331	−	0.564649i	$-0.190988\pi$
$137$	19.3205	1.65066	0.825331	+	0.564649i	$0.190988\pi$
$138$	0	0
$139$	19.8564	1.68420	0.842099	−	0.539323i	$-0.181320\pi$
$139$	19.8564	1.68420	0.842099	+	0.539323i	$0.181320\pi$
$140$	−5.46410	−0.461801
$141$	0	0
$142$	−1.94744	−0.163426
$143$	−1.46410	−0.122434
$144$	0	0
$145$	−4.19615	−0.348471
$146$	8.19615	0.678318
$147$	0	0
$148$	1.75129	0.143955
$149$	−8.66025	−0.709476	−0.354738	−	0.934966i	$-0.615430\pi$
$149$	−8.66025	−0.709476	−0.354738	+	0.934966i	$0.615430\pi$
$150$	0	0
$151$	−8.58846	−0.698919	−0.349459	−	0.936952i	$-0.613635\pi$
$151$	−8.58846	−0.698919	−0.349459	+	0.936952i	$0.613635\pi$
$152$	15.2154	1.23413
$153$	0	0
$154$	−4.00000	−0.322329
$155$	0.732051	0.0587997
$156$	0	0
$157$	−10.7321	−0.856511	−0.428255	−	0.903658i	$-0.640872\pi$
$157$	−10.7321	−0.856511	−0.428255	+	0.903658i	$0.640872\pi$
$158$	−6.58846	−0.524150
$159$	0	0
$160$	−5.85641	−0.462990
$161$	−11.1962	−0.882380
$162$	0	0
$163$	−4.39230	−0.344032	−0.172016	−	0.985094i	$-0.555028\pi$
$163$	−4.39230	−0.344032	−0.172016	+	0.985094i	$0.555028\pi$
$164$	−2.53590	−0.198020
$165$	0	0
$166$	6.53590	0.507284
$167$	2.53590	0.196234	0.0981169	−	0.995175i	$-0.468718\pi$
$167$	2.53590	0.196234	0.0981169	+	0.995175i	$0.468718\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−2.92820	−0.224583
$171$	0	0
$172$	−9.07180	−0.691718
$173$	23.3205	1.77303	0.886513	−	0.462704i	$-0.153121\pi$
$173$	23.3205	1.77303	0.886513	+	0.462704i	$0.153121\pi$
$174$	0	0
$175$	3.73205	0.282117
$176$	1.56922	0.118284
$177$	0	0
$178$	0.875644	0.0656323
$179$	−8.53590	−0.638003	−0.319002	−	0.947754i	$-0.603348\pi$
$179$	−8.53590	−0.638003	−0.319002	+	0.947754i	$0.603348\pi$
$180$	0	0
$181$	4.07180	0.302654	0.151327	−	0.988484i	$-0.451645\pi$
$181$	4.07180	0.302654	0.151327	+	0.988484i	$0.451645\pi$
$182$	2.73205	0.202513
$183$	0	0
$184$	−7.60770	−0.560847
$185$	−1.19615	−0.0879429
$186$	0	0
$187$	5.85641	0.428263
$188$	7.71281	0.562515
$189$	0	0
$190$	−4.39230	−0.318651
$191$	−13.2679	−0.960035	−0.480018	−	0.877259i	$-0.659370\pi$
$191$	−13.2679	−0.960035	−0.480018	+	0.877259i	$0.659370\pi$
$192$	0	0
$193$	−4.66025	−0.335452	−0.167726	−	0.985834i	$-0.553642\pi$
$193$	−4.66025	−0.335452	−0.167726	+	0.985834i	$0.553642\pi$
$194$	−10.5359	−0.756433
$195$	0	0
$196$	−10.1436	−0.724542
$197$	7.26795	0.517820	0.258910	−	0.965901i	$-0.416637\pi$
$197$	7.26795	0.517820	0.258910	+	0.965901i	$0.416637\pi$
$198$	0	0
$199$	−24.3205	−1.72404	−0.862018	−	0.506878i	$-0.830799\pi$
$199$	−24.3205	−1.72404	−0.862018	+	0.506878i	$0.830799\pi$
$200$	2.53590	0.179315
$201$	0	0
$202$	−8.78461	−0.618083
$203$	−15.6603	−1.09913
$204$	0	0
$205$	1.73205	0.120972
$206$	−8.14359	−0.567391
$207$	0	0
$208$	−1.07180	−0.0743157
$209$	8.78461	0.607644
$210$	0	0
$211$	−16.8564	−1.16044	−0.580221	−	0.814459i	$-0.697034\pi$
$211$	−16.8564	−1.16044	−0.580221	+	0.814459i	$0.697034\pi$
$212$	2.24871	0.154442
$213$	0	0
$214$	−4.67949	−0.319883
$215$	6.19615	0.422574
$216$	0	0
$217$	2.73205	0.185464
$218$	−3.60770	−0.244344
$219$	0	0
$220$	−2.14359	−0.144521
$221$	−4.00000	−0.269069
$222$	0	0
$223$	−2.12436	−0.142257	−0.0711287	−	0.997467i	$-0.522660\pi$
$223$	−2.12436	−0.142257	−0.0711287	+	0.997467i	$0.522660\pi$
$224$	−21.8564	−1.46034
$225$	0	0
$226$	15.2679	1.01561
$227$	13.6603	0.906663	0.453331	−	0.891342i	$-0.350235\pi$
$227$	13.6603	0.906663	0.453331	+	0.891342i	$0.350235\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	2.19615	0.144810
$231$	0	0
$232$	−10.6410	−0.698617
$233$	5.00000	0.327561	0.163780	−	0.986497i	$-0.447631\pi$
$233$	5.00000	0.327561	0.163780	+	0.986497i	$0.447631\pi$
$234$	0	0
$235$	−5.26795	−0.343643
$236$	11.3205	0.736902
$237$	0	0
$238$	−10.9282	−0.708370
$239$	18.1244	1.17237	0.586184	−	0.810178i	$-0.300630\pi$
$239$	18.1244	1.17237	0.586184	+	0.810178i	$0.300630\pi$
$240$	0	0
$241$	−18.0526	−1.16287	−0.581434	−	0.813594i	$-0.697508\pi$
$241$	−18.0526	−1.16287	−0.581434	+	0.813594i	$0.697508\pi$
$242$	6.48334	0.416765
$243$	0	0
$244$	−4.28719	−0.274459
$245$	6.92820	0.442627
$246$	0	0
$247$	−6.00000	−0.381771
$248$	1.85641	0.117882
$249$	0	0
$250$	−0.732051	−0.0462990
$251$	−25.5167	−1.61060	−0.805299	−	0.592869i	$-0.797995\pi$
$251$	−25.5167	−1.61060	−0.805299	+	0.592869i	$0.797995\pi$
$252$	0	0
$253$	−4.39230	−0.276142
$254$	−16.1051	−1.01052
$255$	0	0
$256$	−11.7128	−0.732051
$257$	23.3923	1.45917	0.729586	−	0.683889i	$-0.239713\pi$
$257$	23.3923	1.45917	0.729586	+	0.683889i	$0.239713\pi$
$258$	0	0
$259$	−4.46410	−0.277386
$260$	1.46410	0.0907997
$261$	0	0
$262$	−2.67949	−0.165540
$263$	13.9282	0.858850	0.429425	−	0.903103i	$-0.358716\pi$
$263$	13.9282	0.858850	0.429425	+	0.903103i	$0.358716\pi$
$264$	0	0
$265$	−1.53590	−0.0943495
$266$	−16.3923	−1.00508
$267$	0	0
$268$	−10.5359	−0.643582
$269$	13.5167	0.824125	0.412063	−	0.911156i	$-0.364808\pi$
$269$	13.5167	0.824125	0.412063	+	0.911156i	$0.364808\pi$
$270$	0	0
$271$	17.1244	1.04023	0.520115	−	0.854096i	$-0.325889\pi$
$271$	17.1244	1.04023	0.520115	+	0.854096i	$0.325889\pi$
$272$	4.28719	0.259949
$273$	0	0
$274$	−14.1436	−0.854446
$275$	1.46410	0.0882886
$276$	0	0
$277$	−1.66025	−0.0997550	−0.0498775	−	0.998755i	$-0.515883\pi$
$277$	−1.66025	−0.0997550	−0.0498775	+	0.998755i	$0.515883\pi$
$278$	−14.5359	−0.871805
$279$	0	0
$280$	9.46410	0.565588
$281$	−31.0526	−1.85244	−0.926220	−	0.376983i	$-0.876962\pi$
$281$	−31.0526	−1.85244	−0.926220	+	0.376983i	$0.876962\pi$
$282$	0	0
$283$	4.19615	0.249435	0.124718	−	0.992192i	$-0.460197\pi$
$283$	4.19615	0.249435	0.124718	+	0.992192i	$0.460197\pi$
$284$	−3.89488	−0.231119
$285$	0	0
$286$	1.07180	0.0633767
$287$	6.46410	0.381564
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	3.07180	0.180382
$291$	0	0
$292$	16.3923	0.959287
$293$	6.39230	0.373442	0.186721	−	0.982413i	$-0.440214\pi$
$293$	6.39230	0.373442	0.186721	+	0.982413i	$0.440214\pi$
$294$	0	0
$295$	−7.73205	−0.450177
$296$	−3.03332	−0.176308
$297$	0	0
$298$	6.33975	0.367252
$299$	3.00000	0.173494
$300$	0	0
$301$	23.1244	1.33287
$302$	6.28719	0.361787
$303$	0	0
$304$	6.43078	0.368831
$305$	2.92820	0.167668
$306$	0	0
$307$	−26.5167	−1.51339	−0.756693	−	0.653771i	$-0.773186\pi$
$307$	−26.5167	−1.51339	−0.756693	+	0.653771i	$0.773186\pi$
$308$	−8.00000	−0.455842
$309$	0	0
$310$	−0.535898	−0.0304370
$311$	3.32051	0.188289	0.0941444	−	0.995559i	$-0.469988\pi$
$311$	3.32051	0.188289	0.0941444	+	0.995559i	$0.469988\pi$
$312$	0	0
$313$	−7.80385	−0.441100	−0.220550	−	0.975376i	$-0.570785\pi$
$313$	−7.80385	−0.441100	−0.220550	+	0.975376i	$0.570785\pi$
$314$	7.85641	0.443363
$315$	0	0
$316$	−13.1769	−0.741259
$317$	13.8564	0.778253	0.389127	−	0.921184i	$-0.372777\pi$
$317$	13.8564	0.778253	0.389127	+	0.921184i	$0.372777\pi$
$318$	0	0
$319$	−6.14359	−0.343975
$320$	2.14359	0.119831
$321$	0	0
$322$	8.19615	0.456754
$323$	24.0000	1.33540
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	3.21539	0.178084
$327$	0	0
$328$	4.39230	0.242524
$329$	−19.6603	−1.08390
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	13.0718	0.717408
$333$	0	0
$334$	−1.85641	−0.101578
$335$	7.19615	0.393168
$336$	0	0
$337$	−20.3923	−1.11084	−0.555420	−	0.831570i	$-0.687442\pi$
$337$	−20.3923	−1.11084	−0.555420	+	0.831570i	$0.687442\pi$
$338$	−0.732051	−0.0398183
$339$	0	0
$340$	−5.85641	−0.317608
$341$	1.07180	0.0580410
$342$	0	0
$343$	−0.267949	−0.0144679
$344$	15.7128	0.847178
$345$	0	0
$346$	−17.0718	−0.917785
$347$	−6.85641	−0.368071	−0.184036	−	0.982920i	$-0.558916\pi$
$347$	−6.85641	−0.368071	−0.184036	+	0.982920i	$0.558916\pi$
$348$	0	0
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	−2.73205	−0.146034
$351$	0	0
$352$	−8.57437	−0.457016
$353$	−31.5167	−1.67746	−0.838731	−	0.544546i	$-0.816702\pi$
$353$	−31.5167	−1.67746	−0.838731	+	0.544546i	$0.816702\pi$
$354$	0	0
$355$	2.66025	0.141192
$356$	1.75129	0.0928181
$357$	0	0
$358$	6.24871	0.330255
$359$	−28.9090	−1.52576	−0.762878	−	0.646542i	$-0.776215\pi$
$359$	−28.9090	−1.52576	−0.762878	+	0.646542i	$0.776215\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−2.98076	−0.156665
$363$	0	0
$364$	5.46410	0.286397
$365$	−11.1962	−0.586033
$366$	0	0
$367$	30.7846	1.60694	0.803472	−	0.595343i	$-0.202984\pi$
$367$	30.7846	1.60694	0.803472	+	0.595343i	$0.202984\pi$
$368$	−3.21539	−0.167614
$369$	0	0
$370$	0.875644	0.0455226
$371$	−5.73205	−0.297593
$372$	0	0
$373$	0.875644	0.0453391	0.0226696	−	0.999743i	$-0.492783\pi$
$373$	0.875644	0.0453391	0.0226696	+	0.999743i	$0.492783\pi$
$374$	−4.28719	−0.221685
$375$	0	0
$376$	−13.3590	−0.688937
$377$	4.19615	0.216113
$378$	0	0
$379$	−15.6077	−0.801713	−0.400857	−	0.916141i	$-0.631287\pi$
$379$	−15.6077	−0.801713	−0.400857	+	0.916141i	$0.631287\pi$
$380$	−8.78461	−0.450641
$381$	0	0
$382$	9.71281	0.496951
$383$	−3.80385	−0.194368	−0.0971838	−	0.995266i	$-0.530983\pi$
$383$	−3.80385	−0.194368	−0.0971838	+	0.995266i	$0.530983\pi$
$384$	0	0
$385$	5.46410	0.278476
$386$	3.41154	0.173643
$387$	0	0
$388$	−21.0718	−1.06976
$389$	−20.2487	−1.02665	−0.513325	−	0.858194i	$-0.671587\pi$
$389$	−20.2487	−1.02665	−0.513325	+	0.858194i	$0.671587\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	17.5692	0.887380
$393$	0	0
$394$	−5.32051	−0.268043
$395$	9.00000	0.452839
$396$	0	0
$397$	23.1962	1.16418	0.582091	−	0.813124i	$-0.302235\pi$
$397$	23.1962	1.16418	0.582091	+	0.813124i	$0.302235\pi$
$398$	17.8038	0.892426
$399$	0	0
$400$	1.07180	0.0535898
$401$	−22.1244	−1.10484	−0.552419	−	0.833567i	$-0.686295\pi$
$401$	−22.1244	−1.10484	−0.552419	+	0.833567i	$0.686295\pi$
$402$	0	0
$403$	−0.732051	−0.0364660
$404$	−17.5692	−0.874101
$405$	0	0
$406$	11.4641	0.568954
$407$	−1.75129	−0.0868082
$408$	0	0
$409$	23.3205	1.15312	0.576562	−	0.817053i	$-0.304394\pi$
$409$	23.3205	1.15312	0.576562	+	0.817053i	$0.304394\pi$
$410$	−1.26795	−0.0626195
$411$	0	0
$412$	−16.2872	−0.802412
$413$	−28.8564	−1.41993
$414$	0	0
$415$	−8.92820	−0.438268
$416$	5.85641	0.287134
$417$	0	0
$418$	−6.43078	−0.314540
$419$	−35.7128	−1.74469	−0.872343	−	0.488895i	$-0.837400\pi$
$419$	−35.7128	−1.74469	−0.872343	+	0.488895i	$0.837400\pi$
$420$	0	0
$421$	3.07180	0.149710	0.0748551	−	0.997194i	$-0.476151\pi$
$421$	3.07180	0.149710	0.0748551	+	0.997194i	$0.476151\pi$
$422$	12.3397	0.600689
$423$	0	0
$424$	−3.89488	−0.189152
$425$	4.00000	0.194029
$426$	0	0
$427$	10.9282	0.528853
$428$	−9.35898	−0.452384
$429$	0	0
$430$	−4.53590	−0.218740
$431$	26.3923	1.27127	0.635636	−	0.771989i	$-0.280738\pi$
$431$	26.3923	1.27127	0.635636	+	0.771989i	$0.280738\pi$
$432$	0	0
$433$	0.875644	0.0420808	0.0210404	−	0.999779i	$-0.493302\pi$
$433$	0.875644	0.0420808	0.0210404	+	0.999779i	$0.493302\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	−7.21539	−0.345555
$437$	−18.0000	−0.861057
$438$	0	0
$439$	−27.9282	−1.33294	−0.666470	−	0.745532i	$-0.732196\pi$
$439$	−27.9282	−1.33294	−0.666470	+	0.745532i	$0.732196\pi$
$440$	3.71281	0.177001
$441$	0	0
$442$	2.92820	0.139280
$443$	−17.7128	−0.841561	−0.420781	−	0.907162i	$-0.638244\pi$
$443$	−17.7128	−0.841561	−0.420781	+	0.907162i	$0.638244\pi$
$444$	0	0
$445$	−1.19615	−0.0567031
$446$	1.55514	0.0736378
$447$	0	0
$448$	8.00000	0.377964
$449$	24.1244	1.13850	0.569249	−	0.822165i	$-0.307234\pi$
$449$	24.1244	1.13850	0.569249	+	0.822165i	$0.307234\pi$
$450$	0	0
$451$	2.53590	0.119411
$452$	30.5359	1.43629
$453$	0	0
$454$	−10.0000	−0.469323
$455$	−3.73205	−0.174961
$456$	0	0
$457$	31.1962	1.45929	0.729647	−	0.683824i	$-0.239684\pi$
$457$	31.1962	1.45929	0.729647	+	0.683824i	$0.239684\pi$
$458$	7.32051	0.342065
$459$	0	0
$460$	4.39230	0.204792
$461$	−0.411543	−0.0191675	−0.00958373	−	0.999954i	$-0.503051\pi$
$461$	−0.411543	−0.0191675	−0.00958373	+	0.999954i	$0.503051\pi$
$462$	0	0
$463$	−29.4449	−1.36842	−0.684209	−	0.729286i	$-0.739853\pi$
$463$	−29.4449	−1.36842	−0.684209	+	0.729286i	$0.739853\pi$
$464$	−4.49742	−0.208788
$465$	0	0
$466$	−3.66025	−0.169558
$467$	−29.0000	−1.34196	−0.670980	−	0.741475i	$-0.734126\pi$
$467$	−29.0000	−1.34196	−0.670980	+	0.741475i	$0.734126\pi$
$468$	0	0
$469$	26.8564	1.24011
$470$	3.85641	0.177883
$471$	0	0
$472$	−19.6077	−0.902517
$473$	9.07180	0.417122
$474$	0	0
$475$	6.00000	0.275299
$476$	−21.8564	−1.00179
$477$	0	0
$478$	−13.2679	−0.606862
$479$	−18.8038	−0.859170	−0.429585	−	0.903026i	$-0.641340\pi$
$479$	−18.8038	−0.859170	−0.429585	+	0.903026i	$0.641340\pi$
$480$	0	0
$481$	1.19615	0.0545399
$482$	13.2154	0.601945
$483$	0	0
$484$	12.9667	0.589395
$485$	14.3923	0.653521
$486$	0	0
$487$	−20.3923	−0.924064	−0.462032	−	0.886863i	$-0.652879\pi$
$487$	−20.3923	−0.924064	−0.462032	+	0.886863i	$0.652879\pi$
$488$	7.42563	0.336142
$489$	0	0
$490$	−5.07180	−0.229120
$491$	14.5359	0.655996	0.327998	−	0.944678i	$-0.393626\pi$
$491$	14.5359	0.655996	0.327998	+	0.944678i	$0.393626\pi$
$492$	0	0
$493$	−16.7846	−0.755941
$494$	4.39230	0.197619
$495$	0	0
$496$	0.784610	0.0352300
$497$	9.92820	0.445341
$498$	0	0
$499$	43.1244	1.93051	0.965256	−	0.261307i	$-0.0841536\pi$
$499$	43.1244	1.93051	0.965256	+	0.261307i	$0.0841536\pi$
$500$	−1.46410	−0.0654766
$501$	0	0
$502$	18.6795	0.833707
$503$	22.3923	0.998424	0.499212	−	0.866480i	$-0.333623\pi$
$503$	22.3923	0.998424	0.499212	+	0.866480i	$0.333623\pi$
$504$	0	0
$505$	12.0000	0.533993
$506$	3.21539	0.142942
$507$	0	0
$508$	−32.2102	−1.42910
$509$	−15.4641	−0.685434	−0.342717	−	0.939439i	$-0.611347\pi$
$509$	−15.4641	−0.685434	−0.342717	+	0.939439i	$0.611347\pi$
$510$	0	0
$511$	−41.7846	−1.84844
$512$	−11.7128	−0.517638
$513$	0	0
$514$	−17.1244	−0.755323
$515$	11.1244	0.490198
$516$	0	0
$517$	−7.71281	−0.339209
$518$	3.26795	0.143585
$519$	0	0
$520$	−2.53590	−0.111207
$521$	9.46410	0.414630	0.207315	−	0.978274i	$-0.433528\pi$
$521$	9.46410	0.414630	0.207315	+	0.978274i	$0.433528\pi$
$522$	0	0
$523$	31.5167	1.37813	0.689064	−	0.724701i	$-0.258022\pi$
$523$	31.5167	1.37813	0.689064	+	0.724701i	$0.258022\pi$
$524$	−5.35898	−0.234108
$525$	0	0
$526$	−10.1962	−0.444573
$527$	2.92820	0.127555
$528$	0	0
$529$	−14.0000	−0.608696
$530$	1.12436	0.0488389
$531$	0	0
$532$	−32.7846	−1.42139
$533$	−1.73205	−0.0750234
$534$	0	0
$535$	6.39230	0.276363
$536$	18.2487	0.788224
$537$	0	0
$538$	−9.89488	−0.426599
$539$	10.1436	0.436916
$540$	0	0
$541$	−19.1244	−0.822220	−0.411110	−	0.911586i	$-0.634859\pi$
$541$	−19.1244	−0.822220	−0.411110	+	0.911586i	$0.634859\pi$
$542$	−12.5359	−0.538463
$543$	0	0
$544$	−23.4256	−1.00437
$545$	4.92820	0.211101
$546$	0	0
$547$	−2.53590	−0.108427	−0.0542136	−	0.998529i	$-0.517265\pi$
$547$	−2.53590	−0.108427	−0.0542136	+	0.998529i	$0.517265\pi$
$548$	−28.2872	−1.20837
$549$	0	0
$550$	−1.07180	−0.0457016
$551$	−25.1769	−1.07257
$552$	0	0
$553$	33.5885	1.42833
$554$	1.21539	0.0516370
$555$	0	0
$556$	−29.0718	−1.23292
$557$	26.3397	1.11605	0.558025	−	0.829824i	$-0.311559\pi$
$557$	26.3397	1.11605	0.558025	+	0.829824i	$0.311559\pi$
$558$	0	0
$559$	−6.19615	−0.262069
$560$	4.00000	0.169031
$561$	0	0
$562$	22.7321	0.958894
$563$	42.3205	1.78360	0.891798	−	0.452433i	$-0.149444\pi$
$563$	42.3205	1.78360	0.891798	+	0.452433i	$0.149444\pi$
$564$	0	0
$565$	−20.8564	−0.877436
$566$	−3.07180	−0.129117
$567$	0	0
$568$	6.74613	0.283061
$569$	−5.26795	−0.220844	−0.110422	−	0.993885i	$-0.535220\pi$
$569$	−5.26795	−0.220844	−0.110422	+	0.993885i	$0.535220\pi$
$570$	0	0
$571$	27.0000	1.12991	0.564957	−	0.825120i	$-0.308893\pi$
$571$	27.0000	1.12991	0.564957	+	0.825120i	$0.308893\pi$
$572$	2.14359	0.0896281
$573$	0	0
$574$	−4.73205	−0.197512
$575$	−3.00000	−0.125109
$576$	0	0
$577$	−36.7846	−1.53136	−0.765682	−	0.643220i	$-0.777598\pi$
$577$	−36.7846	−1.53136	−0.765682	+	0.643220i	$0.777598\pi$
$578$	0.732051	0.0304493
$579$	0	0
$580$	6.14359	0.255099
$581$	−33.3205	−1.38237
$582$	0	0
$583$	−2.24871	−0.0931321
$584$	−28.3923	−1.17488
$585$	0	0
$586$	−4.67949	−0.193308
$587$	−38.6410	−1.59489	−0.797443	−	0.603395i	$-0.793814\pi$
$587$	−38.6410	−1.59489	−0.797443	+	0.603395i	$0.793814\pi$
$588$	0	0
$589$	4.39230	0.180982
$590$	5.66025	0.233029
$591$	0	0
$592$	−1.28203	−0.0526912
$593$	34.9282	1.43433	0.717165	−	0.696904i	$-0.245440\pi$
$593$	34.9282	1.43433	0.717165	+	0.696904i	$0.245440\pi$
$594$	0	0
$595$	14.9282	0.611997
$596$	12.6795	0.519372
$597$	0	0
$598$	−2.19615	−0.0898074
$599$	−6.05256	−0.247301	−0.123650	−	0.992326i	$-0.539460\pi$
$599$	−6.05256	−0.247301	−0.123650	+	0.992326i	$0.539460\pi$
$600$	0	0
$601$	22.7128	0.926475	0.463237	−	0.886234i	$-0.346688\pi$
$601$	22.7128	0.926475	0.463237	+	0.886234i	$0.346688\pi$
$602$	−16.9282	−0.689942
$603$	0	0
$604$	12.5744	0.511644
$605$	−8.85641	−0.360064
$606$	0	0
$607$	−37.3205	−1.51479	−0.757396	−	0.652955i	$-0.773529\pi$
$607$	−37.3205	−1.51479	−0.757396	+	0.652955i	$0.773529\pi$
$608$	−35.1384	−1.42505
$609$	0	0
$610$	−2.14359	−0.0867916
$611$	5.26795	0.213118
$612$	0	0
$613$	15.4641	0.624589	0.312295	−	0.949985i	$-0.398902\pi$
$613$	15.4641	0.624589	0.312295	+	0.949985i	$0.398902\pi$
$614$	19.4115	0.783386
$615$	0	0
$616$	13.8564	0.558291
$617$	−3.46410	−0.139459	−0.0697297	−	0.997566i	$-0.522214\pi$
$617$	−3.46410	−0.139459	−0.0697297	+	0.997566i	$0.522214\pi$
$618$	0	0
$619$	−22.1962	−0.892139	−0.446069	−	0.894998i	$-0.647177\pi$
$619$	−22.1962	−0.892139	−0.446069	+	0.894998i	$0.647177\pi$
$620$	−1.07180	−0.0430444
$621$	0	0
$622$	−2.43078	−0.0974654
$623$	−4.46410	−0.178850
$624$	0	0
$625$	1.00000	0.0400000
$626$	5.71281	0.228330
$627$	0	0
$628$	15.7128	0.627009
$629$	−4.78461	−0.190775
$630$	0	0
$631$	−9.26795	−0.368951	−0.184476	−	0.982837i	$-0.559059\pi$
$631$	−9.26795	−0.368951	−0.184476	+	0.982837i	$0.559059\pi$
$632$	22.8231	0.907854
$633$	0	0
$634$	−10.1436	−0.402854
$635$	22.0000	0.873043
$636$	0	0
$637$	−6.92820	−0.274505
$638$	4.49742	0.178055
$639$	0	0
$640$	10.1436	0.400961
$641$	−0.143594	−0.00567160	−0.00283580	−	0.999996i	$-0.500903\pi$
$641$	−0.143594	−0.00567160	−0.00283580	+	0.999996i	$0.500903\pi$
$642$	0	0
$643$	−39.3205	−1.55065	−0.775325	−	0.631563i	$-0.782414\pi$
$643$	−39.3205	−1.55065	−0.775325	+	0.631563i	$0.782414\pi$
$644$	16.3923	0.645947
$645$	0	0
$646$	−17.5692	−0.691252
$647$	−44.4641	−1.74806	−0.874032	−	0.485868i	$-0.838504\pi$
$647$	−44.4641	−1.74806	−0.874032	+	0.485868i	$0.838504\pi$
$648$	0	0
$649$	−11.3205	−0.444369
$650$	0.732051	0.0287134
$651$	0	0
$652$	6.43078	0.251849
$653$	−40.8564	−1.59883	−0.799417	−	0.600776i	$-0.794858\pi$
$653$	−40.8564	−1.59883	−0.799417	+	0.600776i	$0.794858\pi$
$654$	0	0
$655$	3.66025	0.143018
$656$	1.85641	0.0724805
$657$	0	0
$658$	14.3923	0.561070
$659$	−44.4449	−1.73133	−0.865663	−	0.500627i	$-0.833103\pi$
$659$	−44.4449	−1.73133	−0.865663	+	0.500627i	$0.833103\pi$
$660$	0	0
$661$	46.1051	1.79328	0.896641	−	0.442759i	$-0.146000\pi$
$661$	46.1051	1.79328	0.896641	+	0.442759i	$0.146000\pi$
$662$	8.78461	0.341424
$663$	0	0
$664$	−22.6410	−0.878642
$665$	22.3923	0.868336
$666$	0	0
$667$	12.5885	0.487427
$668$	−3.71281	−0.143653
$669$	0	0
$670$	−5.26795	−0.203519
$671$	4.28719	0.165505
$672$	0	0
$673$	−25.6077	−0.987104	−0.493552	−	0.869716i	$-0.664302\pi$
$673$	−25.6077	−0.987104	−0.493552	+	0.869716i	$0.664302\pi$
$674$	14.9282	0.575013
$675$	0	0
$676$	−1.46410	−0.0563116
$677$	36.0000	1.38359	0.691796	−	0.722093i	$-0.256820\pi$
$677$	36.0000	1.38359	0.691796	+	0.722093i	$0.256820\pi$
$678$	0	0
$679$	53.7128	2.06131
$680$	10.1436	0.388989
$681$	0	0
$682$	−0.784610	−0.0300443
$683$	33.1769	1.26948	0.634740	−	0.772726i	$-0.281107\pi$
$683$	33.1769	1.26948	0.634740	+	0.772726i	$0.281107\pi$
$684$	0	0
$685$	19.3205	0.738199
$686$	0.196152	0.00748913
$687$	0	0
$688$	6.64102	0.253186
$689$	1.53590	0.0585131
$690$	0	0
$691$	−16.5359	−0.629055	−0.314528	−	0.949248i	$-0.601846\pi$
$691$	−16.5359	−0.629055	−0.314528	+	0.949248i	$0.601846\pi$
$692$	−34.1436	−1.29794
$693$	0	0
$694$	5.01924	0.190528
$695$	19.8564	0.753196
$696$	0	0
$697$	6.92820	0.262424
$698$	−13.1769	−0.498754
$699$	0	0
$700$	−5.46410	−0.206524
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	0	0
$703$	−7.17691	−0.270682
$704$	3.13844	0.118284
$705$	0	0
$706$	23.0718	0.868319
$707$	44.7846	1.68430
$708$	0	0
$709$	49.3731	1.85424	0.927122	−	0.374759i	$-0.122275\pi$
$709$	49.3731	1.85424	0.927122	+	0.374759i	$0.122275\pi$
$710$	−1.94744	−0.0730862
$711$	0	0
$712$	−3.03332	−0.113679
$713$	−2.19615	−0.0822466
$714$	0	0
$715$	−1.46410	−0.0547543
$716$	12.4974	0.467051
$717$	0	0
$718$	21.1628	0.789790
$719$	24.0526	0.897009	0.448505	−	0.893781i	$-0.351957\pi$
$719$	24.0526	0.897009	0.448505	+	0.893781i	$0.351957\pi$
$720$	0	0
$721$	41.5167	1.54616
$722$	−12.4449	−0.463150
$723$	0	0
$724$	−5.96152	−0.221558
$725$	−4.19615	−0.155841
$726$	0	0
$727$	−15.8038	−0.586132	−0.293066	−	0.956092i	$-0.594676\pi$
$727$	−15.8038	−0.586132	−0.293066	+	0.956092i	$0.594676\pi$
$728$	−9.46410	−0.350763
$729$	0	0
$730$	8.19615	0.303353
$731$	24.7846	0.916692
$732$	0	0
$733$	38.3923	1.41805	0.709026	−	0.705182i	$-0.249135\pi$
$733$	38.3923	1.41805	0.709026	+	0.705182i	$0.249135\pi$
$734$	−22.5359	−0.831815
$735$	0	0
$736$	17.5692	0.647610
$737$	10.5359	0.388095
$738$	0	0
$739$	19.2679	0.708783	0.354391	−	0.935097i	$-0.384688\pi$
$739$	19.2679	0.708783	0.354391	+	0.935097i	$0.384688\pi$
$740$	1.75129	0.0643787
$741$	0	0
$742$	4.19615	0.154046
$743$	−19.8564	−0.728461	−0.364230	−	0.931309i	$-0.618668\pi$
$743$	−19.8564	−0.728461	−0.364230	+	0.931309i	$0.618668\pi$
$744$	0	0
$745$	−8.66025	−0.317287
$746$	−0.641016	−0.0234693
$747$	0	0
$748$	−8.57437	−0.313510
$749$	23.8564	0.871694
$750$	0	0
$751$	−24.5359	−0.895328	−0.447664	−	0.894202i	$-0.647744\pi$
$751$	−24.5359	−0.895328	−0.447664	+	0.894202i	$0.647744\pi$
$752$	−5.64617	−0.205895
$753$	0	0
$754$	−3.07180	−0.111868
$755$	−8.58846	−0.312566
$756$	0	0
$757$	−0.784610	−0.0285171	−0.0142586	−	0.999898i	$-0.504539\pi$
$757$	−0.784610	−0.0285171	−0.0142586	+	0.999898i	$0.504539\pi$
$758$	11.4256	0.414997
$759$	0	0
$760$	15.2154	0.551920
$761$	37.0526	1.34315	0.671577	−	0.740935i	$-0.265617\pi$
$761$	37.0526	1.34315	0.671577	+	0.740935i	$0.265617\pi$
$762$	0	0
$763$	18.3923	0.665846
$764$	19.4256	0.702794
$765$	0	0
$766$	2.78461	0.100612
$767$	7.73205	0.279188
$768$	0	0
$769$	−1.80385	−0.0650484	−0.0325242	−	0.999471i	$-0.510355\pi$
$769$	−1.80385	−0.0650484	−0.0325242	+	0.999471i	$0.510355\pi$
$770$	−4.00000	−0.144150
$771$	0	0
$772$	6.82309	0.245568
$773$	46.7846	1.68273	0.841363	−	0.540471i	$-0.181754\pi$
$773$	46.7846	1.68273	0.841363	+	0.540471i	$0.181754\pi$
$774$	0	0
$775$	0.732051	0.0262960
$776$	36.4974	1.31018
$777$	0	0
$778$	14.8231	0.531433
$779$	10.3923	0.372343
$780$	0	0
$781$	3.89488	0.139370
$782$	8.78461	0.314137
$783$	0	0
$784$	7.42563	0.265201
$785$	−10.7321	−0.383043
$786$	0	0
$787$	−18.4115	−0.656301	−0.328150	−	0.944626i	$-0.606425\pi$
$787$	−18.4115	−0.656301	−0.328150	+	0.944626i	$0.606425\pi$
$788$	−10.6410	−0.379071
$789$	0	0
$790$	−6.58846	−0.234407
$791$	−77.8372	−2.76757
$792$	0	0
$793$	−2.92820	−0.103984
$794$	−16.9808	−0.602625
$795$	0	0
$796$	35.6077	1.26208
$797$	−37.6410	−1.33331	−0.666657	−	0.745365i	$-0.732275\pi$
$797$	−37.6410	−1.33331	−0.666657	+	0.745365i	$0.732275\pi$
$798$	0	0
$799$	−21.0718	−0.745467
$800$	−5.85641	−0.207055
$801$	0	0
$802$	16.1962	0.571906
$803$	−16.3923	−0.578472
$804$	0	0
$805$	−11.1962	−0.394613
$806$	0.535898	0.0188762
$807$	0	0
$808$	30.4308	1.07055
$809$	−51.2295	−1.80113	−0.900566	−	0.434719i	$-0.856848\pi$
$809$	−51.2295	−1.80113	−0.900566	+	0.434719i	$0.856848\pi$
$810$	0	0
$811$	32.5359	1.14249	0.571245	−	0.820780i	$-0.306461\pi$
$811$	32.5359	1.14249	0.571245	+	0.820780i	$0.306461\pi$
$812$	22.9282	0.804622
$813$	0	0
$814$	1.28203	0.0449352
$815$	−4.39230	−0.153856
$816$	0	0
$817$	37.1769	1.30066
$818$	−17.0718	−0.596901
$819$	0	0
$820$	−2.53590	−0.0885574
$821$	−8.12436	−0.283542	−0.141771	−	0.989899i	$-0.545280\pi$
$821$	−8.12436	−0.283542	−0.141771	+	0.989899i	$0.545280\pi$
$822$	0	0
$823$	−30.5359	−1.06441	−0.532207	−	0.846614i	$-0.678637\pi$
$823$	−30.5359	−1.06441	−0.532207	+	0.846614i	$0.678637\pi$
$824$	28.2102	0.982750
$825$	0	0
$826$	21.1244	0.735010
$827$	−1.46410	−0.0509118	−0.0254559	−	0.999676i	$-0.508104\pi$
$827$	−1.46410	−0.0509118	−0.0254559	+	0.999676i	$0.508104\pi$
$828$	0	0
$829$	−24.6077	−0.854661	−0.427330	−	0.904096i	$-0.640546\pi$
$829$	−24.6077	−0.854661	−0.427330	+	0.904096i	$0.640546\pi$
$830$	6.53590	0.226864
$831$	0	0
$832$	−2.14359	−0.0743157
$833$	27.7128	0.960192
$834$	0	0
$835$	2.53590	0.0877584
$836$	−12.8616	−0.444826
$837$	0	0
$838$	26.1436	0.903115
$839$	−38.2487	−1.32049	−0.660246	−	0.751049i	$-0.729548\pi$
$839$	−38.2487	−1.32049	−0.660246	+	0.751049i	$0.729548\pi$
$840$	0	0
$841$	−11.3923	−0.392838
$842$	−2.24871	−0.0774957
$843$	0	0
$844$	24.6795	0.849503
$845$	1.00000	0.0344010
$846$	0	0
$847$	−33.0526	−1.13570
$848$	−1.64617	−0.0565297
$849$	0	0
$850$	−2.92820	−0.100437
$851$	3.58846	0.123011
$852$	0	0
$853$	−21.5885	−0.739175	−0.369587	−	0.929196i	$-0.620501\pi$
$853$	−21.5885	−0.739175	−0.369587	+	0.929196i	$0.620501\pi$
$854$	−8.00000	−0.273754
$855$	0	0
$856$	16.2102	0.554054
$857$	11.7846	0.402555	0.201277	−	0.979534i	$-0.435491\pi$
$857$	11.7846	0.402555	0.201277	+	0.979534i	$0.435491\pi$
$858$	0	0
$859$	−20.7128	−0.706712	−0.353356	−	0.935489i	$-0.614960\pi$
$859$	−20.7128	−0.706712	−0.353356	+	0.935489i	$0.614960\pi$
$860$	−9.07180	−0.309346
$861$	0	0
$862$	−19.3205	−0.658059
$863$	23.0718	0.785373	0.392687	−	0.919672i	$-0.371546\pi$
$863$	23.0718	0.785373	0.392687	+	0.919672i	$0.371546\pi$
$864$	0	0
$865$	23.3205	0.792921
$866$	−0.641016	−0.0217826
$867$	0	0
$868$	−4.00000	−0.135769
$869$	13.1769	0.446996
$870$	0	0
$871$	−7.19615	−0.243832
$872$	12.4974	0.423216
$873$	0	0
$874$	13.1769	0.445716
$875$	3.73205	0.126166
$876$	0	0
$877$	48.3731	1.63344	0.816721	−	0.577032i	$-0.195789\pi$
$877$	48.3731	1.63344	0.816721	+	0.577032i	$0.195789\pi$
$878$	20.4449	0.689981
$879$	0	0
$880$	1.56922	0.0528984
$881$	23.8564	0.803743	0.401871	−	0.915696i	$-0.368360\pi$
$881$	23.8564	0.803743	0.401871	+	0.915696i	$0.368360\pi$
$882$	0	0
$883$	−43.0333	−1.44819	−0.724093	−	0.689702i	$-0.757742\pi$
$883$	−43.0333	−1.44819	−0.724093	+	0.689702i	$0.757742\pi$
$884$	5.85641	0.196972
$885$	0	0
$886$	12.9667	0.435624
$887$	−5.39230	−0.181056	−0.0905279	−	0.995894i	$-0.528855\pi$
$887$	−5.39230	−0.181056	−0.0905279	+	0.995894i	$0.528855\pi$
$888$	0	0
$889$	82.1051	2.75372
$890$	0.875644	0.0293517
$891$	0	0
$892$	3.11027	0.104140
$893$	−31.6077	−1.05771
$894$	0	0
$895$	−8.53590	−0.285324
$896$	37.8564	1.26469
$897$	0	0
$898$	−17.6603	−0.589330
$899$	−3.07180	−0.102450
$900$	0	0
$901$	−6.14359	−0.204673
$902$	−1.85641	−0.0618116
$903$	0	0
$904$	−52.8897	−1.75909
$905$	4.07180	0.135351
$906$	0	0
$907$	11.8038	0.391940	0.195970	−	0.980610i	$-0.437214\pi$
$907$	11.8038	0.391940	0.195970	+	0.980610i	$0.437214\pi$
$908$	−20.0000	−0.663723
$909$	0	0
$910$	2.73205	0.0905666
$911$	−1.01924	−0.0337689	−0.0168844	−	0.999857i	$-0.505375\pi$
$911$	−1.01924	−0.0337689	−0.0168844	+	0.999857i	$0.505375\pi$
$912$	0	0
$913$	−13.0718	−0.432613
$914$	−22.8372	−0.755386
$915$	0	0
$916$	14.6410	0.483753
$917$	13.6603	0.451101
$918$	0	0
$919$	−15.5359	−0.512482	−0.256241	−	0.966613i	$-0.582484\pi$
$919$	−15.5359	−0.512482	−0.256241	+	0.966613i	$0.582484\pi$
$920$	−7.60770	−0.250818
$921$	0	0
$922$	0.301270	0.00992181
$923$	−2.66025	−0.0875633
$924$	0	0
$925$	−1.19615	−0.0393292
$926$	21.5551	0.708346
$927$	0	0
$928$	24.5744	0.806693
$929$	−52.0000	−1.70606	−0.853032	−	0.521858i	$-0.825239\pi$
$929$	−52.0000	−1.70606	−0.853032	+	0.521858i	$0.825239\pi$
$930$	0	0
$931$	41.5692	1.36238
$932$	−7.32051	−0.239791
$933$	0	0
$934$	21.2295	0.694650
$935$	5.85641	0.191525
$936$	0	0
$937$	−41.3205	−1.34988	−0.674941	−	0.737872i	$-0.735831\pi$
$937$	−41.3205	−1.34988	−0.674941	+	0.737872i	$0.735831\pi$
$938$	−19.6603	−0.641930
$939$	0	0
$940$	7.71281	0.251564
$941$	−17.8756	−0.582729	−0.291365	−	0.956612i	$-0.594109\pi$
$941$	−17.8756	−0.582729	−0.291365	+	0.956612i	$0.594109\pi$
$942$	0	0
$943$	−5.19615	−0.169210
$944$	−8.28719	−0.269725
$945$	0	0
$946$	−6.64102	−0.215918
$947$	−34.0526	−1.10656	−0.553280	−	0.832996i	$-0.686624\pi$
$947$	−34.0526	−1.10656	−0.553280	+	0.832996i	$0.686624\pi$
$948$	0	0
$949$	11.1962	0.363442
$950$	−4.39230	−0.142505
$951$	0	0
$952$	37.8564	1.22693
$953$	36.4641	1.18119	0.590594	−	0.806969i	$-0.298893\pi$
$953$	36.4641	1.18119	0.590594	+	0.806969i	$0.298893\pi$
$954$	0	0
$955$	−13.2679	−0.429341
$956$	−26.5359	−0.858232
$957$	0	0
$958$	13.7654	0.444739
$959$	72.1051	2.32840
$960$	0	0
$961$	−30.4641	−0.982713
$962$	−0.875644	−0.0282319
$963$	0	0
$964$	26.4308	0.851278
$965$	−4.66025	−0.150019
$966$	0	0
$967$	4.00000	0.128631	0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000	0.128631	0.0643157	+	0.997930i	$0.479514\pi$
$968$	−22.4589	−0.721858
$969$	0	0
$970$	−10.5359	−0.338287
$971$	20.9808	0.673305	0.336652	−	0.941629i	$-0.390705\pi$
$971$	20.9808	0.673305	0.336652	+	0.941629i	$0.390705\pi$
$972$	0	0
$973$	74.1051	2.37570
$974$	14.9282	0.478330
$975$	0	0
$976$	3.13844	0.100459
$977$	34.7321	1.11118	0.555588	−	0.831457i	$-0.312493\pi$
$977$	34.7321	1.11118	0.555588	+	0.831457i	$0.312493\pi$
$978$	0	0
$979$	−1.75129	−0.0559714
$980$	−10.1436	−0.324025
$981$	0	0
$982$	−10.6410	−0.339568
$983$	−9.51666	−0.303534	−0.151767	−	0.988416i	$-0.548496\pi$
$983$	−9.51666	−0.303534	−0.151767	+	0.988416i	$0.548496\pi$
$984$	0	0
$985$	7.26795	0.231576
$986$	12.2872	0.391304
$987$	0	0
$988$	8.78461	0.279476
$989$	−18.5885	−0.591079
$990$	0	0
$991$	48.7128	1.54741	0.773707	−	0.633544i	$-0.218400\pi$
$991$	48.7128	1.54741	0.773707	+	0.633544i	$0.218400\pi$
$992$	−4.28719	−0.136118
$993$	0	0
$994$	−7.26795	−0.230525
$995$	−24.3205	−0.771012
$996$	0	0
$997$	32.9282	1.04285	0.521423	−	0.853298i	$-0.325401\pi$
$997$	32.9282	1.04285	0.521423	+	0.853298i	$0.325401\pi$
$998$	−31.5692	−0.999306
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1755.2.a.l.1.1	yes	2
3.2	odd	2		1755.2.a.h.1.2	✓	2
5.4	even	2		8775.2.a.t.1.2		2
15.14	odd	2		8775.2.a.bf.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1755.2.a.h.1.2	✓	2	3.2	odd	2
1755.2.a.l.1.1	yes	2	1.1	even	1	trivial
8775.2.a.t.1.2		2	5.4	even	2
8775.2.a.bf.1.1		2	15.14	odd	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:	\( N \)	\(=\)	\( 1755 = 3^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1755.a (trivial)

\(f(q)\)	\(=\)	\(q-0.732051 q^{2} -1.46410 q^{4} +1.00000 q^{5} +3.73205 q^{7} +2.53590 q^{8} +O(q^{10})\)	\(q-0.732051 q^{2} -1.46410 q^{4} +1.00000 q^{5} +3.73205 q^{7} +2.53590 q^{8} -0.732051 q^{10} +1.46410 q^{11} -1.00000 q^{13} -2.73205 q^{14} +1.07180 q^{16} +4.00000 q^{17} +6.00000 q^{19} -1.46410 q^{20} -1.07180 q^{22} -3.00000 q^{23} +1.00000 q^{25} +0.732051 q^{26} -5.46410 q^{28} -4.19615 q^{29} +0.732051 q^{31} -5.85641 q^{32} -2.92820 q^{34} +3.73205 q^{35} -1.19615 q^{37} -4.39230 q^{38} +2.53590 q^{40} +1.73205 q^{41} +6.19615 q^{43} -2.14359 q^{44} +2.19615 q^{46} -5.26795 q^{47} +6.92820 q^{49} -0.732051 q^{50} +1.46410 q^{52} -1.53590 q^{53} +1.46410 q^{55} +9.46410 q^{56} +3.07180 q^{58} -7.73205 q^{59} +2.92820 q^{61} -0.535898 q^{62} +2.14359 q^{64} -1.00000 q^{65} +7.19615 q^{67} -5.85641 q^{68} -2.73205 q^{70} +2.66025 q^{71} -11.1962 q^{73} +0.875644 q^{74} -8.78461 q^{76} +5.46410 q^{77} +9.00000 q^{79} +1.07180 q^{80} -1.26795 q^{82} -8.92820 q^{83} +4.00000 q^{85} -4.53590 q^{86} +3.71281 q^{88} -1.19615 q^{89} -3.73205 q^{91} +4.39230 q^{92} +3.85641 q^{94} +6.00000 q^{95} +14.3923 q^{97} -5.07180 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 4 q^{4} + 2 q^{5} + 4 q^{7} + 12 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 4 * q^4 + 2 * q^5 + 4 * q^7 + 12 * q^8	\( 2 q + 2 q^{2} + 4 q^{4} + 2 q^{5} + 4 q^{7} + 12 q^{8} + 2 q^{10} - 4 q^{11} - 2 q^{13} - 2 q^{14} + 16 q^{16} + 8 q^{17} + 12 q^{19} + 4 q^{20} - 16 q^{22} - 6 q^{23} + 2 q^{25} - 2 q^{26} - 4 q^{28} + 2 q^{29} - 2 q^{31} + 16 q^{32} + 8 q^{34} + 4 q^{35} + 8 q^{37} + 12 q^{38} + 12 q^{40} + 2 q^{43} - 32 q^{44} - 6 q^{46} - 14 q^{47} + 2 q^{50} - 4 q^{52} - 10 q^{53} - 4 q^{55} + 12 q^{56} + 20 q^{58} - 12 q^{59} - 8 q^{61} - 8 q^{62} + 32 q^{64} - 2 q^{65} + 4 q^{67} + 16 q^{68} - 2 q^{70} - 12 q^{71} - 12 q^{73} + 26 q^{74} + 24 q^{76} + 4 q^{77} + 18 q^{79} + 16 q^{80} - 6 q^{82} - 4 q^{83} + 8 q^{85} - 16 q^{86} - 48 q^{88} + 8 q^{89} - 4 q^{91} - 12 q^{92} - 20 q^{94} + 12 q^{95} + 8 q^{97} - 24 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 4 * q^4 + 2 * q^5 + 4 * q^7 + 12 * q^8 + 2 * q^10 - 4 * q^11 - 2 * q^13 - 2 * q^14 + 16 * q^16 + 8 * q^17 + 12 * q^19 + 4 * q^20 - 16 * q^22 - 6 * q^23 + 2 * q^25 - 2 * q^26 - 4 * q^28 + 2 * q^29 - 2 * q^31 + 16 * q^32 + 8 * q^34 + 4 * q^35 + 8 * q^37 + 12 * q^38 + 12 * q^40 + 2 * q^43 - 32 * q^44 - 6 * q^46 - 14 * q^47 + 2 * q^50 - 4 * q^52 - 10 * q^53 - 4 * q^55 + 12 * q^56 + 20 * q^58 - 12 * q^59 - 8 * q^61 - 8 * q^62 + 32 * q^64 - 2 * q^65 + 4 * q^67 + 16 * q^68 - 2 * q^70 - 12 * q^71 - 12 * q^73 + 26 * q^74 + 24 * q^76 + 4 * q^77 + 18 * q^79 + 16 * q^80 - 6 * q^82 - 4 * q^83 + 8 * q^85 - 16 * q^86 - 48 * q^88 + 8 * q^89 - 4 * q^91 - 12 * q^92 - 20 * q^94 + 12 * q^95 + 8 * q^97 - 24 * q^98

Introduction

L-functions

Modular forms

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