LMFDB - Embedded newform 8568.2.a.bj.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8568,2,Mod(1,8568)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8568.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8568, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8568 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8568.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,1,0,-4,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$68.4158244518$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.5225.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + x + 11 $ x^4 - x^3 - 8*x^2 + x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 952)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.48718$ of defining polynomial
Character	$\chi$	$=$	8568.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.02435 q^{5} -1.00000 q^{7} +1.23607 q^{11} -6.47214 q^{13} -1.00000 q^{17} +6.97437 q^{19} -7.07433 q^{23} +11.1954 q^{25} +6.31040 q^{29} +4.88824 q^{31} +4.02435 q^{35} -2.63387 q^{37} +10.1954 q^{41} -2.69762 q^{43} +3.67652 q^{47} +1.00000 q^{49} +1.39047 q^{53} -4.97437 q^{55} -5.07433 q^{59} +3.93369 q^{61} +26.0461 q^{65} -5.63259 q^{67} +6.84431 q^{71} +0.246650 q^{73} -1.23607 q^{77} -11.6765 q^{79} +6.65089 q^{83} +4.02435 q^{85} -2.27872 q^{89} +6.47214 q^{91} -28.0673 q^{95} +10.7069 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{5} - 4 q^{7} - 4 q^{11} - 8 q^{13} - 4 q^{17} + 14 q^{19} - 8 q^{23} + 11 q^{25} - 4 q^{29} + 5 q^{31} - q^{35} - 4 q^{37} + 7 q^{41} + 19 q^{43} - 8 q^{47} + 4 q^{49} - 5 q^{53} - 6 q^{55} - 23 q^{61}+ \cdots - 15 q^{97}+O(q^{100}) $ 4 * q + q^5 - 4 * q^7 - 4 * q^11 - 8 * q^13 - 4 * q^17 + 14 * q^19 - 8 * q^23 + 11 * q^25 - 4 * q^29 + 5 * q^31 - q^35 - 4 * q^37 + 7 * q^41 + 19 * q^43 - 8 * q^47 + 4 * q^49 - 5 * q^53 - 6 * q^55 - 23 * q^61 + 8 * q^65 + 15 * q^67 - 2 * q^71 - 5 * q^73 + 4 * q^77 - 24 * q^79 - 10 * q^83 - q^85 + 16 * q^89 + 8 * q^91 - 22 * q^95 - 15 * q^97

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	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−4.02435	−1.79974	−0.899872	−	0.436154i	$-0.856340\pi$
$5$	−4.02435	−1.79974	−0.899872	+	0.436154i	$0.856340\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.23607	0.372689	0.186344	−	0.982485i	$-0.440336\pi$
$11$	1.23607	0.372689	0.186344	+	0.982485i	$0.440336\pi$
$12$	0	0
$13$	−6.47214	−1.79505	−0.897524	−	0.440966i	$-0.854636\pi$
$13$	−6.47214	−1.79505	−0.897524	+	0.440966i	$0.854636\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	6.97437	1.60003	0.800015	−	0.599980i	$-0.204825\pi$
$19$	6.97437	1.60003	0.800015	+	0.599980i	$0.204825\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.07433	−1.47510	−0.737550	−	0.675293i	$-0.764017\pi$
$23$	−7.07433	−1.47510	−0.737550	+	0.675293i	$0.764017\pi$
$24$	0	0
$25$	11.1954	2.23908
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.31040	1.17181	0.585906	−	0.810379i	$-0.300739\pi$
$29$	6.31040	1.17181	0.585906	+	0.810379i	$0.300739\pi$
$30$	0	0
$31$	4.88824	0.877954	0.438977	−	0.898498i	$-0.355341\pi$
$31$	4.88824	0.877954	0.438977	+	0.898498i	$0.355341\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.02435	0.680239
$36$	0	0
$37$	−2.63387	−0.433006	−0.216503	−	0.976282i	$-0.569465\pi$
$37$	−2.63387	−0.433006	−0.216503	+	0.976282i	$0.569465\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10.1954	1.59225	0.796126	−	0.605131i	$-0.206879\pi$
$41$	10.1954	1.59225	0.796126	+	0.605131i	$0.206879\pi$
$42$	0	0
$43$	−2.69762	−0.411384	−0.205692	−	0.978617i	$-0.565944\pi$
$43$	−2.69762	−0.411384	−0.205692	+	0.978617i	$0.565944\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.67652	0.536276	0.268138	−	0.963381i	$-0.413592\pi$
$47$	3.67652	0.536276	0.268138	+	0.963381i	$0.413592\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.39047	0.190996	0.0954982	−	0.995430i	$-0.469556\pi$
$53$	1.39047	0.190996	0.0954982	+	0.995430i	$0.469556\pi$
$54$	0	0
$55$	−4.97437	−0.670744
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.07433	−0.660621	−0.330311	−	0.943872i	$-0.607154\pi$
$59$	−5.07433	−0.660621	−0.330311	+	0.943872i	$0.607154\pi$
$60$	0	0
$61$	3.93369	0.503657	0.251829	−	0.967772i	$-0.418968\pi$
$61$	3.93369	0.503657	0.251829	+	0.967772i	$0.418968\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	26.0461	3.23063
$66$	0	0
$67$	−5.63259	−0.688131	−0.344065	−	0.938946i	$-0.611804\pi$
$67$	−5.63259	−0.688131	−0.344065	+	0.938946i	$0.611804\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.84431	0.812270	0.406135	−	0.913813i	$-0.366876\pi$
$71$	6.84431	0.812270	0.406135	+	0.913813i	$0.366876\pi$
$72$	0	0
$73$	0.246650	0.0288682	0.0144341	−	0.999896i	$-0.495405\pi$
$73$	0.246650	0.0288682	0.0144341	+	0.999896i	$0.495405\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.23607	−0.140863
$78$	0	0
$79$	−11.6765	−1.31371	−0.656856	−	0.754016i	$-0.728114\pi$
$79$	−11.6765	−1.31371	−0.656856	+	0.754016i	$0.728114\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.65089	0.730030	0.365015	−	0.931002i	$-0.381064\pi$
$83$	6.65089	0.730030	0.365015	+	0.931002i	$0.381064\pi$
$84$	0	0
$85$	4.02435	0.436502
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.27872	−0.241543	−0.120772	−	0.992680i	$-0.538537\pi$
$89$	−2.27872	−0.241543	−0.120772	+	0.992680i	$0.538537\pi$
$90$	0	0
$91$	6.47214	0.678464
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−28.0673	−2.87964
$96$	0	0
$97$	10.7069	1.08712	0.543562	−	0.839369i	$-0.317075\pi$
$97$	10.7069	1.08712	0.543562	+	0.839369i	$0.317075\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.7931	1.67097	0.835486	−	0.549512i	$-0.185187\pi$
$101$	16.7931	1.67097	0.835486	+	0.549512i	$0.185187\pi$
$102$	0	0
$103$	9.57656	0.943607	0.471803	−	0.881704i	$-0.343603\pi$
$103$	9.57656	0.943607	0.471803	+	0.881704i	$0.343603\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−16.9613	−1.63971	−0.819855	−	0.572571i	$-0.805946\pi$
$107$	−16.9613	−1.63971	−0.819855	+	0.572571i	$0.805946\pi$
$108$	0	0
$109$	−19.6082	−1.87813	−0.939065	−	0.343741i	$-0.888306\pi$
$109$	−19.6082	−1.87813	−0.939065	+	0.343741i	$0.888306\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	13.4952	1.26952	0.634761	−	0.772709i	$-0.281099\pi$
$113$	13.4952	1.26952	0.634761	+	0.772709i	$0.281099\pi$
$114$	0	0
$115$	28.4696	2.65480
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	−9.47214	−0.861103
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−24.9324	−2.23002
$126$	0	0
$127$	4.36484	0.387317	0.193659	−	0.981069i	$-0.437965\pi$
$127$	4.36484	0.387317	0.193659	+	0.981069i	$0.437965\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−10.6826	−0.933341	−0.466670	−	0.884431i	$-0.654547\pi$
$131$	−10.6826	−0.933341	−0.466670	+	0.884431i	$0.654547\pi$
$132$	0	0
$133$	−6.97437	−0.604755
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−18.1954	−1.55454	−0.777268	−	0.629169i	$-0.783395\pi$
$137$	−18.1954	−1.55454	−0.777268	+	0.629169i	$0.783395\pi$
$138$	0	0
$139$	13.8430	1.17415	0.587075	−	0.809532i	$-0.300279\pi$
$139$	13.8430	1.17415	0.587075	+	0.809532i	$0.300279\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−8.00000	−0.668994
$144$	0	0
$145$	−25.3952	−2.10896
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.42987	0.280986	0.140493	−	0.990082i	$-0.455131\pi$
$149$	3.42987	0.280986	0.140493	+	0.990082i	$0.455131\pi$
$150$	0	0
$151$	−9.16976	−0.746224	−0.373112	−	0.927786i	$-0.621709\pi$
$151$	−9.16976	−0.746224	−0.373112	+	0.927786i	$0.621709\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−19.6720	−1.58009
$156$	0	0
$157$	−6.70215	−0.534890	−0.267445	−	0.963573i	$-0.586179\pi$
$157$	−6.70215	−0.534890	−0.267445	+	0.963573i	$0.586179\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.07433	0.557535
$162$	0	0
$163$	−14.4104	−1.12871	−0.564353	−	0.825533i	$-0.690874\pi$
$163$	−14.4104	−1.12871	−0.564353	+	0.825533i	$0.690874\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.25112	0.251579	0.125789	−	0.992057i	$-0.459854\pi$
$167$	3.25112	0.251579	0.125789	+	0.992057i	$0.459854\pi$
$168$	0	0
$169$	28.8885	2.22220
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−4.25717	−0.323666	−0.161833	−	0.986818i	$-0.551741\pi$
$173$	−4.25717	−0.323666	−0.161833	+	0.986818i	$0.551741\pi$
$174$	0	0
$175$	−11.1954	−0.846292
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2.72772	0.203879	0.101940	−	0.994791i	$-0.467495\pi$
$179$	2.72772	0.203879	0.101940	+	0.994791i	$0.467495\pi$
$180$	0	0
$181$	3.15471	0.234488	0.117244	−	0.993103i	$-0.462594\pi$
$181$	3.15471	0.234488	0.117244	+	0.993103i	$0.462594\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	10.5996	0.779300
$186$	0	0
$187$	−1.23607	−0.0903902
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3.53263	−0.255612	−0.127806	−	0.991799i	$-0.540794\pi$
$191$	−3.53263	−0.255612	−0.127806	+	0.991799i	$0.540794\pi$
$192$	0	0
$193$	16.2275	1.16808	0.584039	−	0.811726i	$-0.301472\pi$
$193$	16.2275	1.16808	0.584039	+	0.811726i	$0.301472\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−17.7569	−1.26513	−0.632563	−	0.774509i	$-0.717997\pi$
$197$	−17.7569	−1.26513	−0.632563	+	0.774509i	$0.717997\pi$
$198$	0	0
$199$	−15.1486	−1.07386	−0.536928	−	0.843628i	$-0.680415\pi$
$199$	−15.1486	−1.07386	−0.536928	+	0.843628i	$0.680415\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−6.31040	−0.442903
$204$	0	0
$205$	−41.0298	−2.86565
$206$	0	0
$207$	0	0
$208$	0	0
$209$	8.62079	0.596313
$210$	0	0
$211$	10.5826	0.728537	0.364269	−	0.931294i	$-0.381319\pi$
$211$	10.5826	0.728537	0.364269	+	0.931294i	$0.381319\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	10.8562	0.740385
$216$	0	0
$217$	−4.88824	−0.331835
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.47214	0.435363
$222$	0	0
$223$	−3.95130	−0.264599	−0.132299	−	0.991210i	$-0.542236\pi$
$223$	−3.95130	−0.264599	−0.132299	+	0.991210i	$0.542236\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−14.1031	−0.936059	−0.468029	−	0.883713i	$-0.655036\pi$
$227$	−14.1031	−0.936059	−0.468029	+	0.883713i	$0.655036\pi$
$228$	0	0
$229$	−17.5766	−1.16149	−0.580746	−	0.814085i	$-0.697239\pi$
$229$	−17.5766	−1.16149	−0.580746	+	0.814085i	$0.697239\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−15.5766	−1.02045	−0.510227	−	0.860040i	$-0.670439\pi$
$233$	−15.5766	−1.02045	−0.510227	+	0.860040i	$0.670439\pi$
$234$	0	0
$235$	−14.7956	−0.965159
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5.16976	0.334404	0.167202	−	0.985923i	$-0.446527\pi$
$239$	5.16976	0.334404	0.167202	+	0.985923i	$0.446527\pi$
$240$	0	0
$241$	−2.45830	−0.158353	−0.0791766	−	0.996861i	$-0.525229\pi$
$241$	−2.45830	−0.158353	−0.0791766	+	0.996861i	$0.525229\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−4.02435	−0.257106
$246$	0	0
$247$	−45.1391	−2.87213
$248$	0	0
$249$	0	0
$250$	0	0
$251$	17.3863	1.09741	0.548707	−	0.836015i	$-0.315120\pi$
$251$	17.3863	1.09741	0.548707	+	0.836015i	$0.315120\pi$
$252$	0	0
$253$	−8.74435	−0.549753
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.2787	0.641169	0.320584	−	0.947220i	$-0.396121\pi$
$257$	10.2787	0.641169	0.320584	+	0.947220i	$0.396121\pi$
$258$	0	0
$259$	2.63387	0.163661
$260$	0	0
$261$	0	0
$262$	0	0
$263$	10.1487	0.625793	0.312897	−	0.949787i	$-0.398701\pi$
$263$	10.1487	0.625793	0.312897	+	0.949787i	$0.398701\pi$
$264$	0	0
$265$	−5.59576	−0.343745
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.47055	0.150632	0.0753161	−	0.997160i	$-0.476003\pi$
$269$	2.47055	0.150632	0.0753161	+	0.997160i	$0.476003\pi$
$270$	0	0
$271$	21.0864	1.28091	0.640455	−	0.767996i	$-0.278746\pi$
$271$	21.0864	1.28091	0.640455	+	0.767996i	$0.278746\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	13.8383	0.834479
$276$	0	0
$277$	25.2335	1.51613	0.758067	−	0.652176i	$-0.226144\pi$
$277$	25.2335	1.51613	0.758067	+	0.652176i	$0.226144\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−27.5391	−1.64285	−0.821423	−	0.570319i	$-0.806820\pi$
$281$	−27.5391	−1.64285	−0.821423	+	0.570319i	$0.806820\pi$
$282$	0	0
$283$	26.9385	1.60133	0.800665	−	0.599113i	$-0.204480\pi$
$283$	26.9385	1.60133	0.800665	+	0.599113i	$0.204480\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.1954	−0.601815
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−9.20439	−0.537726	−0.268863	−	0.963178i	$-0.586648\pi$
$293$	−9.20439	−0.537726	−0.268863	+	0.963178i	$0.586648\pi$
$294$	0	0
$295$	20.4209	1.18895
$296$	0	0
$297$	0	0
$298$	0	0
$299$	45.7860	2.64787
$300$	0	0
$301$	2.69762	0.155488
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−15.8305	−0.906454
$306$	0	0
$307$	−5.84878	−0.333807	−0.166904	−	0.985973i	$-0.553377\pi$
$307$	−5.84878	−0.333807	−0.166904	+	0.985973i	$0.553377\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−30.5282	−1.73109	−0.865547	−	0.500828i	$-0.833029\pi$
$311$	−30.5282	−1.73109	−0.865547	+	0.500828i	$0.833029\pi$
$312$	0	0
$313$	−17.2880	−0.977173	−0.488586	−	0.872515i	$-0.662487\pi$
$313$	−17.2880	−0.977173	−0.488586	+	0.872515i	$0.662487\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4.99395	−0.280488	−0.140244	−	0.990117i	$-0.544789\pi$
$317$	−4.99395	−0.280488	−0.140244	+	0.990117i	$0.544789\pi$
$318$	0	0
$319$	7.80008	0.436721
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−6.97437	−0.388064
$324$	0	0
$325$	−72.4581	−4.01925
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3.67652	−0.202693
$330$	0	0
$331$	27.5211	1.51270	0.756349	−	0.654168i	$-0.226981\pi$
$331$	27.5211	1.51270	0.756349	+	0.654168i	$0.226981\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	22.6675	1.23846
$336$	0	0
$337$	−19.6439	−1.07007	−0.535035	−	0.844830i	$-0.679701\pi$
$337$	−19.6439	−1.07007	−0.535035	+	0.844830i	$0.679701\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6.04220	0.327203
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.53838	−0.189950	−0.0949751	−	0.995480i	$-0.530277\pi$
$347$	−3.53838	−0.189950	−0.0949751	+	0.995480i	$0.530277\pi$
$348$	0	0
$349$	−15.9699	−0.854849	−0.427425	−	0.904051i	$-0.640579\pi$
$349$	−15.9699	−0.854849	−0.427425	+	0.904051i	$0.640579\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	17.3440	0.923127	0.461564	−	0.887107i	$-0.347289\pi$
$353$	17.3440	0.923127	0.461564	+	0.887107i	$0.347289\pi$
$354$	0	0
$355$	−27.5439	−1.46188
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−15.2579	−0.805279	−0.402639	−	0.915359i	$-0.631907\pi$
$359$	−15.2579	−0.805279	−0.402639	+	0.915359i	$0.631907\pi$
$360$	0	0
$361$	29.6418	1.56010
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−0.992604	−0.0519553
$366$	0	0
$367$	−20.8162	−1.08660	−0.543298	−	0.839540i	$-0.682825\pi$
$367$	−20.8162	−1.08660	−0.543298	+	0.839540i	$0.682825\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.39047	−0.0721899
$372$	0	0
$373$	−18.0929	−0.936813	−0.468407	−	0.883513i	$-0.655172\pi$
$373$	−18.0929	−0.936813	−0.468407	+	0.883513i	$0.655172\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−40.8417	−2.10346
$378$	0	0
$379$	−32.2265	−1.65536	−0.827681	−	0.561198i	$-0.810341\pi$
$379$	−32.2265	−1.65536	−0.827681	+	0.561198i	$0.810341\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5.77648	−0.295164	−0.147582	−	0.989050i	$-0.547149\pi$
$383$	−5.77648	−0.295164	−0.147582	+	0.989050i	$0.547149\pi$
$384$	0	0
$385$	4.97437	0.253517
$386$	0	0
$387$	0	0
$388$	0	0
$389$	16.6070	0.842006	0.421003	−	0.907059i	$-0.361678\pi$
$389$	16.6070	0.842006	0.421003	+	0.907059i	$0.361678\pi$
$390$	0	0
$391$	7.07433	0.357764
$392$	0	0
$393$	0	0
$394$	0	0
$395$	46.9904	2.36434
$396$	0	0
$397$	−9.80083	−0.491890	−0.245945	−	0.969284i	$-0.579098\pi$
$397$	−9.80083	−0.491890	−0.245945	+	0.969284i	$0.579098\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−26.4696	−1.32183	−0.660914	−	0.750462i	$-0.729831\pi$
$401$	−26.4696	−1.32183	−0.660914	+	0.750462i	$0.729831\pi$
$402$	0	0
$403$	−31.6374	−1.57597
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3.25565	−0.161376
$408$	0	0
$409$	22.9296	1.13380	0.566898	−	0.823788i	$-0.308143\pi$
$409$	22.9296	1.13380	0.566898	+	0.823788i	$0.308143\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5.07433	0.249691
$414$	0	0
$415$	−26.7655	−1.31387
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−5.11334	−0.249803	−0.124902	−	0.992169i	$-0.539862\pi$
$419$	−5.11334	−0.249803	−0.124902	+	0.992169i	$0.539862\pi$
$420$	0	0
$421$	−22.2928	−1.08648	−0.543242	−	0.839576i	$-0.682803\pi$
$421$	−22.2928	−1.08648	−0.543242	+	0.839576i	$0.682803\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−11.1954	−0.543056
$426$	0	0
$427$	−3.93369	−0.190365
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−16.4510	−0.792415	−0.396208	−	0.918161i	$-0.629674\pi$
$431$	−16.4510	−0.792415	−0.396208	+	0.918161i	$0.629674\pi$
$432$	0	0
$433$	10.1999	0.490177	0.245088	−	0.969501i	$-0.421183\pi$
$433$	10.1999	0.490177	0.245088	+	0.969501i	$0.421183\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−49.3390	−2.36020
$438$	0	0
$439$	18.4074	0.878538	0.439269	−	0.898356i	$-0.355238\pi$
$439$	18.4074	0.878538	0.439269	+	0.898356i	$0.355238\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1.43047	−0.0679635	−0.0339817	−	0.999422i	$-0.510819\pi$
$443$	−1.43047	−0.0679635	−0.0339817	+	0.999422i	$0.510819\pi$
$444$	0	0
$445$	9.17035	0.434716
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−3.72778	−0.175925	−0.0879625	−	0.996124i	$-0.528036\pi$
$449$	−3.72778	−0.175925	−0.0879625	+	0.996124i	$0.528036\pi$
$450$	0	0
$451$	12.6022	0.593414
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−26.0461	−1.22106
$456$	0	0
$457$	19.7601	0.924338	0.462169	−	0.886792i	$-0.347071\pi$
$457$	19.7601	0.924338	0.462169	+	0.886792i	$0.347071\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	4.40227	0.205034	0.102517	−	0.994731i	$-0.467310\pi$
$461$	4.40227	0.205034	0.102517	+	0.994731i	$0.467310\pi$
$462$	0	0
$463$	4.51887	0.210010	0.105005	−	0.994472i	$-0.466514\pi$
$463$	4.51887	0.210010	0.105005	+	0.994472i	$0.466514\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.51893	−0.0702877	−0.0351438	−	0.999382i	$-0.511189\pi$
$467$	−1.51893	−0.0702877	−0.0351438	+	0.999382i	$0.511189\pi$
$468$	0	0
$469$	5.63259	0.260089
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3.33444	−0.153318
$474$	0	0
$475$	78.0808	3.58259
$476$	0	0
$477$	0	0
$478$	0	0
$479$	39.8418	1.82042	0.910209	−	0.414148i	$-0.135920\pi$
$479$	39.8418	1.82042	0.910209	+	0.414148i	$0.135920\pi$
$480$	0	0
$481$	17.0468	0.777267
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−43.0884	−1.95654
$486$	0	0
$487$	9.19532	0.416680	0.208340	−	0.978056i	$-0.433194\pi$
$487$	9.19532	0.416680	0.208340	+	0.978056i	$0.433194\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−26.5282	−1.19720	−0.598600	−	0.801048i	$-0.704276\pi$
$491$	−26.5282	−1.19720	−0.598600	+	0.801048i	$0.704276\pi$
$492$	0	0
$493$	−6.31040	−0.284206
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.84431	−0.307009
$498$	0	0
$499$	28.9312	1.29514	0.647569	−	0.762007i	$-0.275786\pi$
$499$	28.9312	1.29514	0.647569	+	0.762007i	$0.275786\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−31.1977	−1.39103	−0.695517	−	0.718509i	$-0.744825\pi$
$503$	−31.1977	−1.39103	−0.695517	+	0.718509i	$0.744825\pi$
$504$	0	0
$505$	−67.5811	−3.00732
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−3.87888	−0.171928	−0.0859641	−	0.996298i	$-0.527397\pi$
$509$	−3.87888	−0.171928	−0.0859641	+	0.996298i	$0.527397\pi$
$510$	0	0
$511$	−0.246650	−0.0109111
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−38.5394	−1.69825
$516$	0	0
$517$	4.54443	0.199864
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−44.1050	−1.93227	−0.966137	−	0.258030i	$-0.916927\pi$
$521$	−44.1050	−1.93227	−0.966137	+	0.258030i	$0.916927\pi$
$522$	0	0
$523$	−0.735418	−0.0321576	−0.0160788	−	0.999871i	$-0.505118\pi$
$523$	−0.735418	−0.0321576	−0.0160788	+	0.999871i	$0.505118\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.88824	−0.212935
$528$	0	0
$529$	27.0461	1.17592
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−65.9859	−2.85817
$534$	0	0
$535$	68.2582	2.95106
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.23607	0.0532412
$540$	0	0
$541$	5.95682	0.256104	0.128052	−	0.991767i	$-0.459128\pi$
$541$	5.95682	0.256104	0.128052	+	0.991767i	$0.459128\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	78.9104	3.38015
$546$	0	0
$547$	−39.8844	−1.70533	−0.852667	−	0.522455i	$-0.825016\pi$
$547$	−39.8844	−1.70533	−0.852667	+	0.522455i	$0.825016\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	44.0110	1.87493
$552$	0	0
$553$	11.6765	0.496536
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−29.5740	−1.25309	−0.626545	−	0.779385i	$-0.715532\pi$
$557$	−29.5740	−1.25309	−0.626545	+	0.779385i	$0.715532\pi$
$558$	0	0
$559$	17.4594	0.738453
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−43.7900	−1.84553	−0.922763	−	0.385367i	$-0.874075\pi$
$563$	−43.7900	−1.84553	−0.922763	+	0.385367i	$0.874075\pi$
$564$	0	0
$565$	−54.3094	−2.28481
$566$	0	0
$567$	0	0
$568$	0	0
$569$	44.3088	1.85752	0.928761	−	0.370678i	$-0.120875\pi$
$569$	44.3088	1.85752	0.928761	+	0.370678i	$0.120875\pi$
$570$	0	0
$571$	−25.9543	−1.08615	−0.543076	−	0.839684i	$-0.682740\pi$
$571$	−25.9543	−1.08615	−0.543076	+	0.839684i	$0.682740\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−79.1999	−3.30286
$576$	0	0
$577$	−11.1442	−0.463939	−0.231969	−	0.972723i	$-0.574517\pi$
$577$	−11.1442	−0.463939	−0.231969	+	0.972723i	$0.574517\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−6.65089	−0.275926
$582$	0	0
$583$	1.71872	0.0711822
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−19.6676	−0.811768	−0.405884	−	0.913925i	$-0.633036\pi$
$587$	−19.6676	−0.811768	−0.405884	+	0.913925i	$0.633036\pi$
$588$	0	0
$589$	34.0924	1.40475
$590$	0	0
$591$	0	0
$592$	0	0
$593$	28.1396	1.15555	0.577777	−	0.816194i	$-0.303920\pi$
$593$	28.1396	1.15555	0.577777	+	0.816194i	$0.303920\pi$
$594$	0	0
$595$	−4.02435	−0.164982
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−2.43434	−0.0994645	−0.0497322	−	0.998763i	$-0.515837\pi$
$599$	−2.43434	−0.0994645	−0.0497322	+	0.998763i	$0.515837\pi$
$600$	0	0
$601$	−15.8462	−0.646381	−0.323190	−	0.946334i	$-0.604755\pi$
$601$	−15.8462	−0.646381	−0.323190	+	0.946334i	$0.604755\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	38.1192	1.54977
$606$	0	0
$607$	18.1413	0.736334	0.368167	−	0.929760i	$-0.379986\pi$
$607$	18.1413	0.736334	0.368167	+	0.929760i	$0.379986\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−23.7950	−0.962641
$612$	0	0
$613$	14.9702	0.604641	0.302320	−	0.953206i	$-0.402239\pi$
$613$	14.9702	0.604641	0.302320	+	0.953206i	$0.402239\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−7.43440	−0.299298	−0.149649	−	0.988739i	$-0.547814\pi$
$617$	−7.43440	−0.299298	−0.149649	+	0.988739i	$0.547814\pi$
$618$	0	0
$619$	−12.9548	−0.520697	−0.260348	−	0.965515i	$-0.583837\pi$
$619$	−12.9548	−0.520697	−0.260348	+	0.965515i	$0.583837\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.27872	0.0912948
$624$	0	0
$625$	44.3598	1.77439
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.63387	0.105019
$630$	0	0
$631$	0.899738	0.0358180	0.0179090	−	0.999840i	$-0.494299\pi$
$631$	0.899738	0.0358180	0.0179090	+	0.999840i	$0.494299\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−17.5657	−0.697072
$636$	0	0
$637$	−6.47214	−0.256435
$638$	0	0
$639$	0	0
$640$	0	0
$641$	14.8322	0.585837	0.292919	−	0.956137i	$-0.405374\pi$
$641$	14.8322	0.585837	0.292919	+	0.956137i	$0.405374\pi$
$642$	0	0
$643$	−17.5168	−0.690793	−0.345397	−	0.938457i	$-0.612256\pi$
$643$	−17.5168	−0.690793	−0.345397	+	0.938457i	$0.612256\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.05829	−0.0416057	−0.0208028	−	0.999784i	$-0.506622\pi$
$647$	−1.05829	−0.0416057	−0.0208028	+	0.999784i	$0.506622\pi$
$648$	0	0
$649$	−6.27222	−0.246206
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−35.2848	−1.38080	−0.690400	−	0.723428i	$-0.742565\pi$
$653$	−35.2848	−1.38080	−0.690400	+	0.723428i	$0.742565\pi$
$654$	0	0
$655$	42.9904	1.67977
$656$	0	0
$657$	0	0
$658$	0	0
$659$	20.1047	0.783169	0.391585	−	0.920142i	$-0.371927\pi$
$659$	20.1047	0.783169	0.391585	+	0.920142i	$0.371927\pi$
$660$	0	0
$661$	12.6541	0.492186	0.246093	−	0.969246i	$-0.420853\pi$
$661$	12.6541	0.492186	0.246093	+	0.969246i	$0.420853\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	28.0673	1.08840
$666$	0	0
$667$	−44.6418	−1.72854
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4.86231	0.187707
$672$	0	0
$673$	−21.6342	−0.833937	−0.416968	−	0.908921i	$-0.636907\pi$
$673$	−21.6342	−0.833937	−0.416968	+	0.908921i	$0.636907\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−27.9965	−1.07599	−0.537996	−	0.842948i	$-0.680818\pi$
$677$	−27.9965	−1.07599	−0.537996	+	0.842948i	$0.680818\pi$
$678$	0	0
$679$	−10.7069	−0.410894
$680$	0	0
$681$	0	0
$682$	0	0
$683$	40.2225	1.53907	0.769536	−	0.638603i	$-0.220487\pi$
$683$	40.2225	1.53907	0.769536	+	0.638603i	$0.220487\pi$
$684$	0	0
$685$	73.2246	2.79777
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−8.99934	−0.342848
$690$	0	0
$691$	−30.7414	−1.16946	−0.584729	−	0.811229i	$-0.698799\pi$
$691$	−30.7414	−1.16946	−0.584729	+	0.811229i	$0.698799\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−55.7092	−2.11317
$696$	0	0
$697$	−10.1954	−0.386178
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−24.3696	−0.920428	−0.460214	−	0.887808i	$-0.652227\pi$
$701$	−24.3696	−0.920428	−0.460214	+	0.887808i	$0.652227\pi$
$702$	0	0
$703$	−18.3696	−0.692823
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−16.7931	−0.631568
$708$	0	0
$709$	−22.2957	−0.837334	−0.418667	−	0.908140i	$-0.637503\pi$
$709$	−22.2957	−0.837334	−0.418667	+	0.908140i	$0.637503\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−34.5810	−1.29507
$714$	0	0
$715$	32.1948	1.20402
$716$	0	0
$717$	0	0
$718$	0	0
$719$	14.2466	0.531310	0.265655	−	0.964068i	$-0.414412\pi$
$719$	14.2466	0.531310	0.265655	+	0.964068i	$0.414412\pi$
$720$	0	0
$721$	−9.57656	−0.356650
$722$	0	0
$723$	0	0
$724$	0	0
$725$	70.6474	2.62378
$726$	0	0
$727$	26.2762	0.974529	0.487264	−	0.873255i	$-0.337995\pi$
$727$	26.2762	0.974529	0.487264	+	0.873255i	$0.337995\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2.69762	0.0997752
$732$	0	0
$733$	−29.7464	−1.09871	−0.549354	−	0.835590i	$-0.685126\pi$
$733$	−29.7464	−1.09871	−0.549354	+	0.835590i	$0.685126\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.96227	−0.256458
$738$	0	0
$739$	20.0628	0.738021	0.369010	−	0.929425i	$-0.379697\pi$
$739$	20.0628	0.738021	0.369010	+	0.929425i	$0.379697\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	10.0948	0.370344	0.185172	−	0.982706i	$-0.440716\pi$
$743$	10.0948	0.370344	0.185172	+	0.982706i	$0.440716\pi$
$744$	0	0
$745$	−13.8030	−0.505703
$746$	0	0
$747$	0	0
$748$	0	0
$749$	16.9613	0.619752
$750$	0	0
$751$	−2.58103	−0.0941831	−0.0470916	−	0.998891i	$-0.514995\pi$
$751$	−2.58103	−0.0941831	−0.0470916	+	0.998891i	$0.514995\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	36.9023	1.34301
$756$	0	0
$757$	42.6072	1.54859	0.774293	−	0.632828i	$-0.218106\pi$
$757$	42.6072	1.54859	0.774293	+	0.632828i	$0.218106\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	6.81361	0.246993	0.123497	−	0.992345i	$-0.460589\pi$
$761$	6.81361	0.246993	0.123497	+	0.992345i	$0.460589\pi$
$762$	0	0
$763$	19.6082	0.709866
$764$	0	0
$765$	0	0
$766$	0	0
$767$	32.8417	1.18585
$768$	0	0
$769$	−9.62782	−0.347188	−0.173594	−	0.984817i	$-0.555538\pi$
$769$	−9.62782	−0.347188	−0.173594	+	0.984817i	$0.555538\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	32.6842	1.17557	0.587784	−	0.809018i	$-0.300001\pi$
$773$	32.6842	1.17557	0.587784	+	0.809018i	$0.300001\pi$
$774$	0	0
$775$	54.7258	1.96581
$776$	0	0
$777$	0	0
$778$	0	0
$779$	71.1064	2.54765
$780$	0	0
$781$	8.46003	0.302724
$782$	0	0
$783$	0	0
$784$	0	0
$785$	26.9718	0.962665
$786$	0	0
$787$	−40.6653	−1.44956	−0.724782	−	0.688979i	$-0.758059\pi$
$787$	−40.6653	−1.44956	−0.724782	+	0.688979i	$0.758059\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−13.4952	−0.479834
$792$	0	0
$793$	−25.4594	−0.904089
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−19.0019	−0.673082	−0.336541	−	0.941669i	$-0.609257\pi$
$797$	−19.0019	−0.673082	−0.336541	+	0.941669i	$0.609257\pi$
$798$	0	0
$799$	−3.67652	−0.130066
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0.304876	0.0107588
$804$	0	0
$805$	−28.4696	−1.00342
$806$	0	0
$807$	0	0
$808$	0	0
$809$	7.95386	0.279643	0.139821	−	0.990177i	$-0.455347\pi$
$809$	7.95386	0.279643	0.139821	+	0.990177i	$0.455347\pi$
$810$	0	0
$811$	−40.6097	−1.42600	−0.712999	−	0.701165i	$-0.752664\pi$
$811$	−40.6097	−1.42600	−0.712999	+	0.701165i	$0.752664\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	57.9923	2.03138
$816$	0	0
$817$	−18.8142	−0.658226
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−50.7051	−1.76962	−0.884810	−	0.465951i	$-0.845712\pi$
$821$	−50.7051	−1.76962	−0.884810	+	0.465951i	$0.845712\pi$
$822$	0	0
$823$	46.3922	1.61713	0.808564	−	0.588408i	$-0.200245\pi$
$823$	46.3922	1.61713	0.808564	+	0.588408i	$0.200245\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	34.6166	1.20374	0.601869	−	0.798595i	$-0.294423\pi$
$827$	34.6166	1.20374	0.601869	+	0.798595i	$0.294423\pi$
$828$	0	0
$829$	2.04614	0.0710653	0.0355326	−	0.999369i	$-0.488687\pi$
$829$	2.04614	0.0710653	0.0355326	+	0.999369i	$0.488687\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	−13.0836	−0.452778
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−2.11462	−0.0730049	−0.0365024	−	0.999334i	$-0.511622\pi$
$839$	−2.11462	−0.0730049	−0.0365024	+	0.999334i	$0.511622\pi$
$840$	0	0
$841$	10.8211	0.373142
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−116.258	−3.99938
$846$	0	0
$847$	9.47214	0.325466
$848$	0	0
$849$	0	0
$850$	0	0
$851$	18.6329	0.638727
$852$	0	0
$853$	22.7338	0.778392	0.389196	−	0.921155i	$-0.372753\pi$
$853$	22.7338	0.778392	0.389196	+	0.921155i	$0.372753\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.03743	−0.0695973	−0.0347986	−	0.999394i	$-0.511079\pi$
$857$	−2.03743	−0.0695973	−0.0347986	+	0.999394i	$0.511079\pi$
$858$	0	0
$859$	−15.8648	−0.541301	−0.270650	−	0.962678i	$-0.587239\pi$
$859$	−15.8648	−0.541301	−0.270650	+	0.962678i	$0.587239\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	14.8857	0.506714	0.253357	−	0.967373i	$-0.418465\pi$
$863$	14.8857	0.506714	0.253357	+	0.967373i	$0.418465\pi$
$864$	0	0
$865$	17.1323	0.582517
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−14.4330	−0.489605
$870$	0	0
$871$	36.4549	1.23523
$872$	0	0
$873$	0	0
$874$	0	0
$875$	24.9324	0.842869
$876$	0	0
$877$	−3.66844	−0.123874	−0.0619372	−	0.998080i	$-0.519728\pi$
$877$	−3.66844	−0.123874	−0.0619372	+	0.998080i	$0.519728\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	39.9695	1.34661	0.673304	−	0.739366i	$-0.264874\pi$
$881$	39.9695	1.34661	0.673304	+	0.739366i	$0.264874\pi$
$882$	0	0
$883$	−17.1028	−0.575556	−0.287778	−	0.957697i	$-0.592917\pi$
$883$	−17.1028	−0.575556	−0.287778	+	0.957697i	$0.592917\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	26.5622	0.891871	0.445936	−	0.895065i	$-0.352871\pi$
$887$	26.5622	0.891871	0.445936	+	0.895065i	$0.352871\pi$
$888$	0	0
$889$	−4.36484	−0.146392
$890$	0	0
$891$	0	0
$892$	0	0
$893$	25.6414	0.858058
$894$	0	0
$895$	−10.9773	−0.366931
$896$	0	0
$897$	0	0
$898$	0	0
$899$	30.8467	1.02880
$900$	0	0
$901$	−1.39047	−0.0463234
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−12.6957	−0.422018
$906$	0	0
$907$	42.8523	1.42289	0.711443	−	0.702744i	$-0.248042\pi$
$907$	42.8523	1.42289	0.711443	+	0.702744i	$0.248042\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	37.1141	1.22964	0.614822	−	0.788666i	$-0.289228\pi$
$911$	37.1141	1.22964	0.614822	+	0.788666i	$0.289228\pi$
$912$	0	0
$913$	8.22096	0.272074
$914$	0	0
$915$	0	0
$916$	0	0
$917$	10.6826	0.352770
$918$	0	0
$919$	23.7655	0.783950	0.391975	−	0.919976i	$-0.371792\pi$
$919$	23.7655	0.783950	0.391975	+	0.919976i	$0.371792\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−44.2973	−1.45806
$924$	0	0
$925$	−29.4872	−0.969535
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−24.7467	−0.811912	−0.405956	−	0.913893i	$-0.633061\pi$
$929$	−24.7467	−0.811912	−0.405956	+	0.913893i	$0.633061\pi$
$930$	0	0
$931$	6.97437	0.228576
$932$	0	0
$933$	0	0
$934$	0	0
$935$	4.97437	0.162679
$936$	0	0
$937$	26.9577	0.880671	0.440336	−	0.897833i	$-0.354859\pi$
$937$	26.9577	0.880671	0.440336	+	0.897833i	$0.354859\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	30.6992	1.00077	0.500383	−	0.865804i	$-0.333193\pi$
$941$	30.6992	1.00077	0.500383	+	0.865804i	$0.333193\pi$
$942$	0	0
$943$	−72.1255	−2.34873
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−28.3892	−0.922525	−0.461262	−	0.887264i	$-0.652603\pi$
$947$	−28.3892	−0.922525	−0.461262	+	0.887264i	$0.652603\pi$
$948$	0	0
$949$	−1.59635	−0.0518197
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−39.1278	−1.26747	−0.633737	−	0.773549i	$-0.718480\pi$
$953$	−39.1278	−1.26747	−0.633737	+	0.773549i	$0.718480\pi$
$954$	0	0
$955$	14.2166	0.460037
$956$	0	0
$957$	0	0
$958$	0	0
$959$	18.1954	0.587560
$960$	0	0
$961$	−7.10510	−0.229197
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−65.3050	−2.10224
$966$	0	0
$967$	39.8719	1.28219	0.641097	−	0.767460i	$-0.278480\pi$
$967$	39.8719	1.28219	0.641097	+	0.767460i	$0.278480\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−16.8891	−0.541996	−0.270998	−	0.962580i	$-0.587354\pi$
$971$	−16.8891	−0.541996	−0.270998	+	0.962580i	$0.587354\pi$
$972$	0	0
$973$	−13.8430	−0.443787
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−21.4487	−0.686205	−0.343103	−	0.939298i	$-0.611478\pi$
$977$	−21.4487	−0.686205	−0.343103	+	0.939298i	$0.611478\pi$
$978$	0	0
$979$	−2.81665	−0.0900205
$980$	0	0
$981$	0	0
$982$	0	0
$983$	48.8004	1.55649	0.778245	−	0.627960i	$-0.216110\pi$
$983$	48.8004	1.55649	0.778245	+	0.627960i	$0.216110\pi$
$984$	0	0
$985$	71.4600	2.27690
$986$	0	0
$987$	0	0
$988$	0	0
$989$	19.0839	0.606832
$990$	0	0
$991$	5.00203	0.158895	0.0794474	−	0.996839i	$-0.474684\pi$
$991$	5.00203	0.158895	0.0794474	+	0.996839i	$0.474684\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	60.9632	1.93266
$996$	0	0
$997$	−54.7966	−1.73543	−0.867713	−	0.497066i	$-0.834411\pi$
$997$	−54.7966	−1.73543	−0.867713	+	0.497066i	$0.834411\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8568.2.a.bj.1.1		4
3.2	odd	2		952.2.a.g.1.3	✓	4
12.11	even	2		1904.2.a.q.1.2		4
21.20	even	2		6664.2.a.o.1.2		4
24.5	odd	2		7616.2.a.bj.1.2		4
24.11	even	2		7616.2.a.bp.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
952.2.a.g.1.3	✓	4	3.2	odd	2
1904.2.a.q.1.2		4	12.11	even	2
6664.2.a.o.1.2		4	21.20	even	2
7616.2.a.bj.1.2		4	24.5	odd	2
7616.2.a.bp.1.3		4	24.11	even	2
8568.2.a.bj.1.1		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8568 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8568.a (trivial)

\(f(q)\)	\(=\)	\(q-4.02435 q^{5} -1.00000 q^{7} +1.23607 q^{11} -6.47214 q^{13} -1.00000 q^{17} +6.97437 q^{19} -7.07433 q^{23} +11.1954 q^{25} +6.31040 q^{29} +4.88824 q^{31} +4.02435 q^{35} -2.63387 q^{37} +10.1954 q^{41} -2.69762 q^{43} +3.67652 q^{47} +1.00000 q^{49} +1.39047 q^{53} -4.97437 q^{55} -5.07433 q^{59} +3.93369 q^{61} +26.0461 q^{65} -5.63259 q^{67} +6.84431 q^{71} +0.246650 q^{73} -1.23607 q^{77} -11.6765 q^{79} +6.65089 q^{83} +4.02435 q^{85} -2.27872 q^{89} +6.47214 q^{91} -28.0673 q^{95} +10.7069 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{5} - 4 q^{7} - 4 q^{11} - 8 q^{13} - 4 q^{17} + 14 q^{19} - 8 q^{23} + 11 q^{25} - 4 q^{29} + 5 q^{31} - q^{35} - 4 q^{37} + 7 q^{41} + 19 q^{43} - 8 q^{47} + 4 q^{49} - 5 q^{53} - 6 q^{55} - 23 q^{61}+ \cdots - 15 q^{97}+O(q^{100}) \) 4 * q + q^5 - 4 * q^7 - 4 * q^11 - 8 * q^13 - 4 * q^17 + 14 * q^19 - 8 * q^23 + 11 * q^25 - 4 * q^29 + 5 * q^31 - q^35 - 4 * q^37 + 7 * q^41 + 19 * q^43 - 8 * q^47 + 4 * q^49 - 5 * q^53 - 6 * q^55 - 23 * q^61 + 8 * q^65 + 15 * q^67 - 2 * q^71 - 5 * q^73 + 4 * q^77 - 24 * q^79 - 10 * q^83 - q^85 + 16 * q^89 + 8 * q^91 - 22 * q^95 - 15 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more