LMFDB - Embedded newform 1287.2.a.f.1.1

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:	$1$
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:	no (minimal twist has level 429)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ 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)

$f(q)$	$=$	$q-1.73205 q^{2} +1.00000 q^{4} +2.73205 q^{5} -2.00000 q^{7} +1.73205 q^{8} +O(q^{10})$	\(q-1.73205 q^{2} +1.00000 q^{4} +2.73205 q^{5} -2.00000 q^{7} +1.73205 q^{8} -4.73205 q^{10} +1.00000 q^{11} -1.00000 q^{13} +3.46410 q^{14} -5.00000 q^{16} -3.26795 q^{17} -7.46410 q^{19} +2.73205 q^{20} -1.73205 q^{22} +2.00000 q^{23} +2.46410 q^{25} +1.73205 q^{26} -2.00000 q^{28} -6.19615 q^{29} +2.19615 q^{31} +5.19615 q^{32} +5.66025 q^{34} -5.46410 q^{35} -2.00000 q^{37} +12.9282 q^{38} +4.73205 q^{40} -1.46410 q^{41} +4.19615 q^{43} +1.00000 q^{44} -3.46410 q^{46} -5.46410 q^{47} -3.00000 q^{49} -4.26795 q^{50} -1.00000 q^{52} -2.00000 q^{53} +2.73205 q^{55} -3.46410 q^{56} +10.7321 q^{58} +6.53590 q^{59} -8.92820 q^{61} -3.80385 q^{62} +1.00000 q^{64} -2.73205 q^{65} -1.80385 q^{67} -3.26795 q^{68} +9.46410 q^{70} +6.92820 q^{71} -10.9282 q^{73} +3.46410 q^{74} -7.46410 q^{76} -2.00000 q^{77} +13.6603 q^{79} -13.6603 q^{80} +2.53590 q^{82} +13.8564 q^{83} -8.92820 q^{85} -7.26795 q^{86} +1.73205 q^{88} -1.66025 q^{89} +2.00000 q^{91} +2.00000 q^{92} +9.46410 q^{94} -20.3923 q^{95} -4.92820 q^{97} +5.19615 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 2 q^{5} - 4 q^{7}+O(q^{10}) $ 2 * q + 2 * q^4 + 2 * q^5 - 4 * q^7	$ 2 q + 2 q^{4} + 2 q^{5} - 4 q^{7} - 6 q^{10} + 2 q^{11} - 2 q^{13} - 10 q^{16} - 10 q^{17} - 8 q^{19} + 2 q^{20} + 4 q^{23} - 2 q^{25} - 4 q^{28} - 2 q^{29} - 6 q^{31} - 6 q^{34} - 4 q^{35} - 4 q^{37} + 12 q^{38} + 6 q^{40} + 4 q^{41} - 2 q^{43} + 2 q^{44} - 4 q^{47} - 6 q^{49} - 12 q^{50} - 2 q^{52} - 4 q^{53} + 2 q^{55} + 18 q^{58} + 20 q^{59} - 4 q^{61} - 18 q^{62} + 2 q^{64} - 2 q^{65} - 14 q^{67} - 10 q^{68} + 12 q^{70} - 8 q^{73} - 8 q^{76} - 4 q^{77} + 10 q^{79} - 10 q^{80} + 12 q^{82} - 4 q^{85} - 18 q^{86} + 14 q^{89} + 4 q^{91} + 4 q^{92} + 12 q^{94} - 20 q^{95} + 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^4 + 2 * q^5 - 4 * q^7 - 6 * q^10 + 2 * q^11 - 2 * q^13 - 10 * q^16 - 10 * q^17 - 8 * q^19 + 2 * q^20 + 4 * q^23 - 2 * q^25 - 4 * q^28 - 2 * q^29 - 6 * q^31 - 6 * q^34 - 4 * q^35 - 4 * q^37 + 12 * q^38 + 6 * q^40 + 4 * q^41 - 2 * q^43 + 2 * q^44 - 4 * q^47 - 6 * q^49 - 12 * q^50 - 2 * q^52 - 4 * q^53 + 2 * q^55 + 18 * q^58 + 20 * q^59 - 4 * q^61 - 18 * q^62 + 2 * q^64 - 2 * q^65 - 14 * q^67 - 10 * q^68 + 12 * q^70 - 8 * q^73 - 8 * q^76 - 4 * q^77 + 10 * q^79 - 10 * q^80 + 12 * q^82 - 4 * q^85 - 18 * q^86 + 14 * q^89 + 4 * q^91 + 4 * q^92 + 12 * q^94 - 20 * q^95 + 4 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.73205	−1.22474	−0.612372	−	0.790569i	$-0.709785\pi$
$2$	−1.73205	−1.22474	−0.612372	+	0.790569i	$0.709785\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	2.73205	1.22181	0.610905	−	0.791704i	$-0.290806\pi$
$5$	2.73205	1.22181	0.610905	+	0.791704i	$0.290806\pi$
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	1.73205	0.612372
$9$	0	0
$10$	−4.73205	−1.49641
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	3.46410	0.925820
$15$	0	0
$16$	−5.00000	−1.25000
$17$	−3.26795	−0.792594	−0.396297	−	0.918122i	$-0.629705\pi$
$17$	−3.26795	−0.792594	−0.396297	+	0.918122i	$0.629705\pi$
$18$	0	0
$19$	−7.46410	−1.71238	−0.856191	−	0.516659i	$-0.827175\pi$
$19$	−7.46410	−1.71238	−0.856191	+	0.516659i	$0.827175\pi$
$20$	2.73205	0.610905
$21$	0	0
$22$	−1.73205	−0.369274
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	2.46410	0.492820
$26$	1.73205	0.339683
$27$	0	0
$28$	−2.00000	−0.377964
$29$	−6.19615	−1.15060	−0.575298	−	0.817944i	$-0.695114\pi$
$29$	−6.19615	−1.15060	−0.575298	+	0.817944i	$0.695114\pi$
$30$	0	0
$31$	2.19615	0.394441	0.197220	−	0.980359i	$-0.436809\pi$
$31$	2.19615	0.394441	0.197220	+	0.980359i	$0.436809\pi$
$32$	5.19615	0.918559
$33$	0	0
$34$	5.66025	0.970726
$35$	−5.46410	−0.923602
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	12.9282	2.09723
$39$	0	0
$40$	4.73205	0.748203
$41$	−1.46410	−0.228654	−0.114327	−	0.993443i	$-0.536471\pi$
$41$	−1.46410	−0.228654	−0.114327	+	0.993443i	$0.536471\pi$
$42$	0	0
$43$	4.19615	0.639907	0.319954	−	0.947433i	$-0.396333\pi$
$43$	4.19615	0.639907	0.319954	+	0.947433i	$0.396333\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−3.46410	−0.510754
$47$	−5.46410	−0.797021	−0.398511	−	0.917164i	$-0.630473\pi$
$47$	−5.46410	−0.797021	−0.398511	+	0.917164i	$0.630473\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	−4.26795	−0.603579
$51$	0	0
$52$	−1.00000	−0.138675
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	2.73205	0.368390
$56$	−3.46410	−0.462910
$57$	0	0
$58$	10.7321	1.40919
$59$	6.53590	0.850901	0.425451	−	0.904982i	$-0.360116\pi$
$59$	6.53590	0.850901	0.425451	+	0.904982i	$0.360116\pi$
$60$	0	0
$61$	−8.92820	−1.14314	−0.571570	−	0.820554i	$-0.693665\pi$
$61$	−8.92820	−1.14314	−0.571570	+	0.820554i	$0.693665\pi$
$62$	−3.80385	−0.483089
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.73205	−0.338869
$66$	0	0
$67$	−1.80385	−0.220375	−0.110188	−	0.993911i	$-0.535145\pi$
$67$	−1.80385	−0.220375	−0.110188	+	0.993911i	$0.535145\pi$
$68$	−3.26795	−0.396297
$69$	0	0
$70$	9.46410	1.13118
$71$	6.92820	0.822226	0.411113	−	0.911584i	$-0.365140\pi$
$71$	6.92820	0.822226	0.411113	+	0.911584i	$0.365140\pi$
$72$	0	0
$73$	−10.9282	−1.27905	−0.639525	−	0.768771i	$-0.720869\pi$
$73$	−10.9282	−1.27905	−0.639525	+	0.768771i	$0.720869\pi$
$74$	3.46410	0.402694
$75$	0	0
$76$	−7.46410	−0.856191
$77$	−2.00000	−0.227921
$78$	0	0
$79$	13.6603	1.53690	0.768449	−	0.639911i	$-0.221029\pi$
$79$	13.6603	1.53690	0.768449	+	0.639911i	$0.221029\pi$
$80$	−13.6603	−1.52726
$81$	0	0
$82$	2.53590	0.280043
$83$	13.8564	1.52094	0.760469	−	0.649374i	$-0.224969\pi$
$83$	13.8564	1.52094	0.760469	+	0.649374i	$0.224969\pi$
$84$	0	0
$85$	−8.92820	−0.968400
$86$	−7.26795	−0.783723
$87$	0	0
$88$	1.73205	0.184637
$89$	−1.66025	−0.175987	−0.0879933	−	0.996121i	$-0.528045\pi$
$89$	−1.66025	−0.175987	−0.0879933	+	0.996121i	$0.528045\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	2.00000	0.208514
$93$	0	0
$94$	9.46410	0.976148
$95$	−20.3923	−2.09221
$96$	0	0
$97$	−4.92820	−0.500383	−0.250192	−	0.968196i	$-0.580494\pi$
$97$	−4.92820	−0.500383	−0.250192	+	0.968196i	$0.580494\pi$
$98$	5.19615	0.524891
$99$	0	0
$100$	2.46410	0.246410
$101$	2.19615	0.218525	0.109263	−	0.994013i	$-0.465151\pi$
$101$	2.19615	0.218525	0.109263	+	0.994013i	$0.465151\pi$
$102$	0	0
$103$	−15.3205	−1.50957	−0.754787	−	0.655970i	$-0.772260\pi$
$103$	−15.3205	−1.50957	−0.754787	+	0.655970i	$0.772260\pi$
$104$	−1.73205	−0.169842
$105$	0	0
$106$	3.46410	0.336463
$107$	−12.3923	−1.19801	−0.599005	−	0.800746i	$-0.704437\pi$
$107$	−12.3923	−1.19801	−0.599005	+	0.800746i	$0.704437\pi$
$108$	0	0
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	−4.73205	−0.451183
$111$	0	0
$112$	10.0000	0.944911
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	5.46410	0.509530
$116$	−6.19615	−0.575298
$117$	0	0
$118$	−11.3205	−1.04214
$119$	6.53590	0.599145
$120$	0	0
$121$	1.00000	0.0909091
$122$	15.4641	1.40005
$123$	0	0
$124$	2.19615	0.197220
$125$	−6.92820	−0.619677
$126$	0	0
$127$	−11.1244	−0.987127	−0.493563	−	0.869710i	$-0.664306\pi$
$127$	−11.1244	−0.987127	−0.493563	+	0.869710i	$0.664306\pi$
$128$	−12.1244	−1.07165
$129$	0	0
$130$	4.73205	0.415028
$131$	18.9282	1.65376	0.826882	−	0.562375i	$-0.190112\pi$
$131$	18.9282	1.65376	0.826882	+	0.562375i	$0.190112\pi$
$132$	0	0
$133$	14.9282	1.29444
$134$	3.12436	0.269903
$135$	0	0
$136$	−5.66025	−0.485363
$137$	−16.5885	−1.41725	−0.708624	−	0.705587i	$-0.750684\pi$
$137$	−16.5885	−1.41725	−0.708624	+	0.705587i	$0.750684\pi$
$138$	0	0
$139$	−20.1962	−1.71302	−0.856508	−	0.516134i	$-0.827371\pi$
$139$	−20.1962	−1.71302	−0.856508	+	0.516134i	$0.827371\pi$
$140$	−5.46410	−0.461801
$141$	0	0
$142$	−12.0000	−1.00702
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−16.9282	−1.40581
$146$	18.9282	1.56651
$147$	0	0
$148$	−2.00000	−0.164399
$149$	−8.00000	−0.655386	−0.327693	−	0.944784i	$-0.606271\pi$
$149$	−8.00000	−0.655386	−0.327693	+	0.944784i	$0.606271\pi$
$150$	0	0
$151$	−11.4641	−0.932935	−0.466468	−	0.884538i	$-0.654474\pi$
$151$	−11.4641	−0.932935	−0.466468	+	0.884538i	$0.654474\pi$
$152$	−12.9282	−1.04862
$153$	0	0
$154$	3.46410	0.279145
$155$	6.00000	0.481932
$156$	0	0
$157$	19.3205	1.54194	0.770972	−	0.636869i	$-0.219771\pi$
$157$	19.3205	1.54194	0.770972	+	0.636869i	$0.219771\pi$
$158$	−23.6603	−1.88231
$159$	0	0
$160$	14.1962	1.12230
$161$	−4.00000	−0.315244
$162$	0	0
$163$	−20.7321	−1.62386	−0.811930	−	0.583755i	$-0.801583\pi$
$163$	−20.7321	−1.62386	−0.811930	+	0.583755i	$0.801583\pi$
$164$	−1.46410	−0.114327
$165$	0	0
$166$	−24.0000	−1.86276
$167$	−17.8564	−1.38177	−0.690885	−	0.722965i	$-0.742779\pi$
$167$	−17.8564	−1.38177	−0.690885	+	0.722965i	$0.742779\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	15.4641	1.18604
$171$	0	0
$172$	4.19615	0.319954
$173$	5.12436	0.389598	0.194799	−	0.980843i	$-0.437595\pi$
$173$	5.12436	0.389598	0.194799	+	0.980843i	$0.437595\pi$
$174$	0	0
$175$	−4.92820	−0.372537
$176$	−5.00000	−0.376889
$177$	0	0
$178$	2.87564	0.215539
$179$	19.8564	1.48414	0.742069	−	0.670324i	$-0.233845\pi$
$179$	19.8564	1.48414	0.742069	+	0.670324i	$0.233845\pi$
$180$	0	0
$181$	−12.3923	−0.921113	−0.460556	−	0.887630i	$-0.652350\pi$
$181$	−12.3923	−0.921113	−0.460556	+	0.887630i	$0.652350\pi$
$182$	−3.46410	−0.256776
$183$	0	0
$184$	3.46410	0.255377
$185$	−5.46410	−0.401729
$186$	0	0
$187$	−3.26795	−0.238976
$188$	−5.46410	−0.398511
$189$	0	0
$190$	35.3205	2.56242
$191$	5.07180	0.366982	0.183491	−	0.983021i	$-0.441260\pi$
$191$	5.07180	0.366982	0.183491	+	0.983021i	$0.441260\pi$
$192$	0	0
$193$	6.92820	0.498703	0.249351	−	0.968413i	$-0.419783\pi$
$193$	6.92820	0.498703	0.249351	+	0.968413i	$0.419783\pi$
$194$	8.53590	0.612842
$195$	0	0
$196$	−3.00000	−0.214286
$197$	−8.00000	−0.569976	−0.284988	−	0.958531i	$-0.591990\pi$
$197$	−8.00000	−0.569976	−0.284988	+	0.958531i	$0.591990\pi$
$198$	0	0
$199$	17.4641	1.23800	0.618999	−	0.785392i	$-0.287539\pi$
$199$	17.4641	1.23800	0.618999	+	0.785392i	$0.287539\pi$
$200$	4.26795	0.301790
$201$	0	0
$202$	−3.80385	−0.267638
$203$	12.3923	0.869769
$204$	0	0
$205$	−4.00000	−0.279372
$206$	26.5359	1.84884
$207$	0	0
$208$	5.00000	0.346688
$209$	−7.46410	−0.516303
$210$	0	0
$211$	22.7321	1.56494	0.782469	−	0.622689i	$-0.213960\pi$
$211$	22.7321	1.56494	0.782469	+	0.622689i	$0.213960\pi$
$212$	−2.00000	−0.137361
$213$	0	0
$214$	21.4641	1.46726
$215$	11.4641	0.781845
$216$	0	0
$217$	−4.39230	−0.298169
$218$	13.8564	0.938474
$219$	0	0
$220$	2.73205	0.184195
$221$	3.26795	0.219826
$222$	0	0
$223$	21.5167	1.44086	0.720431	−	0.693527i	$-0.243944\pi$
$223$	21.5167	1.44086	0.720431	+	0.693527i	$0.243944\pi$
$224$	−10.3923	−0.694365
$225$	0	0
$226$	17.3205	1.15214
$227$	−10.3923	−0.689761	−0.344881	−	0.938647i	$-0.612081\pi$
$227$	−10.3923	−0.689761	−0.344881	+	0.938647i	$0.612081\pi$
$228$	0	0
$229$	−8.53590	−0.564068	−0.282034	−	0.959404i	$-0.591009\pi$
$229$	−8.53590	−0.564068	−0.282034	+	0.959404i	$0.591009\pi$
$230$	−9.46410	−0.624044
$231$	0	0
$232$	−10.7321	−0.704594
$233$	6.19615	0.405923	0.202962	−	0.979187i	$-0.434943\pi$
$233$	6.19615	0.405923	0.202962	+	0.979187i	$0.434943\pi$
$234$	0	0
$235$	−14.9282	−0.973809
$236$	6.53590	0.425451
$237$	0	0
$238$	−11.3205	−0.733800
$239$	−1.60770	−0.103993	−0.0519966	−	0.998647i	$-0.516558\pi$
$239$	−1.60770	−0.103993	−0.0519966	+	0.998647i	$0.516558\pi$
$240$	0	0
$241$	22.7846	1.46769	0.733843	−	0.679319i	$-0.237725\pi$
$241$	22.7846	1.46769	0.733843	+	0.679319i	$0.237725\pi$
$242$	−1.73205	−0.111340
$243$	0	0
$244$	−8.92820	−0.571570
$245$	−8.19615	−0.523633
$246$	0	0
$247$	7.46410	0.474929
$248$	3.80385	0.241545
$249$	0	0
$250$	12.0000	0.758947
$251$	30.9282	1.95217	0.976085	−	0.217387i	$-0.0697534\pi$
$251$	30.9282	1.95217	0.976085	+	0.217387i	$0.0697534\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	19.2679	1.20898
$255$	0	0
$256$	19.0000	1.18750
$257$	17.3205	1.08042	0.540212	−	0.841529i	$-0.318344\pi$
$257$	17.3205	1.08042	0.540212	+	0.841529i	$0.318344\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	−2.73205	−0.169435
$261$	0	0
$262$	−32.7846	−2.02544
$263$	−2.53590	−0.156370	−0.0781851	−	0.996939i	$-0.524913\pi$
$263$	−2.53590	−0.156370	−0.0781851	+	0.996939i	$0.524913\pi$
$264$	0	0
$265$	−5.46410	−0.335657
$266$	−25.8564	−1.58536
$267$	0	0
$268$	−1.80385	−0.110188
$269$	−17.3205	−1.05605	−0.528025	−	0.849229i	$-0.677067\pi$
$269$	−17.3205	−1.05605	−0.528025	+	0.849229i	$0.677067\pi$
$270$	0	0
$271$	−25.3205	−1.53811	−0.769056	−	0.639182i	$-0.779273\pi$
$271$	−25.3205	−1.53811	−0.769056	+	0.639182i	$0.779273\pi$
$272$	16.3397	0.990743
$273$	0	0
$274$	28.7321	1.73577
$275$	2.46410	0.148591
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	34.9808	2.09801
$279$	0	0
$280$	−9.46410	−0.565588
$281$	11.3205	0.675325	0.337662	−	0.941267i	$-0.390364\pi$
$281$	11.3205	0.675325	0.337662	+	0.941267i	$0.390364\pi$
$282$	0	0
$283$	20.1962	1.20054	0.600268	−	0.799799i	$-0.295060\pi$
$283$	20.1962	1.20054	0.600268	+	0.799799i	$0.295060\pi$
$284$	6.92820	0.411113
$285$	0	0
$286$	1.73205	0.102418
$287$	2.92820	0.172846
$288$	0	0
$289$	−6.32051	−0.371795
$290$	29.3205	1.72176
$291$	0	0
$292$	−10.9282	−0.639525
$293$	−26.9282	−1.57316	−0.786581	−	0.617487i	$-0.788151\pi$
$293$	−26.9282	−1.57316	−0.786581	+	0.617487i	$0.788151\pi$
$294$	0	0
$295$	17.8564	1.03964
$296$	−3.46410	−0.201347
$297$	0	0
$298$	13.8564	0.802680
$299$	−2.00000	−0.115663
$300$	0	0
$301$	−8.39230	−0.483724
$302$	19.8564	1.14261
$303$	0	0
$304$	37.3205	2.14048
$305$	−24.3923	−1.39670
$306$	0	0
$307$	−27.4641	−1.56746	−0.783730	−	0.621102i	$-0.786685\pi$
$307$	−27.4641	−1.56746	−0.783730	+	0.621102i	$0.786685\pi$
$308$	−2.00000	−0.113961
$309$	0	0
$310$	−10.3923	−0.590243
$311$	16.9282	0.959910	0.479955	−	0.877293i	$-0.340653\pi$
$311$	16.9282	0.959910	0.479955	+	0.877293i	$0.340653\pi$
$312$	0	0
$313$	−2.53590	−0.143337	−0.0716687	−	0.997428i	$-0.522832\pi$
$313$	−2.53590	−0.143337	−0.0716687	+	0.997428i	$0.522832\pi$
$314$	−33.4641	−1.88849
$315$	0	0
$316$	13.6603	0.768449
$317$	7.12436	0.400144	0.200072	−	0.979781i	$-0.435882\pi$
$317$	7.12436	0.400144	0.200072	+	0.979781i	$0.435882\pi$
$318$	0	0
$319$	−6.19615	−0.346918
$320$	2.73205	0.152726
$321$	0	0
$322$	6.92820	0.386094
$323$	24.3923	1.35722
$324$	0	0
$325$	−2.46410	−0.136684
$326$	35.9090	1.98881
$327$	0	0
$328$	−2.53590	−0.140022
$329$	10.9282	0.602491
$330$	0	0
$331$	−21.5167	−1.18266	−0.591331	−	0.806429i	$-0.701397\pi$
$331$	−21.5167	−1.18266	−0.591331	+	0.806429i	$0.701397\pi$
$332$	13.8564	0.760469
$333$	0	0
$334$	30.9282	1.69232
$335$	−4.92820	−0.269257
$336$	0	0
$337$	−25.3205	−1.37930	−0.689648	−	0.724145i	$-0.742235\pi$
$337$	−25.3205	−1.37930	−0.689648	+	0.724145i	$0.742235\pi$
$338$	−1.73205	−0.0942111
$339$	0	0
$340$	−8.92820	−0.484200
$341$	2.19615	0.118928
$342$	0	0
$343$	20.0000	1.07990
$344$	7.26795	0.391862
$345$	0	0
$346$	−8.87564	−0.477158
$347$	−16.3923	−0.879985	−0.439993	−	0.898001i	$-0.645019\pi$
$347$	−16.3923	−0.879985	−0.439993	+	0.898001i	$0.645019\pi$
$348$	0	0
$349$	23.8564	1.27700	0.638502	−	0.769620i	$-0.279554\pi$
$349$	23.8564	1.27700	0.638502	+	0.769620i	$0.279554\pi$
$350$	8.53590	0.456263
$351$	0	0
$352$	5.19615	0.276956
$353$	28.9808	1.54249	0.771245	−	0.636538i	$-0.219634\pi$
$353$	28.9808	1.54249	0.771245	+	0.636538i	$0.219634\pi$
$354$	0	0
$355$	18.9282	1.00460
$356$	−1.66025	−0.0879933
$357$	0	0
$358$	−34.3923	−1.81769
$359$	23.4641	1.23839	0.619194	−	0.785238i	$-0.287459\pi$
$359$	23.4641	1.23839	0.619194	+	0.785238i	$0.287459\pi$
$360$	0	0
$361$	36.7128	1.93225
$362$	21.4641	1.12813
$363$	0	0
$364$	2.00000	0.104828
$365$	−29.8564	−1.56276
$366$	0	0
$367$	17.0718	0.891141	0.445570	−	0.895247i	$-0.353001\pi$
$367$	17.0718	0.891141	0.445570	+	0.895247i	$0.353001\pi$
$368$	−10.0000	−0.521286
$369$	0	0
$370$	9.46410	0.492015
$371$	4.00000	0.207670
$372$	0	0
$373$	32.2487	1.66977	0.834887	−	0.550421i	$-0.185533\pi$
$373$	32.2487	1.66977	0.834887	+	0.550421i	$0.185533\pi$
$374$	5.66025	0.292685
$375$	0	0
$376$	−9.46410	−0.488074
$377$	6.19615	0.319118
$378$	0	0
$379$	1.80385	0.0926574	0.0463287	−	0.998926i	$-0.485248\pi$
$379$	1.80385	0.0926574	0.0463287	+	0.998926i	$0.485248\pi$
$380$	−20.3923	−1.04610
$381$	0	0
$382$	−8.78461	−0.449460
$383$	−3.60770	−0.184345	−0.0921723	−	0.995743i	$-0.529381\pi$
$383$	−3.60770	−0.184345	−0.0921723	+	0.995743i	$0.529381\pi$
$384$	0	0
$385$	−5.46410	−0.278476
$386$	−12.0000	−0.610784
$387$	0	0
$388$	−4.92820	−0.250192
$389$	−15.4641	−0.784061	−0.392031	−	0.919952i	$-0.628227\pi$
$389$	−15.4641	−0.784061	−0.392031	+	0.919952i	$0.628227\pi$
$390$	0	0
$391$	−6.53590	−0.330535
$392$	−5.19615	−0.262445
$393$	0	0
$394$	13.8564	0.698076
$395$	37.3205	1.87780
$396$	0	0
$397$	11.0718	0.555678	0.277839	−	0.960628i	$-0.410382\pi$
$397$	11.0718	0.555678	0.277839	+	0.960628i	$0.410382\pi$
$398$	−30.2487	−1.51623
$399$	0	0
$400$	−12.3205	−0.616025
$401$	−15.8038	−0.789206	−0.394603	−	0.918852i	$-0.629118\pi$
$401$	−15.8038	−0.789206	−0.394603	+	0.918852i	$0.629118\pi$
$402$	0	0
$403$	−2.19615	−0.109398
$404$	2.19615	0.109263
$405$	0	0
$406$	−21.4641	−1.06525
$407$	−2.00000	−0.0991363
$408$	0	0
$409$	27.7128	1.37031	0.685155	−	0.728397i	$-0.259734\pi$
$409$	27.7128	1.37031	0.685155	+	0.728397i	$0.259734\pi$
$410$	6.92820	0.342160
$411$	0	0
$412$	−15.3205	−0.754787
$413$	−13.0718	−0.643221
$414$	0	0
$415$	37.8564	1.85830
$416$	−5.19615	−0.254762
$417$	0	0
$418$	12.9282	0.632339
$419$	−26.0000	−1.27018	−0.635092	−	0.772437i	$-0.719038\pi$
$419$	−26.0000	−1.27018	−0.635092	+	0.772437i	$0.719038\pi$
$420$	0	0
$421$	−4.92820	−0.240186	−0.120093	−	0.992763i	$-0.538319\pi$
$421$	−4.92820	−0.240186	−0.120093	+	0.992763i	$0.538319\pi$
$422$	−39.3731	−1.91665
$423$	0	0
$424$	−3.46410	−0.168232
$425$	−8.05256	−0.390606
$426$	0	0
$427$	17.8564	0.864132
$428$	−12.3923	−0.599005
$429$	0	0
$430$	−19.8564	−0.957561
$431$	−28.7846	−1.38651	−0.693253	−	0.720694i	$-0.743823\pi$
$431$	−28.7846	−1.38651	−0.693253	+	0.720694i	$0.743823\pi$
$432$	0	0
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	7.60770	0.365181
$435$	0	0
$436$	−8.00000	−0.383131
$437$	−14.9282	−0.714113
$438$	0	0
$439$	35.9090	1.71384	0.856921	−	0.515448i	$-0.172375\pi$
$439$	35.9090	1.71384	0.856921	+	0.515448i	$0.172375\pi$
$440$	4.73205	0.225592
$441$	0	0
$442$	−5.66025	−0.269231
$443$	−36.7846	−1.74769	−0.873845	−	0.486205i	$-0.838381\pi$
$443$	−36.7846	−1.74769	−0.873845	+	0.486205i	$0.838381\pi$
$444$	0	0
$445$	−4.53590	−0.215022
$446$	−37.2679	−1.76469
$447$	0	0
$448$	−2.00000	−0.0944911
$449$	−8.19615	−0.386800	−0.193400	−	0.981120i	$-0.561952\pi$
$449$	−8.19615	−0.386800	−0.193400	+	0.981120i	$0.561952\pi$
$450$	0	0
$451$	−1.46410	−0.0689419
$452$	−10.0000	−0.470360
$453$	0	0
$454$	18.0000	0.844782
$455$	5.46410	0.256161
$456$	0	0
$457$	10.7846	0.504483	0.252241	−	0.967664i	$-0.418832\pi$
$457$	10.7846	0.504483	0.252241	+	0.967664i	$0.418832\pi$
$458$	14.7846	0.690839
$459$	0	0
$460$	5.46410	0.254765
$461$	10.5359	0.490706	0.245353	−	0.969434i	$-0.421096\pi$
$461$	10.5359	0.490706	0.245353	+	0.969434i	$0.421096\pi$
$462$	0	0
$463$	−25.5167	−1.18586	−0.592930	−	0.805254i	$-0.702029\pi$
$463$	−25.5167	−1.18586	−0.592930	+	0.805254i	$0.702029\pi$
$464$	30.9808	1.43825
$465$	0	0
$466$	−10.7321	−0.497153
$467$	19.0718	0.882538	0.441269	−	0.897375i	$-0.354529\pi$
$467$	19.0718	0.882538	0.441269	+	0.897375i	$0.354529\pi$
$468$	0	0
$469$	3.60770	0.166588
$470$	25.8564	1.19267
$471$	0	0
$472$	11.3205	0.521069
$473$	4.19615	0.192939
$474$	0	0
$475$	−18.3923	−0.843897
$476$	6.53590	0.299572
$477$	0	0
$478$	2.78461	0.127365
$479$	−27.4641	−1.25487	−0.627433	−	0.778670i	$-0.715895\pi$
$479$	−27.4641	−1.25487	−0.627433	+	0.778670i	$0.715895\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	−39.4641	−1.79754
$483$	0	0
$484$	1.00000	0.0454545
$485$	−13.4641	−0.611373
$486$	0	0
$487$	−8.33975	−0.377910	−0.188955	−	0.981986i	$-0.560510\pi$
$487$	−8.33975	−0.377910	−0.188955	+	0.981986i	$0.560510\pi$
$488$	−15.4641	−0.700027
$489$	0	0
$490$	14.1962	0.641317
$491$	−2.53590	−0.114443	−0.0572217	−	0.998361i	$-0.518224\pi$
$491$	−2.53590	−0.114443	−0.0572217	+	0.998361i	$0.518224\pi$
$492$	0	0
$493$	20.2487	0.911956
$494$	−12.9282	−0.581667
$495$	0	0
$496$	−10.9808	−0.493051
$497$	−13.8564	−0.621545
$498$	0	0
$499$	−17.8038	−0.797010	−0.398505	−	0.917166i	$-0.630471\pi$
$499$	−17.8038	−0.797010	−0.398505	+	0.917166i	$0.630471\pi$
$500$	−6.92820	−0.309839
$501$	0	0
$502$	−53.5692	−2.39091
$503$	21.4641	0.957037	0.478518	−	0.878077i	$-0.341174\pi$
$503$	21.4641	0.957037	0.478518	+	0.878077i	$0.341174\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	−3.46410	−0.153998
$507$	0	0
$508$	−11.1244	−0.493563
$509$	41.6603	1.84656	0.923279	−	0.384130i	$-0.125498\pi$
$509$	41.6603	1.84656	0.923279	+	0.384130i	$0.125498\pi$
$510$	0	0
$511$	21.8564	0.966870
$512$	−8.66025	−0.382733
$513$	0	0
$514$	−30.0000	−1.32324
$515$	−41.8564	−1.84441
$516$	0	0
$517$	−5.46410	−0.240311
$518$	−6.92820	−0.304408
$519$	0	0
$520$	−4.73205	−0.207514
$521$	40.9282	1.79310	0.896549	−	0.442945i	$-0.146066\pi$
$521$	40.9282	1.79310	0.896549	+	0.442945i	$0.146066\pi$
$522$	0	0
$523$	−21.2679	−0.929982	−0.464991	−	0.885315i	$-0.653943\pi$
$523$	−21.2679	−0.929982	−0.464991	+	0.885315i	$0.653943\pi$
$524$	18.9282	0.826882
$525$	0	0
$526$	4.39230	0.191514
$527$	−7.17691	−0.312631
$528$	0	0
$529$	−19.0000	−0.826087
$530$	9.46410	0.411094
$531$	0	0
$532$	14.9282	0.647220
$533$	1.46410	0.0634173
$534$	0	0
$535$	−33.8564	−1.46374
$536$	−3.12436	−0.134952
$537$	0	0
$538$	30.0000	1.29339
$539$	−3.00000	−0.129219
$540$	0	0
$541$	−14.7846	−0.635640	−0.317820	−	0.948151i	$-0.602951\pi$
$541$	−14.7846	−0.635640	−0.317820	+	0.948151i	$0.602951\pi$
$542$	43.8564	1.88379
$543$	0	0
$544$	−16.9808	−0.728044
$545$	−21.8564	−0.936226
$546$	0	0
$547$	12.9808	0.555017	0.277509	−	0.960723i	$-0.410491\pi$
$547$	12.9808	0.555017	0.277509	+	0.960723i	$0.410491\pi$
$548$	−16.5885	−0.708624
$549$	0	0
$550$	−4.26795	−0.181986
$551$	46.2487	1.97026
$552$	0	0
$553$	−27.3205	−1.16179
$554$	17.3205	0.735878
$555$	0	0
$556$	−20.1962	−0.856508
$557$	24.0000	1.01691	0.508456	−	0.861088i	$-0.330216\pi$
$557$	24.0000	1.01691	0.508456	+	0.861088i	$0.330216\pi$
$558$	0	0
$559$	−4.19615	−0.177478
$560$	27.3205	1.15450
$561$	0	0
$562$	−19.6077	−0.827101
$563$	39.7128	1.67370	0.836848	−	0.547436i	$-0.184396\pi$
$563$	39.7128	1.67370	0.836848	+	0.547436i	$0.184396\pi$
$564$	0	0
$565$	−27.3205	−1.14938
$566$	−34.9808	−1.47035
$567$	0	0
$568$	12.0000	0.503509
$569$	−15.6603	−0.656512	−0.328256	−	0.944589i	$-0.606461\pi$
$569$	−15.6603	−0.656512	−0.328256	+	0.944589i	$0.606461\pi$
$570$	0	0
$571$	16.5885	0.694205	0.347103	−	0.937827i	$-0.387166\pi$
$571$	16.5885	0.694205	0.347103	+	0.937827i	$0.387166\pi$
$572$	−1.00000	−0.0418121
$573$	0	0
$574$	−5.07180	−0.211693
$575$	4.92820	0.205520
$576$	0	0
$577$	−8.92820	−0.371686	−0.185843	−	0.982579i	$-0.559502\pi$
$577$	−8.92820	−0.371686	−0.185843	+	0.982579i	$0.559502\pi$
$578$	10.9474	0.455354
$579$	0	0
$580$	−16.9282	−0.702905
$581$	−27.7128	−1.14972
$582$	0	0
$583$	−2.00000	−0.0828315
$584$	−18.9282	−0.783255
$585$	0	0
$586$	46.6410	1.92672
$587$	−2.53590	−0.104668	−0.0523339	−	0.998630i	$-0.516666\pi$
$587$	−2.53590	−0.104668	−0.0523339	+	0.998630i	$0.516666\pi$
$588$	0	0
$589$	−16.3923	−0.675433
$590$	−30.9282	−1.27329
$591$	0	0
$592$	10.0000	0.410997
$593$	−36.0000	−1.47834	−0.739171	−	0.673517i	$-0.764783\pi$
$593$	−36.0000	−1.47834	−0.739171	+	0.673517i	$0.764783\pi$
$594$	0	0
$595$	17.8564	0.732041
$596$	−8.00000	−0.327693
$597$	0	0
$598$	3.46410	0.141658
$599$	27.7128	1.13231	0.566157	−	0.824297i	$-0.308429\pi$
$599$	27.7128	1.13231	0.566157	+	0.824297i	$0.308429\pi$
$600$	0	0
$601$	0.143594	0.00585730	0.00292865	−	0.999996i	$-0.499068\pi$
$601$	0.143594	0.00585730	0.00292865	+	0.999996i	$0.499068\pi$
$602$	14.5359	0.592439
$603$	0	0
$604$	−11.4641	−0.466468
$605$	2.73205	0.111074
$606$	0	0
$607$	23.8038	0.966168	0.483084	−	0.875574i	$-0.339517\pi$
$607$	23.8038	0.966168	0.483084	+	0.875574i	$0.339517\pi$
$608$	−38.7846	−1.57292
$609$	0	0
$610$	42.2487	1.71060
$611$	5.46410	0.221054
$612$	0	0
$613$	−9.85641	−0.398097	−0.199048	−	0.979990i	$-0.563785\pi$
$613$	−9.85641	−0.398097	−0.199048	+	0.979990i	$0.563785\pi$
$614$	47.5692	1.91974
$615$	0	0
$616$	−3.46410	−0.139573
$617$	−6.33975	−0.255229	−0.127614	−	0.991824i	$-0.540732\pi$
$617$	−6.33975	−0.255229	−0.127614	+	0.991824i	$0.540732\pi$
$618$	0	0
$619$	2.58846	0.104039	0.0520194	−	0.998646i	$-0.483434\pi$
$619$	2.58846	0.104039	0.0520194	+	0.998646i	$0.483434\pi$
$620$	6.00000	0.240966
$621$	0	0
$622$	−29.3205	−1.17565
$623$	3.32051	0.133033
$624$	0	0
$625$	−31.2487	−1.24995
$626$	4.39230	0.175552
$627$	0	0
$628$	19.3205	0.770972
$629$	6.53590	0.260603
$630$	0	0
$631$	−23.6603	−0.941900	−0.470950	−	0.882160i	$-0.656089\pi$
$631$	−23.6603	−0.941900	−0.470950	+	0.882160i	$0.656089\pi$
$632$	23.6603	0.941154
$633$	0	0
$634$	−12.3397	−0.490074
$635$	−30.3923	−1.20608
$636$	0	0
$637$	3.00000	0.118864
$638$	10.7321	0.424886
$639$	0	0
$640$	−33.1244	−1.30936
$641$	7.85641	0.310309	0.155155	−	0.987890i	$-0.450412\pi$
$641$	7.85641	0.310309	0.155155	+	0.987890i	$0.450412\pi$
$642$	0	0
$643$	−8.05256	−0.317562	−0.158781	−	0.987314i	$-0.550756\pi$
$643$	−8.05256	−0.317562	−0.158781	+	0.987314i	$0.550756\pi$
$644$	−4.00000	−0.157622
$645$	0	0
$646$	−42.2487	−1.66225
$647$	−18.9282	−0.744144	−0.372072	−	0.928204i	$-0.621353\pi$
$647$	−18.9282	−0.744144	−0.372072	+	0.928204i	$0.621353\pi$
$648$	0	0
$649$	6.53590	0.256556
$650$	4.26795	0.167403
$651$	0	0
$652$	−20.7321	−0.811930
$653$	−2.67949	−0.104857	−0.0524283	−	0.998625i	$-0.516696\pi$
$653$	−2.67949	−0.104857	−0.0524283	+	0.998625i	$0.516696\pi$
$654$	0	0
$655$	51.7128	2.02059
$656$	7.32051	0.285818
$657$	0	0
$658$	−18.9282	−0.737898
$659$	38.9282	1.51643	0.758214	−	0.652006i	$-0.226072\pi$
$659$	38.9282	1.51643	0.758214	+	0.652006i	$0.226072\pi$
$660$	0	0
$661$	−11.4641	−0.445902	−0.222951	−	0.974830i	$-0.571569\pi$
$661$	−11.4641	−0.445902	−0.222951	+	0.974830i	$0.571569\pi$
$662$	37.2679	1.44846
$663$	0	0
$664$	24.0000	0.931381
$665$	40.7846	1.58156
$666$	0	0
$667$	−12.3923	−0.479832
$668$	−17.8564	−0.690885
$669$	0	0
$670$	8.53590	0.329771
$671$	−8.92820	−0.344669
$672$	0	0
$673$	−16.5359	−0.637412	−0.318706	−	0.947854i	$-0.603248\pi$
$673$	−16.5359	−0.637412	−0.318706	+	0.947854i	$0.603248\pi$
$674$	43.8564	1.68929
$675$	0	0
$676$	1.00000	0.0384615
$677$	−46.3013	−1.77950	−0.889751	−	0.456446i	$-0.849122\pi$
$677$	−46.3013	−1.77950	−0.889751	+	0.456446i	$0.849122\pi$
$678$	0	0
$679$	9.85641	0.378254
$680$	−15.4641	−0.593021
$681$	0	0
$682$	−3.80385	−0.145657
$683$	−16.7846	−0.642245	−0.321123	−	0.947038i	$-0.604060\pi$
$683$	−16.7846	−0.642245	−0.321123	+	0.947038i	$0.604060\pi$
$684$	0	0
$685$	−45.3205	−1.73161
$686$	−34.6410	−1.32260
$687$	0	0
$688$	−20.9808	−0.799884
$689$	2.00000	0.0761939
$690$	0	0
$691$	−32.4449	−1.23426	−0.617130	−	0.786861i	$-0.711705\pi$
$691$	−32.4449	−1.23426	−0.617130	+	0.786861i	$0.711705\pi$
$692$	5.12436	0.194799
$693$	0	0
$694$	28.3923	1.07776
$695$	−55.1769	−2.09298
$696$	0	0
$697$	4.78461	0.181230
$698$	−41.3205	−1.56400
$699$	0	0
$700$	−4.92820	−0.186269
$701$	44.0526	1.66384	0.831921	−	0.554894i	$-0.187241\pi$
$701$	44.0526	1.66384	0.831921	+	0.554894i	$0.187241\pi$
$702$	0	0
$703$	14.9282	0.563028
$704$	1.00000	0.0376889
$705$	0	0
$706$	−50.1962	−1.88916
$707$	−4.39230	−0.165190
$708$	0	0
$709$	−11.4641	−0.430543	−0.215272	−	0.976554i	$-0.569064\pi$
$709$	−11.4641	−0.430543	−0.215272	+	0.976554i	$0.569064\pi$
$710$	−32.7846	−1.23038
$711$	0	0
$712$	−2.87564	−0.107769
$713$	4.39230	0.164493
$714$	0	0
$715$	−2.73205	−0.102173
$716$	19.8564	0.742069
$717$	0	0
$718$	−40.6410	−1.51671
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	0	0
$721$	30.6410	1.14113
$722$	−63.5885	−2.36652
$723$	0	0
$724$	−12.3923	−0.460556
$725$	−15.2679	−0.567037
$726$	0	0
$727$	4.67949	0.173553	0.0867764	−	0.996228i	$-0.472343\pi$
$727$	4.67949	0.173553	0.0867764	+	0.996228i	$0.472343\pi$
$728$	3.46410	0.128388
$729$	0	0
$730$	51.7128	1.91398
$731$	−13.7128	−0.507187
$732$	0	0
$733$	−9.07180	−0.335074	−0.167537	−	0.985866i	$-0.553581\pi$
$733$	−9.07180	−0.335074	−0.167537	+	0.985866i	$0.553581\pi$
$734$	−29.5692	−1.09142
$735$	0	0
$736$	10.3923	0.383065
$737$	−1.80385	−0.0664456
$738$	0	0
$739$	−2.00000	−0.0735712	−0.0367856	−	0.999323i	$-0.511712\pi$
$739$	−2.00000	−0.0735712	−0.0367856	+	0.999323i	$0.511712\pi$
$740$	−5.46410	−0.200864
$741$	0	0
$742$	−6.92820	−0.254342
$743$	−10.1436	−0.372132	−0.186066	−	0.982537i	$-0.559574\pi$
$743$	−10.1436	−0.372132	−0.186066	+	0.982537i	$0.559574\pi$
$744$	0	0
$745$	−21.8564	−0.800757
$746$	−55.8564	−2.04505
$747$	0	0
$748$	−3.26795	−0.119488
$749$	24.7846	0.905610
$750$	0	0
$751$	−44.7846	−1.63421	−0.817107	−	0.576486i	$-0.804423\pi$
$751$	−44.7846	−1.63421	−0.817107	+	0.576486i	$0.804423\pi$
$752$	27.3205	0.996276
$753$	0	0
$754$	−10.7321	−0.390838
$755$	−31.3205	−1.13987
$756$	0	0
$757$	−2.53590	−0.0921688	−0.0460844	−	0.998938i	$-0.514674\pi$
$757$	−2.53590	−0.0921688	−0.0460844	+	0.998938i	$0.514674\pi$
$758$	−3.12436	−0.113482
$759$	0	0
$760$	−35.3205	−1.28121
$761$	14.6410	0.530736	0.265368	−	0.964147i	$-0.414506\pi$
$761$	14.6410	0.530736	0.265368	+	0.964147i	$0.414506\pi$
$762$	0	0
$763$	16.0000	0.579239
$764$	5.07180	0.183491
$765$	0	0
$766$	6.24871	0.225775
$767$	−6.53590	−0.235998
$768$	0	0
$769$	43.8564	1.58150	0.790751	−	0.612138i	$-0.209690\pi$
$769$	43.8564	1.58150	0.790751	+	0.612138i	$0.209690\pi$
$770$	9.46410	0.341063
$771$	0	0
$772$	6.92820	0.249351
$773$	4.48334	0.161255	0.0806273	−	0.996744i	$-0.474308\pi$
$773$	4.48334	0.161255	0.0806273	+	0.996744i	$0.474308\pi$
$774$	0	0
$775$	5.41154	0.194388
$776$	−8.53590	−0.306421
$777$	0	0
$778$	26.7846	0.960275
$779$	10.9282	0.391544
$780$	0	0
$781$	6.92820	0.247911
$782$	11.3205	0.404821
$783$	0	0
$784$	15.0000	0.535714
$785$	52.7846	1.88396
$786$	0	0
$787$	−23.0718	−0.822421	−0.411210	−	0.911540i	$-0.634894\pi$
$787$	−23.0718	−0.822421	−0.411210	+	0.911540i	$0.634894\pi$
$788$	−8.00000	−0.284988
$789$	0	0
$790$	−64.6410	−2.29982
$791$	20.0000	0.711118
$792$	0	0
$793$	8.92820	0.317050
$794$	−19.1769	−0.680563
$795$	0	0
$796$	17.4641	0.618999
$797$	29.6077	1.04876	0.524379	−	0.851485i	$-0.324297\pi$
$797$	29.6077	1.04876	0.524379	+	0.851485i	$0.324297\pi$
$798$	0	0
$799$	17.8564	0.631714
$800$	12.8038	0.452684
$801$	0	0
$802$	27.3731	0.966577
$803$	−10.9282	−0.385648
$804$	0	0
$805$	−10.9282	−0.385169
$806$	3.80385	0.133985
$807$	0	0
$808$	3.80385	0.133819
$809$	36.8372	1.29513	0.647563	−	0.762012i	$-0.275788\pi$
$809$	36.8372	1.29513	0.647563	+	0.762012i	$0.275788\pi$
$810$	0	0
$811$	1.60770	0.0564538	0.0282269	−	0.999602i	$-0.491014\pi$
$811$	1.60770	0.0564538	0.0282269	+	0.999602i	$0.491014\pi$
$812$	12.3923	0.434885
$813$	0	0
$814$	3.46410	0.121417
$815$	−56.6410	−1.98405
$816$	0	0
$817$	−31.3205	−1.09577
$818$	−48.0000	−1.67828
$819$	0	0
$820$	−4.00000	−0.139686
$821$	47.3205	1.65150	0.825749	−	0.564038i	$-0.190753\pi$
$821$	47.3205	1.65150	0.825749	+	0.564038i	$0.190753\pi$
$822$	0	0
$823$	32.3923	1.12912	0.564562	−	0.825390i	$-0.309045\pi$
$823$	32.3923	1.12912	0.564562	+	0.825390i	$0.309045\pi$
$824$	−26.5359	−0.924422
$825$	0	0
$826$	22.6410	0.787782
$827$	28.2487	0.982304	0.491152	−	0.871074i	$-0.336576\pi$
$827$	28.2487	0.982304	0.491152	+	0.871074i	$0.336576\pi$
$828$	0	0
$829$	−20.1436	−0.699616	−0.349808	−	0.936821i	$-0.613753\pi$
$829$	−20.1436	−0.699616	−0.349808	+	0.936821i	$0.613753\pi$
$830$	−65.5692	−2.27594
$831$	0	0
$832$	−1.00000	−0.0346688
$833$	9.80385	0.339683
$834$	0	0
$835$	−48.7846	−1.68826
$836$	−7.46410	−0.258151
$837$	0	0
$838$	45.0333	1.55565
$839$	31.7128	1.09485	0.547424	−	0.836855i	$-0.315609\pi$
$839$	31.7128	1.09485	0.547424	+	0.836855i	$0.315609\pi$
$840$	0	0
$841$	9.39230	0.323873
$842$	8.53590	0.294166
$843$	0	0
$844$	22.7321	0.782469
$845$	2.73205	0.0939854
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	10.0000	0.343401
$849$	0	0
$850$	13.9474	0.478393
$851$	−4.00000	−0.137118
$852$	0	0
$853$	−39.8564	−1.36466	−0.682329	−	0.731046i	$-0.739033\pi$
$853$	−39.8564	−1.36466	−0.682329	+	0.731046i	$0.739033\pi$
$854$	−30.9282	−1.05834
$855$	0	0
$856$	−21.4641	−0.733628
$857$	−42.1962	−1.44139	−0.720697	−	0.693251i	$-0.756178\pi$
$857$	−42.1962	−1.44139	−0.720697	+	0.693251i	$0.756178\pi$
$858$	0	0
$859$	−53.9615	−1.84114	−0.920572	−	0.390574i	$-0.872277\pi$
$859$	−53.9615	−1.84114	−0.920572	+	0.390574i	$0.872277\pi$
$860$	11.4641	0.390923
$861$	0	0
$862$	49.8564	1.69812
$863$	23.3205	0.793839	0.396920	−	0.917853i	$-0.370079\pi$
$863$	23.3205	0.793839	0.396920	+	0.917853i	$0.370079\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	10.3923	0.353145
$867$	0	0
$868$	−4.39230	−0.149085
$869$	13.6603	0.463392
$870$	0	0
$871$	1.80385	0.0611210
$872$	−13.8564	−0.469237
$873$	0	0
$874$	25.8564	0.874606
$875$	13.8564	0.468432
$876$	0	0
$877$	−11.8564	−0.400362	−0.200181	−	0.979759i	$-0.564153\pi$
$877$	−11.8564	−0.400362	−0.200181	+	0.979759i	$0.564153\pi$
$878$	−62.1962	−2.09902
$879$	0	0
$880$	−13.6603	−0.460487
$881$	−40.6410	−1.36923	−0.684615	−	0.728905i	$-0.740030\pi$
$881$	−40.6410	−1.36923	−0.684615	+	0.728905i	$0.740030\pi$
$882$	0	0
$883$	46.2487	1.55639	0.778197	−	0.628021i	$-0.216135\pi$
$883$	46.2487	1.55639	0.778197	+	0.628021i	$0.216135\pi$
$884$	3.26795	0.109913
$885$	0	0
$886$	63.7128	2.14047
$887$	−10.6410	−0.357290	−0.178645	−	0.983914i	$-0.557171\pi$
$887$	−10.6410	−0.357290	−0.178645	+	0.983914i	$0.557171\pi$
$888$	0	0
$889$	22.2487	0.746198
$890$	7.85641	0.263347
$891$	0	0
$892$	21.5167	0.720431
$893$	40.7846	1.36480
$894$	0	0
$895$	54.2487	1.81333
$896$	24.2487	0.810093
$897$	0	0
$898$	14.1962	0.473732
$899$	−13.6077	−0.453842
$900$	0	0
$901$	6.53590	0.217742
$902$	2.53590	0.0844362
$903$	0	0
$904$	−17.3205	−0.576072
$905$	−33.8564	−1.12543
$906$	0	0
$907$	10.9282	0.362865	0.181433	−	0.983403i	$-0.441927\pi$
$907$	10.9282	0.362865	0.181433	+	0.983403i	$0.441927\pi$
$908$	−10.3923	−0.344881
$909$	0	0
$910$	−9.46410	−0.313732
$911$	−2.14359	−0.0710204	−0.0355102	−	0.999369i	$-0.511306\pi$
$911$	−2.14359	−0.0710204	−0.0355102	+	0.999369i	$0.511306\pi$
$912$	0	0
$913$	13.8564	0.458580
$914$	−18.6795	−0.617863
$915$	0	0
$916$	−8.53590	−0.282034
$917$	−37.8564	−1.25013
$918$	0	0
$919$	0.875644	0.0288848	0.0144424	−	0.999896i	$-0.495403\pi$
$919$	0.875644	0.0288848	0.0144424	+	0.999896i	$0.495403\pi$
$920$	9.46410	0.312022
$921$	0	0
$922$	−18.2487	−0.600989
$923$	−6.92820	−0.228045
$924$	0	0
$925$	−4.92820	−0.162038
$926$	44.1962	1.45238
$927$	0	0
$928$	−32.1962	−1.05689
$929$	51.1244	1.67734	0.838668	−	0.544643i	$-0.183335\pi$
$929$	51.1244	1.67734	0.838668	+	0.544643i	$0.183335\pi$
$930$	0	0
$931$	22.3923	0.733878
$932$	6.19615	0.202962
$933$	0	0
$934$	−33.0333	−1.08088
$935$	−8.92820	−0.291983
$936$	0	0
$937$	33.0333	1.07915	0.539576	−	0.841937i	$-0.318585\pi$
$937$	33.0333	1.07915	0.539576	+	0.841937i	$0.318585\pi$
$938$	−6.24871	−0.204028
$939$	0	0
$940$	−14.9282	−0.486904
$941$	16.3923	0.534374	0.267187	−	0.963645i	$-0.413906\pi$
$941$	16.3923	0.534374	0.267187	+	0.963645i	$0.413906\pi$
$942$	0	0
$943$	−2.92820	−0.0953554
$944$	−32.6795	−1.06363
$945$	0	0
$946$	−7.26795	−0.236301
$947$	−0.679492	−0.0220805	−0.0110403	−	0.999939i	$-0.503514\pi$
$947$	−0.679492	−0.0220805	−0.0110403	+	0.999939i	$0.503514\pi$
$948$	0	0
$949$	10.9282	0.354744
$950$	31.8564	1.03356
$951$	0	0
$952$	11.3205	0.366900
$953$	−7.26795	−0.235432	−0.117716	−	0.993047i	$-0.537557\pi$
$953$	−7.26795	−0.235432	−0.117716	+	0.993047i	$0.537557\pi$
$954$	0	0
$955$	13.8564	0.448383
$956$	−1.60770	−0.0519966
$957$	0	0
$958$	47.5692	1.53689
$959$	33.1769	1.07134
$960$	0	0
$961$	−26.1769	−0.844417
$962$	−3.46410	−0.111687
$963$	0	0
$964$	22.7846	0.733843
$965$	18.9282	0.609320
$966$	0	0
$967$	27.0718	0.870570	0.435285	−	0.900293i	$-0.356648\pi$
$967$	27.0718	0.870570	0.435285	+	0.900293i	$0.356648\pi$
$968$	1.73205	0.0556702
$969$	0	0
$970$	23.3205	0.748776
$971$	−3.85641	−0.123758	−0.0618790	−	0.998084i	$-0.519709\pi$
$971$	−3.85641	−0.123758	−0.0618790	+	0.998084i	$0.519709\pi$
$972$	0	0
$973$	40.3923	1.29492
$974$	14.4449	0.462843
$975$	0	0
$976$	44.6410	1.42892
$977$	17.6603	0.565002	0.282501	−	0.959267i	$-0.408836\pi$
$977$	17.6603	0.565002	0.282501	+	0.959267i	$0.408836\pi$
$978$	0	0
$979$	−1.66025	−0.0530619
$980$	−8.19615	−0.261816
$981$	0	0
$982$	4.39230	0.140164
$983$	33.0718	1.05483	0.527413	−	0.849609i	$-0.323162\pi$
$983$	33.0718	1.05483	0.527413	+	0.849609i	$0.323162\pi$
$984$	0	0
$985$	−21.8564	−0.696403
$986$	−35.0718	−1.11691
$987$	0	0
$988$	7.46410	0.237465
$989$	8.39230	0.266860
$990$	0	0
$991$	−5.17691	−0.164450	−0.0822251	−	0.996614i	$-0.526203\pi$
$991$	−5.17691	−0.164450	−0.0822251	+	0.996614i	$0.526203\pi$
$992$	11.4115	0.362317
$993$	0	0
$994$	24.0000	0.761234
$995$	47.7128	1.51260
$996$	0	0
$997$	41.3205	1.30863	0.654317	−	0.756221i	$-0.272956\pi$
$997$	41.3205	1.30863	0.654317	+	0.756221i	$0.272956\pi$
$998$	30.8372	0.976134
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1287.2.a.f.1.1		2
3.2	odd	2		429.2.a.d.1.2	✓	2
12.11	even	2		6864.2.a.bk.1.1		2
33.32	even	2		4719.2.a.n.1.1		2
39.38	odd	2		5577.2.a.h.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.a.d.1.2	✓	2	3.2	odd	2
1287.2.a.f.1.1		2	1.1	even	1	trivial
4719.2.a.n.1.1		2	33.32	even	2
5577.2.a.h.1.1		2	39.38	odd	2
6864.2.a.bk.1.1		2	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:	\( N \)	\(=\)	\( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1287.a (trivial)

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} +1.00000 q^{4} +2.73205 q^{5} -2.00000 q^{7} +1.73205 q^{8} +O(q^{10})\)	\(q-1.73205 q^{2} +1.00000 q^{4} +2.73205 q^{5} -2.00000 q^{7} +1.73205 q^{8} -4.73205 q^{10} +1.00000 q^{11} -1.00000 q^{13} +3.46410 q^{14} -5.00000 q^{16} -3.26795 q^{17} -7.46410 q^{19} +2.73205 q^{20} -1.73205 q^{22} +2.00000 q^{23} +2.46410 q^{25} +1.73205 q^{26} -2.00000 q^{28} -6.19615 q^{29} +2.19615 q^{31} +5.19615 q^{32} +5.66025 q^{34} -5.46410 q^{35} -2.00000 q^{37} +12.9282 q^{38} +4.73205 q^{40} -1.46410 q^{41} +4.19615 q^{43} +1.00000 q^{44} -3.46410 q^{46} -5.46410 q^{47} -3.00000 q^{49} -4.26795 q^{50} -1.00000 q^{52} -2.00000 q^{53} +2.73205 q^{55} -3.46410 q^{56} +10.7321 q^{58} +6.53590 q^{59} -8.92820 q^{61} -3.80385 q^{62} +1.00000 q^{64} -2.73205 q^{65} -1.80385 q^{67} -3.26795 q^{68} +9.46410 q^{70} +6.92820 q^{71} -10.9282 q^{73} +3.46410 q^{74} -7.46410 q^{76} -2.00000 q^{77} +13.6603 q^{79} -13.6603 q^{80} +2.53590 q^{82} +13.8564 q^{83} -8.92820 q^{85} -7.26795 q^{86} +1.73205 q^{88} -1.66025 q^{89} +2.00000 q^{91} +2.00000 q^{92} +9.46410 q^{94} -20.3923 q^{95} -4.92820 q^{97} +5.19615 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 2 q^{5} - 4 q^{7}+O(q^{10}) \) 2 * q + 2 * q^4 + 2 * q^5 - 4 * q^7	\( 2 q + 2 q^{4} + 2 q^{5} - 4 q^{7} - 6 q^{10} + 2 q^{11} - 2 q^{13} - 10 q^{16} - 10 q^{17} - 8 q^{19} + 2 q^{20} + 4 q^{23} - 2 q^{25} - 4 q^{28} - 2 q^{29} - 6 q^{31} - 6 q^{34} - 4 q^{35} - 4 q^{37} + 12 q^{38} + 6 q^{40} + 4 q^{41} - 2 q^{43} + 2 q^{44} - 4 q^{47} - 6 q^{49} - 12 q^{50} - 2 q^{52} - 4 q^{53} + 2 q^{55} + 18 q^{58} + 20 q^{59} - 4 q^{61} - 18 q^{62} + 2 q^{64} - 2 q^{65} - 14 q^{67} - 10 q^{68} + 12 q^{70} - 8 q^{73} - 8 q^{76} - 4 q^{77} + 10 q^{79} - 10 q^{80} + 12 q^{82} - 4 q^{85} - 18 q^{86} + 14 q^{89} + 4 q^{91} + 4 q^{92} + 12 q^{94} - 20 q^{95} + 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^4 + 2 * q^5 - 4 * q^7 - 6 * q^10 + 2 * q^11 - 2 * q^13 - 10 * q^16 - 10 * q^17 - 8 * q^19 + 2 * q^20 + 4 * q^23 - 2 * q^25 - 4 * q^28 - 2 * q^29 - 6 * q^31 - 6 * q^34 - 4 * q^35 - 4 * q^37 + 12 * q^38 + 6 * q^40 + 4 * q^41 - 2 * q^43 + 2 * q^44 - 4 * q^47 - 6 * q^49 - 12 * q^50 - 2 * q^52 - 4 * q^53 + 2 * q^55 + 18 * q^58 + 20 * q^59 - 4 * q^61 - 18 * q^62 + 2 * q^64 - 2 * q^65 - 14 * q^67 - 10 * q^68 + 12 * q^70 - 8 * q^73 - 8 * q^76 - 4 * q^77 + 10 * q^79 - 10 * q^80 + 12 * q^82 - 4 * q^85 - 18 * q^86 + 14 * q^89 + 4 * q^91 + 4 * q^92 + 12 * q^94 - 20 * q^95 + 4 * q^97

Introduction

L-functions

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Database

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