LMFDB - Embedded newform 5712.2.a.bs.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5712,2,Mod(1,5712)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5712, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5712.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5712 = 2^{4} \cdot 3 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5712.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-3,0,-2,0,3,0,3,0,2,0,-2,0,2,0,-3,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	yes
Analytic conductor:	$45.6105496346$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2856)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.34292$ of defining polynomial
Character	$\chi$	$=$	5712.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.34292 q^{5} +1.00000 q^{7} +1.00000 q^{9} +2.19656 q^{11} -3.63565 q^{13} -1.34292 q^{15} -1.00000 q^{17} -1.63565 q^{19} -1.00000 q^{21} +7.17513 q^{23} -3.19656 q^{25} -1.00000 q^{27} -6.97858 q^{29} +2.85363 q^{31} -2.19656 q^{33} +1.34292 q^{35} -5.83221 q^{37} +3.63565 q^{39} +10.7146 q^{41} +6.19656 q^{43} +1.34292 q^{45} +2.85363 q^{47} +1.00000 q^{49} +1.00000 q^{51} +8.51806 q^{53} +2.94981 q^{55} +1.63565 q^{57} +1.14637 q^{59} +6.81079 q^{61} +1.00000 q^{63} -4.88240 q^{65} -2.68585 q^{67} -7.17513 q^{69} +7.27131 q^{71} -4.29273 q^{73} +3.19656 q^{75} +2.19656 q^{77} +9.53948 q^{79} +1.00000 q^{81} +10.3503 q^{83} -1.34292 q^{85} +6.97858 q^{87} -18.6430 q^{89} -3.63565 q^{91} -2.85363 q^{93} -2.19656 q^{95} +12.3503 q^{97} +2.19656 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 2 q^{5} + 3 q^{7} + 3 q^{9} + 2 q^{11} - 2 q^{13} + 2 q^{15} - 3 q^{17} + 4 q^{19} - 3 q^{21} + 2 q^{23} - 5 q^{25} - 3 q^{27} - 6 q^{29} + 10 q^{31} - 2 q^{33} - 2 q^{35} - 4 q^{37} + 2 q^{39}+ \cdots + 2 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 2 * q^5 + 3 * q^7 + 3 * q^9 + 2 * q^11 - 2 * q^13 + 2 * q^15 - 3 * q^17 + 4 * q^19 - 3 * q^21 + 2 * q^23 - 5 * q^25 - 3 * q^27 - 6 * q^29 + 10 * q^31 - 2 * q^33 - 2 * q^35 - 4 * q^37 + 2 * q^39 + 2 * q^41 + 14 * q^43 - 2 * q^45 + 10 * q^47 + 3 * q^49 + 3 * q^51 + 12 * q^55 - 4 * q^57 + 2 * q^59 - 8 * q^61 + 3 * q^63 + 2 * q^65 + 4 * q^67 - 2 * q^69 + 4 * q^71 - 10 * q^73 + 5 * q^75 + 2 * q^77 + 18 * q^79 + 3 * q^81 - 8 * q^83 + 2 * q^85 + 6 * q^87 - 14 * q^89 - 2 * q^91 - 10 * q^93 - 2 * q^95 - 2 * q^97 + 2 * q^99

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.34292	0.600573	0.300287	−	0.953849i	$-0.402918\pi$
$5$	1.34292	0.600573	0.300287	+	0.953849i	$0.402918\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.19656	0.662287	0.331144	−	0.943580i	$-0.392566\pi$
$11$	2.19656	0.662287	0.331144	+	0.943580i	$0.392566\pi$
$12$	0	0
$13$	−3.63565	−1.00835	−0.504175	−	0.863602i	$-0.668203\pi$
$13$	−3.63565	−1.00835	−0.504175	+	0.863602i	$0.668203\pi$
$14$	0	0
$15$	−1.34292	−0.346741
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−1.63565	−0.375245	−0.187622	−	0.982241i	$-0.560078\pi$
$19$	−1.63565	−0.375245	−0.187622	+	0.982241i	$0.560078\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	7.17513	1.49612	0.748060	−	0.663632i	$-0.230986\pi$
$23$	7.17513	1.49612	0.748060	+	0.663632i	$0.230986\pi$
$24$	0	0
$25$	−3.19656	−0.639312
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.97858	−1.29589	−0.647945	−	0.761687i	$-0.724371\pi$
$29$	−6.97858	−1.29589	−0.647945	+	0.761687i	$0.724371\pi$
$30$	0	0
$31$	2.85363	0.512528	0.256264	−	0.966607i	$-0.417508\pi$
$31$	2.85363	0.512528	0.256264	+	0.966607i	$0.417508\pi$
$32$	0	0
$33$	−2.19656	−0.382372
$34$	0	0
$35$	1.34292	0.226995
$36$	0	0
$37$	−5.83221	−0.958810	−0.479405	−	0.877594i	$-0.659147\pi$
$37$	−5.83221	−0.958810	−0.479405	+	0.877594i	$0.659147\pi$
$38$	0	0
$39$	3.63565	0.582171
$40$	0	0
$41$	10.7146	1.67334	0.836671	−	0.547706i	$-0.184499\pi$
$41$	10.7146	1.67334	0.836671	+	0.547706i	$0.184499\pi$
$42$	0	0
$43$	6.19656	0.944966	0.472483	−	0.881340i	$-0.343358\pi$
$43$	6.19656	0.944966	0.472483	+	0.881340i	$0.343358\pi$
$44$	0	0
$45$	1.34292	0.200191
$46$	0	0
$47$	2.85363	0.416245	0.208123	−	0.978103i	$-0.433265\pi$
$47$	2.85363	0.416245	0.208123	+	0.978103i	$0.433265\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.00000	0.140028
$52$	0	0
$53$	8.51806	1.17005	0.585023	−	0.811017i	$-0.301086\pi$
$53$	8.51806	1.17005	0.585023	+	0.811017i	$0.301086\pi$
$54$	0	0
$55$	2.94981	0.397752
$56$	0	0
$57$	1.63565	0.216648
$58$	0	0
$59$	1.14637	0.149244	0.0746220	−	0.997212i	$-0.476225\pi$
$59$	1.14637	0.149244	0.0746220	+	0.997212i	$0.476225\pi$
$60$	0	0
$61$	6.81079	0.872032	0.436016	−	0.899939i	$-0.356389\pi$
$61$	6.81079	0.872032	0.436016	+	0.899939i	$0.356389\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−4.88240	−0.605588
$66$	0	0
$67$	−2.68585	−0.328128	−0.164064	−	0.986450i	$-0.552460\pi$
$67$	−2.68585	−0.328128	−0.164064	+	0.986450i	$0.552460\pi$
$68$	0	0
$69$	−7.17513	−0.863785
$70$	0	0
$71$	7.27131	0.862946	0.431473	−	0.902126i	$-0.357994\pi$
$71$	7.27131	0.862946	0.431473	+	0.902126i	$0.357994\pi$
$72$	0	0
$73$	−4.29273	−0.502426	−0.251213	−	0.967932i	$-0.580829\pi$
$73$	−4.29273	−0.502426	−0.251213	+	0.967932i	$0.580829\pi$
$74$	0	0
$75$	3.19656	0.369107
$76$	0	0
$77$	2.19656	0.250321
$78$	0	0
$79$	9.53948	1.07328	0.536638	−	0.843813i	$-0.319694\pi$
$79$	9.53948	1.07328	0.536638	+	0.843813i	$0.319694\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.3503	1.13609	0.568045	−	0.822998i	$-0.307700\pi$
$83$	10.3503	1.13609	0.568045	+	0.822998i	$0.307700\pi$
$84$	0	0
$85$	−1.34292	−0.145660
$86$	0	0
$87$	6.97858	0.748182
$88$	0	0
$89$	−18.6430	−1.97615	−0.988077	−	0.153960i	$-0.950797\pi$
$89$	−18.6430	−1.97615	−0.988077	+	0.153960i	$0.950797\pi$
$90$	0	0
$91$	−3.63565	−0.381120
$92$	0	0
$93$	−2.85363	−0.295908
$94$	0	0
$95$	−2.19656	−0.225362
$96$	0	0
$97$	12.3503	1.25398	0.626990	−	0.779027i	$-0.284287\pi$
$97$	12.3503	1.25398	0.626990	+	0.779027i	$0.284287\pi$
$98$	0	0
$99$	2.19656	0.220762
$100$	0	0
$101$	−8.85363	−0.880970	−0.440485	−	0.897760i	$-0.645193\pi$
$101$	−8.85363	−0.880970	−0.440485	+	0.897760i	$0.645193\pi$
$102$	0	0
$103$	−16.3790	−1.61387	−0.806937	−	0.590637i	$-0.798876\pi$
$103$	−16.3790	−1.61387	−0.806937	+	0.590637i	$0.798876\pi$
$104$	0	0
$105$	−1.34292	−0.131056
$106$	0	0
$107$	−16.7392	−1.61824	−0.809119	−	0.587646i	$-0.800055\pi$
$107$	−16.7392	−1.61824	−0.809119	+	0.587646i	$0.800055\pi$
$108$	0	0
$109$	9.49663	0.909613	0.454806	−	0.890590i	$-0.349708\pi$
$109$	9.49663	0.909613	0.454806	+	0.890590i	$0.349708\pi$
$110$	0	0
$111$	5.83221	0.553569
$112$	0	0
$113$	14.7392	1.38654	0.693272	−	0.720676i	$-0.256168\pi$
$113$	14.7392	1.38654	0.693272	+	0.720676i	$0.256168\pi$
$114$	0	0
$115$	9.63565	0.898529
$116$	0	0
$117$	−3.63565	−0.336116
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−6.17513	−0.561376
$122$	0	0
$123$	−10.7146	−0.966104
$124$	0	0
$125$	−11.0073	−0.984527
$126$	0	0
$127$	14.7820	1.31169	0.655846	−	0.754895i	$-0.272312\pi$
$127$	14.7820	1.31169	0.655846	+	0.754895i	$0.272312\pi$
$128$	0	0
$129$	−6.19656	−0.545576
$130$	0	0
$131$	8.90696	0.778205	0.389102	−	0.921194i	$-0.372785\pi$
$131$	8.90696	0.778205	0.389102	+	0.921194i	$0.372785\pi$
$132$	0	0
$133$	−1.63565	−0.141829
$134$	0	0
$135$	−1.34292	−0.115580
$136$	0	0
$137$	17.1035	1.46125	0.730626	−	0.682778i	$-0.239228\pi$
$137$	17.1035	1.46125	0.730626	+	0.682778i	$0.239228\pi$
$138$	0	0
$139$	−6.10038	−0.517428	−0.258714	−	0.965954i	$-0.583299\pi$
$139$	−6.10038	−0.517428	−0.258714	+	0.965954i	$0.583299\pi$
$140$	0	0
$141$	−2.85363	−0.240319
$142$	0	0
$143$	−7.98592	−0.667816
$144$	0	0
$145$	−9.37169	−0.778277
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	3.70727	0.303711	0.151856	−	0.988403i	$-0.451475\pi$
$149$	3.70727	0.303711	0.151856	+	0.988403i	$0.451475\pi$
$150$	0	0
$151$	−0.978577	−0.0796355	−0.0398177	−	0.999207i	$-0.512678\pi$
$151$	−0.978577	−0.0796355	−0.0398177	+	0.999207i	$0.512678\pi$
$152$	0	0
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	3.83221	0.307811
$156$	0	0
$157$	2.71462	0.216650	0.108325	−	0.994116i	$-0.465451\pi$
$157$	2.71462	0.216650	0.108325	+	0.994116i	$0.465451\pi$
$158$	0	0
$159$	−8.51806	−0.675526
$160$	0	0
$161$	7.17513	0.565480
$162$	0	0
$163$	22.9357	1.79647	0.898233	−	0.439520i	$-0.144852\pi$
$163$	22.9357	1.79647	0.898233	+	0.439520i	$0.144852\pi$
$164$	0	0
$165$	−2.94981	−0.229642
$166$	0	0
$167$	−8.51385	−0.658821	−0.329411	−	0.944187i	$-0.606850\pi$
$167$	−8.51385	−0.658821	−0.329411	+	0.944187i	$0.606850\pi$
$168$	0	0
$169$	0.217980	0.0167677
$170$	0	0
$171$	−1.63565	−0.125082
$172$	0	0
$173$	24.6718	1.87576	0.937880	−	0.346960i	$-0.112786\pi$
$173$	24.6718	1.87576	0.937880	+	0.346960i	$0.112786\pi$
$174$	0	0
$175$	−3.19656	−0.241637
$176$	0	0
$177$	−1.14637	−0.0861661
$178$	0	0
$179$	2.68585	0.200750	0.100375	−	0.994950i	$-0.467996\pi$
$179$	2.68585	0.200750	0.100375	+	0.994950i	$0.467996\pi$
$180$	0	0
$181$	−0.292731	−0.0217585	−0.0108793	−	0.999941i	$-0.503463\pi$
$181$	−0.292731	−0.0217585	−0.0108793	+	0.999941i	$0.503463\pi$
$182$	0	0
$183$	−6.81079	−0.503468
$184$	0	0
$185$	−7.83221	−0.575836
$186$	0	0
$187$	−2.19656	−0.160628
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−0.978577	−0.0708074	−0.0354037	−	0.999373i	$-0.511272\pi$
$191$	−0.978577	−0.0708074	−0.0354037	+	0.999373i	$0.511272\pi$
$192$	0	0
$193$	−12.7679	−0.919057	−0.459528	−	0.888163i	$-0.651982\pi$
$193$	−12.7679	−0.919057	−0.459528	+	0.888163i	$0.651982\pi$
$194$	0	0
$195$	4.88240	0.349636
$196$	0	0
$197$	−3.46787	−0.247075	−0.123538	−	0.992340i	$-0.539424\pi$
$197$	−3.46787	−0.247075	−0.123538	+	0.992340i	$0.539424\pi$
$198$	0	0
$199$	18.7104	1.32635	0.663173	−	0.748466i	$-0.269209\pi$
$199$	18.7104	1.32635	0.663173	+	0.748466i	$0.269209\pi$
$200$	0	0
$201$	2.68585	0.189445
$202$	0	0
$203$	−6.97858	−0.489800
$204$	0	0
$205$	14.3889	1.00496
$206$	0	0
$207$	7.17513	0.498706
$208$	0	0
$209$	−3.59281	−0.248520
$210$	0	0
$211$	24.6184	1.69480	0.847402	−	0.530952i	$-0.178166\pi$
$211$	24.6184	1.69480	0.847402	+	0.530952i	$0.178166\pi$
$212$	0	0
$213$	−7.27131	−0.498222
$214$	0	0
$215$	8.32150	0.567522
$216$	0	0
$217$	2.85363	0.193717
$218$	0	0
$219$	4.29273	0.290076
$220$	0	0
$221$	3.63565	0.244561
$222$	0	0
$223$	−0.906962	−0.0607347	−0.0303673	−	0.999539i	$-0.509668\pi$
$223$	−0.906962	−0.0607347	−0.0303673	+	0.999539i	$0.509668\pi$
$224$	0	0
$225$	−3.19656	−0.213104
$226$	0	0
$227$	−6.94981	−0.461275	−0.230637	−	0.973040i	$-0.574081\pi$
$227$	−6.94981	−0.461275	−0.230637	+	0.973040i	$0.574081\pi$
$228$	0	0
$229$	−17.1365	−1.13241	−0.566206	−	0.824264i	$-0.691589\pi$
$229$	−17.1365	−1.13241	−0.566206	+	0.824264i	$0.691589\pi$
$230$	0	0
$231$	−2.19656	−0.144523
$232$	0	0
$233$	13.9038	0.910870	0.455435	−	0.890269i	$-0.349484\pi$
$233$	13.9038	0.910870	0.455435	+	0.890269i	$0.349484\pi$
$234$	0	0
$235$	3.83221	0.249986
$236$	0	0
$237$	−9.53948	−0.619656
$238$	0	0
$239$	−19.3043	−1.24869	−0.624345	−	0.781148i	$-0.714634\pi$
$239$	−19.3043	−1.24869	−0.624345	+	0.781148i	$0.714634\pi$
$240$	0	0
$241$	7.76481	0.500175	0.250088	−	0.968223i	$-0.419541\pi$
$241$	7.76481	0.500175	0.250088	+	0.968223i	$0.419541\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.34292	0.0857962
$246$	0	0
$247$	5.94667	0.378378
$248$	0	0
$249$	−10.3503	−0.655922
$250$	0	0
$251$	−20.4177	−1.28875	−0.644376	−	0.764709i	$-0.722883\pi$
$251$	−20.4177	−1.28875	−0.644376	+	0.764709i	$0.722883\pi$
$252$	0	0
$253$	15.7606	0.990860
$254$	0	0
$255$	1.34292	0.0840971
$256$	0	0
$257$	−14.4752	−0.902939	−0.451469	−	0.892287i	$-0.649100\pi$
$257$	−14.4752	−0.902939	−0.451469	+	0.892287i	$0.649100\pi$
$258$	0	0
$259$	−5.83221	−0.362396
$260$	0	0
$261$	−6.97858	−0.431963
$262$	0	0
$263$	11.0790	0.683158	0.341579	−	0.939853i	$-0.389038\pi$
$263$	11.0790	0.683158	0.341579	+	0.939853i	$0.389038\pi$
$264$	0	0
$265$	11.4391	0.702698
$266$	0	0
$267$	18.6430	1.14093
$268$	0	0
$269$	−6.65708	−0.405889	−0.202945	−	0.979190i	$-0.565051\pi$
$269$	−6.65708	−0.405889	−0.202945	+	0.979190i	$0.565051\pi$
$270$	0	0
$271$	26.8641	1.63188	0.815939	−	0.578137i	$-0.196220\pi$
$271$	26.8641	1.63188	0.815939	+	0.578137i	$0.196220\pi$
$272$	0	0
$273$	3.63565	0.220040
$274$	0	0
$275$	−7.02142	−0.423408
$276$	0	0
$277$	−20.4078	−1.22619	−0.613093	−	0.790011i	$-0.710075\pi$
$277$	−20.4078	−1.22619	−0.613093	+	0.790011i	$0.710075\pi$
$278$	0	0
$279$	2.85363	0.170843
$280$	0	0
$281$	−2.05754	−0.122742	−0.0613712	−	0.998115i	$-0.519547\pi$
$281$	−2.05754	−0.122742	−0.0613712	+	0.998115i	$0.519547\pi$
$282$	0	0
$283$	−14.1825	−0.843061	−0.421530	−	0.906814i	$-0.638507\pi$
$283$	−14.1825	−0.843061	−0.421530	+	0.906814i	$0.638507\pi$
$284$	0	0
$285$	2.19656	0.130113
$286$	0	0
$287$	10.7146	0.632464
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−12.3503	−0.723986
$292$	0	0
$293$	2.19235	0.128078	0.0640391	−	0.997947i	$-0.479602\pi$
$293$	2.19235	0.128078	0.0640391	+	0.997947i	$0.479602\pi$
$294$	0	0
$295$	1.53948	0.0896320
$296$	0	0
$297$	−2.19656	−0.127457
$298$	0	0
$299$	−26.0863	−1.50861
$300$	0	0
$301$	6.19656	0.357164
$302$	0	0
$303$	8.85363	0.508628
$304$	0	0
$305$	9.14637	0.523719
$306$	0	0
$307$	5.37169	0.306579	0.153289	−	0.988181i	$-0.451013\pi$
$307$	5.37169	0.306579	0.153289	+	0.988181i	$0.451013\pi$
$308$	0	0
$309$	16.3790	0.931771
$310$	0	0
$311$	18.7434	1.06284	0.531420	−	0.847109i	$-0.321659\pi$
$311$	18.7434	1.06284	0.531420	+	0.847109i	$0.321659\pi$
$312$	0	0
$313$	−5.63986	−0.318784	−0.159392	−	0.987215i	$-0.550953\pi$
$313$	−5.63986	−0.318784	−0.159392	+	0.987215i	$0.550953\pi$
$314$	0	0
$315$	1.34292	0.0756651
$316$	0	0
$317$	−5.32885	−0.299298	−0.149649	−	0.988739i	$-0.547814\pi$
$317$	−5.32885	−0.299298	−0.149649	+	0.988739i	$0.547814\pi$
$318$	0	0
$319$	−15.3288	−0.858251
$320$	0	0
$321$	16.7392	0.934290
$322$	0	0
$323$	1.63565	0.0910102
$324$	0	0
$325$	11.6216	0.644649
$326$	0	0
$327$	−9.49663	−0.525165
$328$	0	0
$329$	2.85363	0.157326
$330$	0	0
$331$	19.9038	1.09401	0.547007	−	0.837128i	$-0.315767\pi$
$331$	19.9038	1.09401	0.547007	+	0.837128i	$0.315767\pi$
$332$	0	0
$333$	−5.83221	−0.319603
$334$	0	0
$335$	−3.60688	−0.197065
$336$	0	0
$337$	30.8108	1.67837	0.839185	−	0.543846i	$-0.183032\pi$
$337$	30.8108	1.67837	0.839185	+	0.543846i	$0.183032\pi$
$338$	0	0
$339$	−14.7392	−0.800522
$340$	0	0
$341$	6.26817	0.339441
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−9.63565	−0.518766
$346$	0	0
$347$	14.7350	0.791014	0.395507	−	0.918463i	$-0.370569\pi$
$347$	14.7350	0.791014	0.395507	+	0.918463i	$0.370569\pi$
$348$	0	0
$349$	19.6932	1.05415	0.527076	−	0.849818i	$-0.323288\pi$
$349$	19.6932	1.05415	0.527076	+	0.849818i	$0.323288\pi$
$350$	0	0
$351$	3.63565	0.194057
$352$	0	0
$353$	27.9044	1.48520	0.742602	−	0.669733i	$-0.233592\pi$
$353$	27.9044	1.48520	0.742602	+	0.669733i	$0.233592\pi$
$354$	0	0
$355$	9.76481	0.518262
$356$	0	0
$357$	1.00000	0.0529256
$358$	0	0
$359$	15.4391	0.814844	0.407422	−	0.913240i	$-0.366428\pi$
$359$	15.4391	0.814844	0.407422	+	0.913240i	$0.366428\pi$
$360$	0	0
$361$	−16.3246	−0.859191
$362$	0	0
$363$	6.17513	0.324111
$364$	0	0
$365$	−5.76481	−0.301744
$366$	0	0
$367$	35.5296	1.85463	0.927315	−	0.374281i	$-0.122110\pi$
$367$	35.5296	1.85463	0.927315	+	0.374281i	$0.122110\pi$
$368$	0	0
$369$	10.7146	0.557781
$370$	0	0
$371$	8.51806	0.442236
$372$	0	0
$373$	22.6430	1.17241	0.586205	−	0.810163i	$-0.300621\pi$
$373$	22.6430	1.17241	0.586205	+	0.810163i	$0.300621\pi$
$374$	0	0
$375$	11.0073	0.568417
$376$	0	0
$377$	25.3717	1.30671
$378$	0	0
$379$	5.95715	0.305998	0.152999	−	0.988226i	$-0.451107\pi$
$379$	5.95715	0.305998	0.152999	+	0.988226i	$0.451107\pi$
$380$	0	0
$381$	−14.7820	−0.757306
$382$	0	0
$383$	−6.57560	−0.335997	−0.167999	−	0.985787i	$-0.553730\pi$
$383$	−6.57560	−0.335997	−0.167999	+	0.985787i	$0.553730\pi$
$384$	0	0
$385$	2.94981	0.150336
$386$	0	0
$387$	6.19656	0.314989
$388$	0	0
$389$	−10.6430	−0.539622	−0.269811	−	0.962913i	$-0.586961\pi$
$389$	−10.6430	−0.539622	−0.269811	+	0.962913i	$0.586961\pi$
$390$	0	0
$391$	−7.17513	−0.362862
$392$	0	0
$393$	−8.90696	−0.449297
$394$	0	0
$395$	12.8108	0.644581
$396$	0	0
$397$	19.3963	0.973470	0.486735	−	0.873550i	$-0.338188\pi$
$397$	19.3963	0.973470	0.486735	+	0.873550i	$0.338188\pi$
$398$	0	0
$399$	1.63565	0.0818851
$400$	0	0
$401$	30.2113	1.50868	0.754339	−	0.656485i	$-0.227958\pi$
$401$	30.2113	1.50868	0.754339	+	0.656485i	$0.227958\pi$
$402$	0	0
$403$	−10.3748	−0.516807
$404$	0	0
$405$	1.34292	0.0667304
$406$	0	0
$407$	−12.8108	−0.635007
$408$	0	0
$409$	−2.52227	−0.124718	−0.0623591	−	0.998054i	$-0.519862\pi$
$409$	−2.52227	−0.124718	−0.0623591	+	0.998054i	$0.519862\pi$
$410$	0	0
$411$	−17.1035	−0.843654
$412$	0	0
$413$	1.14637	0.0564090
$414$	0	0
$415$	13.8996	0.682305
$416$	0	0
$417$	6.10038	0.298737
$418$	0	0
$419$	−22.8866	−1.11808	−0.559042	−	0.829139i	$-0.688831\pi$
$419$	−22.8866	−1.11808	−0.559042	+	0.829139i	$0.688831\pi$
$420$	0	0
$421$	2.29694	0.111946	0.0559730	−	0.998432i	$-0.482174\pi$
$421$	2.29694	0.111946	0.0559730	+	0.998432i	$0.482174\pi$
$422$	0	0
$423$	2.85363	0.138748
$424$	0	0
$425$	3.19656	0.155056
$426$	0	0
$427$	6.81079	0.329597
$428$	0	0
$429$	7.98592	0.385564
$430$	0	0
$431$	−36.1151	−1.73960	−0.869801	−	0.493403i	$-0.835753\pi$
$431$	−36.1151	−1.73960	−0.869801	+	0.493403i	$0.835753\pi$
$432$	0	0
$433$	−27.6932	−1.33085	−0.665425	−	0.746465i	$-0.731750\pi$
$433$	−27.6932	−1.33085	−0.665425	+	0.746465i	$0.731750\pi$
$434$	0	0
$435$	9.37169	0.449338
$436$	0	0
$437$	−11.7360	−0.561411
$438$	0	0
$439$	5.00314	0.238787	0.119393	−	0.992847i	$-0.461905\pi$
$439$	5.00314	0.238787	0.119393	+	0.992847i	$0.461905\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	6.37483	0.302877	0.151439	−	0.988467i	$-0.451609\pi$
$443$	6.37483	0.302877	0.151439	+	0.988467i	$0.451609\pi$
$444$	0	0
$445$	−25.0361	−1.18683
$446$	0	0
$447$	−3.70727	−0.175348
$448$	0	0
$449$	36.8009	1.73674	0.868371	−	0.495914i	$-0.165167\pi$
$449$	36.8009	1.73674	0.868371	+	0.495914i	$0.165167\pi$
$450$	0	0
$451$	23.5353	1.10823
$452$	0	0
$453$	0.978577	0.0459776
$454$	0	0
$455$	−4.88240	−0.228891
$456$	0	0
$457$	−0.0533277	−0.00249456	−0.00124728	−	0.999999i	$-0.500397\pi$
$457$	−0.0533277	−0.00249456	−0.00124728	+	0.999999i	$0.500397\pi$
$458$	0	0
$459$	1.00000	0.0466760
$460$	0	0
$461$	−26.2829	−1.22412	−0.612058	−	0.790813i	$-0.709658\pi$
$461$	−26.2829	−1.22412	−0.612058	+	0.790813i	$0.709658\pi$
$462$	0	0
$463$	22.3503	1.03871	0.519353	−	0.854560i	$-0.326173\pi$
$463$	22.3503	1.03871	0.519353	+	0.854560i	$0.326173\pi$
$464$	0	0
$465$	−3.83221	−0.177715
$466$	0	0
$467$	−18.0147	−0.833621	−0.416810	−	0.908994i	$-0.636852\pi$
$467$	−18.0147	−0.833621	−0.416810	+	0.908994i	$0.636852\pi$
$468$	0	0
$469$	−2.68585	−0.124021
$470$	0	0
$471$	−2.71462	−0.125083
$472$	0	0
$473$	13.6111	0.625839
$474$	0	0
$475$	5.22846	0.239898
$476$	0	0
$477$	8.51806	0.390015
$478$	0	0
$479$	26.7293	1.22129	0.610647	−	0.791903i	$-0.290910\pi$
$479$	26.7293	1.22129	0.610647	+	0.791903i	$0.290910\pi$
$480$	0	0
$481$	21.2039	0.966815
$482$	0	0
$483$	−7.17513	−0.326480
$484$	0	0
$485$	16.5855	0.753107
$486$	0	0
$487$	−8.97858	−0.406858	−0.203429	−	0.979090i	$-0.565209\pi$
$487$	−8.97858	−0.406858	−0.203429	+	0.979090i	$0.565209\pi$
$488$	0	0
$489$	−22.9357	−1.03719
$490$	0	0
$491$	−23.7220	−1.07056	−0.535279	−	0.844676i	$-0.679793\pi$
$491$	−23.7220	−1.07056	−0.535279	+	0.844676i	$0.679793\pi$
$492$	0	0
$493$	6.97858	0.314299
$494$	0	0
$495$	2.94981	0.132584
$496$	0	0
$497$	7.27131	0.326163
$498$	0	0
$499$	42.2400	1.89092	0.945461	−	0.325734i	$-0.105611\pi$
$499$	42.2400	1.89092	0.945461	+	0.325734i	$0.105611\pi$
$500$	0	0
$501$	8.51385	0.380371
$502$	0	0
$503$	32.4366	1.44628	0.723138	−	0.690704i	$-0.242699\pi$
$503$	32.4366	1.44628	0.723138	+	0.690704i	$0.242699\pi$
$504$	0	0
$505$	−11.8898	−0.529087
$506$	0	0
$507$	−0.217980	−0.00968085
$508$	0	0
$509$	−13.5212	−0.599316	−0.299658	−	0.954047i	$-0.596873\pi$
$509$	−13.5212	−0.599316	−0.299658	+	0.954047i	$0.596873\pi$
$510$	0	0
$511$	−4.29273	−0.189899
$512$	0	0
$513$	1.63565	0.0722159
$514$	0	0
$515$	−21.9958	−0.969250
$516$	0	0
$517$	6.26817	0.275674
$518$	0	0
$519$	−24.6718	−1.08297
$520$	0	0
$521$	−10.9927	−0.481597	−0.240798	−	0.970575i	$-0.577409\pi$
$521$	−10.9927	−0.481597	−0.240798	+	0.970575i	$0.577409\pi$
$522$	0	0
$523$	19.8652	0.868644	0.434322	−	0.900758i	$-0.356988\pi$
$523$	19.8652	0.868644	0.434322	+	0.900758i	$0.356988\pi$
$524$	0	0
$525$	3.19656	0.139509
$526$	0	0
$527$	−2.85363	−0.124306
$528$	0	0
$529$	28.4826	1.23837
$530$	0	0
$531$	1.14637	0.0497480
$532$	0	0
$533$	−38.9546	−1.68731
$534$	0	0
$535$	−22.4794	−0.971870
$536$	0	0
$537$	−2.68585	−0.115903
$538$	0	0
$539$	2.19656	0.0946124
$540$	0	0
$541$	−34.7581	−1.49437	−0.747183	−	0.664618i	$-0.768594\pi$
$541$	−34.7581	−1.49437	−0.747183	+	0.664618i	$0.768594\pi$
$542$	0	0
$543$	0.292731	0.0125623
$544$	0	0
$545$	12.7533	0.546289
$546$	0	0
$547$	−39.0178	−1.66828	−0.834141	−	0.551551i	$-0.814036\pi$
$547$	−39.0178	−1.66828	−0.834141	+	0.551551i	$0.814036\pi$
$548$	0	0
$549$	6.81079	0.290677
$550$	0	0
$551$	11.4145	0.486276
$552$	0	0
$553$	9.53948	0.405660
$554$	0	0
$555$	7.83221	0.332459
$556$	0	0
$557$	−17.0544	−0.722618	−0.361309	−	0.932446i	$-0.617670\pi$
$557$	−17.0544	−0.722618	−0.361309	+	0.932446i	$0.617670\pi$
$558$	0	0
$559$	−22.5285	−0.952856
$560$	0	0
$561$	2.19656	0.0927387
$562$	0	0
$563$	12.1432	0.511776	0.255888	−	0.966706i	$-0.417632\pi$
$563$	12.1432	0.511776	0.255888	+	0.966706i	$0.417632\pi$
$564$	0	0
$565$	19.7936	0.832722
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	24.0147	1.00675	0.503374	−	0.864069i	$-0.332092\pi$
$569$	24.0147	1.00675	0.503374	+	0.864069i	$0.332092\pi$
$570$	0	0
$571$	−29.9572	−1.25367	−0.626834	−	0.779153i	$-0.715650\pi$
$571$	−29.9572	−1.25367	−0.626834	+	0.779153i	$0.715650\pi$
$572$	0	0
$573$	0.978577	0.0408806
$574$	0	0
$575$	−22.9357	−0.956486
$576$	0	0
$577$	−35.6932	−1.48593	−0.742964	−	0.669332i	$-0.766580\pi$
$577$	−35.6932	−1.48593	−0.742964	+	0.669332i	$0.766580\pi$
$578$	0	0
$579$	12.7679	0.530618
$580$	0	0
$581$	10.3503	0.429402
$582$	0	0
$583$	18.7104	0.774906
$584$	0	0
$585$	−4.88240	−0.201863
$586$	0	0
$587$	−45.2039	−1.86576	−0.932882	−	0.360181i	$-0.882715\pi$
$587$	−45.2039	−1.86576	−0.932882	+	0.360181i	$0.882715\pi$
$588$	0	0
$589$	−4.66756	−0.192323
$590$	0	0
$591$	3.46787	0.142649
$592$	0	0
$593$	38.8683	1.59613	0.798065	−	0.602571i	$-0.205857\pi$
$593$	38.8683	1.59613	0.798065	+	0.602571i	$0.205857\pi$
$594$	0	0
$595$	−1.34292	−0.0550545
$596$	0	0
$597$	−18.7104	−0.765766
$598$	0	0
$599$	−9.76481	−0.398979	−0.199490	−	0.979900i	$-0.563928\pi$
$599$	−9.76481	−0.398979	−0.199490	+	0.979900i	$0.563928\pi$
$600$	0	0
$601$	21.4109	0.873371	0.436685	−	0.899614i	$-0.356152\pi$
$601$	21.4109	0.873371	0.436685	+	0.899614i	$0.356152\pi$
$602$	0	0
$603$	−2.68585	−0.109376
$604$	0	0
$605$	−8.29273	−0.337147
$606$	0	0
$607$	−6.31729	−0.256411	−0.128205	−	0.991748i	$-0.540922\pi$
$607$	−6.31729	−0.256411	−0.128205	+	0.991748i	$0.540922\pi$
$608$	0	0
$609$	6.97858	0.282786
$610$	0	0
$611$	−10.3748	−0.419721
$612$	0	0
$613$	37.5682	1.51737	0.758684	−	0.651459i	$-0.225843\pi$
$613$	37.5682	1.51737	0.758684	+	0.651459i	$0.225843\pi$
$614$	0	0
$615$	−14.3889	−0.580217
$616$	0	0
$617$	28.3503	1.14134	0.570669	−	0.821180i	$-0.306684\pi$
$617$	28.3503	1.14134	0.570669	+	0.821180i	$0.306684\pi$
$618$	0	0
$619$	11.4391	0.459776	0.229888	−	0.973217i	$-0.426164\pi$
$619$	11.4391	0.459776	0.229888	+	0.973217i	$0.426164\pi$
$620$	0	0
$621$	−7.17513	−0.287928
$622$	0	0
$623$	−18.6430	−0.746916
$624$	0	0
$625$	1.20077	0.0480307
$626$	0	0
$627$	3.59281	0.143483
$628$	0	0
$629$	5.83221	0.232546
$630$	0	0
$631$	−29.7753	−1.18534	−0.592668	−	0.805447i	$-0.701925\pi$
$631$	−29.7753	−1.18534	−0.592668	+	0.805447i	$0.701925\pi$
$632$	0	0
$633$	−24.6184	−0.978495
$634$	0	0
$635$	19.8511	0.787767
$636$	0	0
$637$	−3.63565	−0.144050
$638$	0	0
$639$	7.27131	0.287649
$640$	0	0
$641$	−30.6044	−1.20880	−0.604400	−	0.796681i	$-0.706587\pi$
$641$	−30.6044	−1.20880	−0.604400	+	0.796681i	$0.706587\pi$
$642$	0	0
$643$	−16.2008	−0.638896	−0.319448	−	0.947604i	$-0.603497\pi$
$643$	−16.2008	−0.638896	−0.319448	+	0.947604i	$0.603497\pi$
$644$	0	0
$645$	−8.32150	−0.327659
$646$	0	0
$647$	−28.8438	−1.13397	−0.566983	−	0.823730i	$-0.691889\pi$
$647$	−28.8438	−1.13397	−0.566983	+	0.823730i	$0.691889\pi$
$648$	0	0
$649$	2.51806	0.0988424
$650$	0	0
$651$	−2.85363	−0.111843
$652$	0	0
$653$	27.3331	1.06963	0.534813	−	0.844971i	$-0.320382\pi$
$653$	27.3331	1.06963	0.534813	+	0.844971i	$0.320382\pi$
$654$	0	0
$655$	11.9614	0.467369
$656$	0	0
$657$	−4.29273	−0.167475
$658$	0	0
$659$	−5.87506	−0.228860	−0.114430	−	0.993431i	$-0.536504\pi$
$659$	−5.87506	−0.228860	−0.114430	+	0.993431i	$0.536504\pi$
$660$	0	0
$661$	−15.2144	−0.591771	−0.295886	−	0.955223i	$-0.595615\pi$
$661$	−15.2144	−0.591771	−0.295886	+	0.955223i	$0.595615\pi$
$662$	0	0
$663$	−3.63565	−0.141197
$664$	0	0
$665$	−2.19656	−0.0851788
$666$	0	0
$667$	−50.0722	−1.93880
$668$	0	0
$669$	0.906962	0.0350652
$670$	0	0
$671$	14.9603	0.577536
$672$	0	0
$673$	0.829076	0.0319585	0.0159793	−	0.999872i	$-0.494913\pi$
$673$	0.829076	0.0319585	0.0159793	+	0.999872i	$0.494913\pi$
$674$	0	0
$675$	3.19656	0.123036
$676$	0	0
$677$	20.8066	0.799662	0.399831	−	0.916589i	$-0.369069\pi$
$677$	20.8066	0.799662	0.399831	+	0.916589i	$0.369069\pi$
$678$	0	0
$679$	12.3503	0.473960
$680$	0	0
$681$	6.94981	0.266317
$682$	0	0
$683$	−25.4103	−0.972299	−0.486150	−	0.873876i	$-0.661599\pi$
$683$	−25.4103	−0.972299	−0.486150	+	0.873876i	$0.661599\pi$
$684$	0	0
$685$	22.9687	0.877589
$686$	0	0
$687$	17.1365	0.653798
$688$	0	0
$689$	−30.9687	−1.17981
$690$	0	0
$691$	32.7826	1.24711	0.623555	−	0.781779i	$-0.285688\pi$
$691$	32.7826	1.24711	0.623555	+	0.781779i	$0.285688\pi$
$692$	0	0
$693$	2.19656	0.0834403
$694$	0	0
$695$	−8.19235	−0.310753
$696$	0	0
$697$	−10.7146	−0.405845
$698$	0	0
$699$	−13.9038	−0.525891
$700$	0	0
$701$	24.6002	0.929135	0.464568	−	0.885538i	$-0.346210\pi$
$701$	24.6002	0.929135	0.464568	+	0.885538i	$0.346210\pi$
$702$	0	0
$703$	9.53948	0.359788
$704$	0	0
$705$	−3.83221	−0.144329
$706$	0	0
$707$	−8.85363	−0.332975
$708$	0	0
$709$	35.0852	1.31765	0.658827	−	0.752295i	$-0.271053\pi$
$709$	35.0852	1.31765	0.658827	+	0.752295i	$0.271053\pi$
$710$	0	0
$711$	9.53948	0.357758
$712$	0	0
$713$	20.4752	0.766803
$714$	0	0
$715$	−10.7245	−0.401073
$716$	0	0
$717$	19.3043	0.720932
$718$	0	0
$719$	5.64408	0.210489	0.105244	−	0.994446i	$-0.466438\pi$
$719$	5.64408	0.210489	0.105244	+	0.994446i	$0.466438\pi$
$720$	0	0
$721$	−16.3790	−0.609987
$722$	0	0
$723$	−7.76481	−0.288776
$724$	0	0
$725$	22.3074	0.828477
$726$	0	0
$727$	32.3650	1.20035	0.600175	−	0.799869i	$-0.295098\pi$
$727$	32.3650	1.20035	0.600175	+	0.799869i	$0.295098\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−6.19656	−0.229188
$732$	0	0
$733$	7.90804	0.292090	0.146045	−	0.989278i	$-0.453346\pi$
$733$	7.90804	0.292090	0.146045	+	0.989278i	$0.453346\pi$
$734$	0	0
$735$	−1.34292	−0.0495345
$736$	0	0
$737$	−5.89962	−0.217315
$738$	0	0
$739$	−8.15371	−0.299939	−0.149970	−	0.988691i	$-0.547918\pi$
$739$	−8.15371	−0.299939	−0.149970	+	0.988691i	$0.547918\pi$
$740$	0	0
$741$	−5.94667	−0.218456
$742$	0	0
$743$	33.6791	1.23557	0.617783	−	0.786348i	$-0.288031\pi$
$743$	33.6791	1.23557	0.617783	+	0.786348i	$0.288031\pi$
$744$	0	0
$745$	4.97858	0.182401
$746$	0	0
$747$	10.3503	0.378697
$748$	0	0
$749$	−16.7392	−0.611636
$750$	0	0
$751$	−1.53948	−0.0561764	−0.0280882	−	0.999605i	$-0.508942\pi$
$751$	−1.53948	−0.0561764	−0.0280882	+	0.999605i	$0.508942\pi$
$752$	0	0
$753$	20.4177	0.744061
$754$	0	0
$755$	−1.31415	−0.0478270
$756$	0	0
$757$	49.4334	1.79669	0.898344	−	0.439292i	$-0.144771\pi$
$757$	49.4334	1.79669	0.898344	+	0.439292i	$0.144771\pi$
$758$	0	0
$759$	−15.7606	−0.572073
$760$	0	0
$761$	−44.7434	−1.62195	−0.810973	−	0.585083i	$-0.801062\pi$
$761$	−44.7434	−1.62195	−0.810973	+	0.585083i	$0.801062\pi$
$762$	0	0
$763$	9.49663	0.343801
$764$	0	0
$765$	−1.34292	−0.0485535
$766$	0	0
$767$	−4.16779	−0.150490
$768$	0	0
$769$	19.7507	0.712230	0.356115	−	0.934442i	$-0.384101\pi$
$769$	19.7507	0.712230	0.356115	+	0.934442i	$0.384101\pi$
$770$	0	0
$771$	14.4752	0.521312
$772$	0	0
$773$	−38.3895	−1.38078	−0.690388	−	0.723440i	$-0.742560\pi$
$773$	−38.3895	−1.38078	−0.690388	+	0.723440i	$0.742560\pi$
$774$	0	0
$775$	−9.12181	−0.327665
$776$	0	0
$777$	5.83221	0.209229
$778$	0	0
$779$	−17.5254	−0.627913
$780$	0	0
$781$	15.9718	0.571518
$782$	0	0
$783$	6.97858	0.249394
$784$	0	0
$785$	3.64552	0.130114
$786$	0	0
$787$	−15.8322	−0.564357	−0.282179	−	0.959362i	$-0.591057\pi$
$787$	−15.8322	−0.564357	−0.282179	+	0.959362i	$0.591057\pi$
$788$	0	0
$789$	−11.0790	−0.394421
$790$	0	0
$791$	14.7392	0.524065
$792$	0	0
$793$	−24.7617	−0.879313
$794$	0	0
$795$	−11.4391	−0.405703
$796$	0	0
$797$	38.5082	1.36403	0.682015	−	0.731338i	$-0.261104\pi$
$797$	38.5082	1.36403	0.682015	+	0.731338i	$0.261104\pi$
$798$	0	0
$799$	−2.85363	−0.100954
$800$	0	0
$801$	−18.6430	−0.658718
$802$	0	0
$803$	−9.42923	−0.332750
$804$	0	0
$805$	9.63565	0.339612
$806$	0	0
$807$	6.65708	0.234340
$808$	0	0
$809$	−11.9101	−0.418737	−0.209368	−	0.977837i	$-0.567141\pi$
$809$	−11.9101	−0.418737	−0.209368	+	0.977837i	$0.567141\pi$
$810$	0	0
$811$	−20.3074	−0.713090	−0.356545	−	0.934278i	$-0.616045\pi$
$811$	−20.3074	−0.713090	−0.356545	+	0.934278i	$0.616045\pi$
$812$	0	0
$813$	−26.8641	−0.942166
$814$	0	0
$815$	30.8009	1.07891
$816$	0	0
$817$	−10.1354	−0.354594
$818$	0	0
$819$	−3.63565	−0.127040
$820$	0	0
$821$	−42.2688	−1.47519	−0.737595	−	0.675243i	$-0.764039\pi$
$821$	−42.2688	−1.47519	−0.737595	+	0.675243i	$0.764039\pi$
$822$	0	0
$823$	9.06067	0.315835	0.157918	−	0.987452i	$-0.449522\pi$
$823$	9.06067	0.315835	0.157918	+	0.987452i	$0.449522\pi$
$824$	0	0
$825$	7.02142	0.244455
$826$	0	0
$827$	−24.6901	−0.858557	−0.429279	−	0.903172i	$-0.641232\pi$
$827$	−24.6901	−0.858557	−0.429279	+	0.903172i	$0.641232\pi$
$828$	0	0
$829$	−10.6712	−0.370624	−0.185312	−	0.982680i	$-0.559330\pi$
$829$	−10.6712	−0.370624	−0.185312	+	0.982680i	$0.559330\pi$
$830$	0	0
$831$	20.4078	0.707939
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	−11.4334	−0.395671
$836$	0	0
$837$	−2.85363	−0.0986360
$838$	0	0
$839$	−26.6142	−0.918825	−0.459413	−	0.888223i	$-0.651940\pi$
$839$	−26.6142	−0.918825	−0.459413	+	0.888223i	$0.651940\pi$
$840$	0	0
$841$	19.7005	0.679329
$842$	0	0
$843$	2.05754	0.0708654
$844$	0	0
$845$	0.292731	0.0100703
$846$	0	0
$847$	−6.17513	−0.212180
$848$	0	0
$849$	14.1825	0.486741
$850$	0	0
$851$	−41.8469	−1.43449
$852$	0	0
$853$	−33.9964	−1.16401	−0.582007	−	0.813184i	$-0.697733\pi$
$853$	−33.9964	−1.16401	−0.582007	+	0.813184i	$0.697733\pi$
$854$	0	0
$855$	−2.19656	−0.0751207
$856$	0	0
$857$	−53.5787	−1.83021	−0.915107	−	0.403210i	$-0.867894\pi$
$857$	−53.5787	−1.83021	−0.915107	+	0.403210i	$0.867894\pi$
$858$	0	0
$859$	−22.2070	−0.757694	−0.378847	−	0.925459i	$-0.623679\pi$
$859$	−22.2070	−0.757694	−0.378847	+	0.925459i	$0.623679\pi$
$860$	0	0
$861$	−10.7146	−0.365153
$862$	0	0
$863$	18.9028	0.643457	0.321729	−	0.946832i	$-0.395736\pi$
$863$	18.9028	0.643457	0.321729	+	0.946832i	$0.395736\pi$
$864$	0	0
$865$	33.1323	1.12653
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	20.9540	0.710816
$870$	0	0
$871$	9.76481	0.330868
$872$	0	0
$873$	12.3503	0.417993
$874$	0	0
$875$	−11.0073	−0.372116
$876$	0	0
$877$	−15.5970	−0.526674	−0.263337	−	0.964704i	$-0.584823\pi$
$877$	−15.5970	−0.526674	−0.263337	+	0.964704i	$0.584823\pi$
$878$	0	0
$879$	−2.19235	−0.0739460
$880$	0	0
$881$	−47.5359	−1.60152	−0.800762	−	0.598982i	$-0.795572\pi$
$881$	−47.5359	−1.60152	−0.800762	+	0.598982i	$0.795572\pi$
$882$	0	0
$883$	32.1537	1.08206	0.541029	−	0.841004i	$-0.318035\pi$
$883$	32.1537	1.08206	0.541029	+	0.841004i	$0.318035\pi$
$884$	0	0
$885$	−1.53948	−0.0517491
$886$	0	0
$887$	−44.2358	−1.48529	−0.742647	−	0.669684i	$-0.766430\pi$
$887$	−44.2358	−1.48529	−0.742647	+	0.669684i	$0.766430\pi$
$888$	0	0
$889$	14.7820	0.495773
$890$	0	0
$891$	2.19656	0.0735874
$892$	0	0
$893$	−4.66756	−0.156194
$894$	0	0
$895$	3.60688	0.120565
$896$	0	0
$897$	26.0863	0.870996
$898$	0	0
$899$	−19.9143	−0.664179
$900$	0	0
$901$	−8.51806	−0.283778
$902$	0	0
$903$	−6.19656	−0.206209
$904$	0	0
$905$	−0.393115	−0.0130676
$906$	0	0
$907$	−49.6300	−1.64794	−0.823969	−	0.566636i	$-0.808245\pi$
$907$	−49.6300	−1.64794	−0.823969	+	0.566636i	$0.808245\pi$
$908$	0	0
$909$	−8.85363	−0.293657
$910$	0	0
$911$	13.4763	0.446489	0.223245	−	0.974762i	$-0.428335\pi$
$911$	13.4763	0.446489	0.223245	+	0.974762i	$0.428335\pi$
$912$	0	0
$913$	22.7350	0.752417
$914$	0	0
$915$	−9.14637	−0.302370
$916$	0	0
$917$	8.90696	0.294134
$918$	0	0
$919$	22.5897	0.745165	0.372582	−	0.927999i	$-0.378472\pi$
$919$	22.5897	0.745165	0.372582	+	0.927999i	$0.378472\pi$
$920$	0	0
$921$	−5.37169	−0.177003
$922$	0	0
$923$	−26.4360	−0.870150
$924$	0	0
$925$	18.6430	0.612978
$926$	0	0
$927$	−16.3790	−0.537958
$928$	0	0
$929$	−44.8725	−1.47222	−0.736110	−	0.676862i	$-0.763339\pi$
$929$	−44.8725	−1.47222	−0.736110	+	0.676862i	$0.763339\pi$
$930$	0	0
$931$	−1.63565	−0.0536064
$932$	0	0
$933$	−18.7434	−0.613631
$934$	0	0
$935$	−2.94981	−0.0964690
$936$	0	0
$937$	−40.5510	−1.32474	−0.662372	−	0.749175i	$-0.730450\pi$
$937$	−40.5510	−1.32474	−0.662372	+	0.749175i	$0.730450\pi$
$938$	0	0
$939$	5.63986	0.184050
$940$	0	0
$941$	10.1151	0.329742	0.164871	−	0.986315i	$-0.447279\pi$
$941$	10.1151	0.329742	0.164871	+	0.986315i	$0.447279\pi$
$942$	0	0
$943$	76.8788	2.50352
$944$	0	0
$945$	−1.34292	−0.0436853
$946$	0	0
$947$	10.5510	0.342863	0.171431	−	0.985196i	$-0.445161\pi$
$947$	10.5510	0.342863	0.171431	+	0.985196i	$0.445161\pi$
$948$	0	0
$949$	15.6069	0.506621
$950$	0	0
$951$	5.32885	0.172800
$952$	0	0
$953$	20.3257	0.658414	0.329207	−	0.944258i	$-0.393219\pi$
$953$	20.3257	0.658414	0.329207	+	0.944258i	$0.393219\pi$
$954$	0	0
$955$	−1.31415	−0.0425250
$956$	0	0
$957$	15.3288	0.495511
$958$	0	0
$959$	17.1035	0.552301
$960$	0	0
$961$	−22.8568	−0.737315
$962$	0	0
$963$	−16.7392	−0.539412
$964$	0	0
$965$	−17.1464	−0.551961
$966$	0	0
$967$	36.1256	1.16172	0.580860	−	0.814004i	$-0.302716\pi$
$967$	36.1256	1.16172	0.580860	+	0.814004i	$0.302716\pi$
$968$	0	0
$969$	−1.63565	−0.0525448
$970$	0	0
$971$	11.6926	0.375233	0.187616	−	0.982242i	$-0.439924\pi$
$971$	11.6926	0.375233	0.187616	+	0.982242i	$0.439924\pi$
$972$	0	0
$973$	−6.10038	−0.195569
$974$	0	0
$975$	−11.6216	−0.372188
$976$	0	0
$977$	−38.5657	−1.23383	−0.616914	−	0.787031i	$-0.711617\pi$
$977$	−38.5657	−1.23383	−0.616914	+	0.787031i	$0.711617\pi$
$978$	0	0
$979$	−40.9504	−1.30878
$980$	0	0
$981$	9.49663	0.303204
$982$	0	0
$983$	8.06319	0.257176	0.128588	−	0.991698i	$-0.458956\pi$
$983$	8.06319	0.257176	0.128588	+	0.991698i	$0.458956\pi$
$984$	0	0
$985$	−4.65708	−0.148387
$986$	0	0
$987$	−2.85363	−0.0908322
$988$	0	0
$989$	44.4611	1.41378
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	0	0
$993$	−19.9038	−0.631629
$994$	0	0
$995$	25.1266	0.796568
$996$	0	0
$997$	−2.58546	−0.0818824	−0.0409412	−	0.999162i	$-0.513036\pi$
$997$	−2.58546	−0.0818824	−0.0409412	+	0.999162i	$0.513036\pi$
$998$	0	0
$999$	5.83221	0.184523

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5712.2.a.bs.1.3		3
4.3	odd	2		2856.2.a.s.1.3	✓	3
12.11	even	2		8568.2.a.bc.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2856.2.a.s.1.3	✓	3	4.3	odd	2
5712.2.a.bs.1.3		3	1.1	even	1	trivial
8568.2.a.bc.1.1		3	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5712 = 2^{4} \cdot 3 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5712.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.34292 q^{5} +1.00000 q^{7} +1.00000 q^{9} +2.19656 q^{11} -3.63565 q^{13} -1.34292 q^{15} -1.00000 q^{17} -1.63565 q^{19} -1.00000 q^{21} +7.17513 q^{23} -3.19656 q^{25} -1.00000 q^{27} -6.97858 q^{29} +2.85363 q^{31} -2.19656 q^{33} +1.34292 q^{35} -5.83221 q^{37} +3.63565 q^{39} +10.7146 q^{41} +6.19656 q^{43} +1.34292 q^{45} +2.85363 q^{47} +1.00000 q^{49} +1.00000 q^{51} +8.51806 q^{53} +2.94981 q^{55} +1.63565 q^{57} +1.14637 q^{59} +6.81079 q^{61} +1.00000 q^{63} -4.88240 q^{65} -2.68585 q^{67} -7.17513 q^{69} +7.27131 q^{71} -4.29273 q^{73} +3.19656 q^{75} +2.19656 q^{77} +9.53948 q^{79} +1.00000 q^{81} +10.3503 q^{83} -1.34292 q^{85} +6.97858 q^{87} -18.6430 q^{89} -3.63565 q^{91} -2.85363 q^{93} -2.19656 q^{95} +12.3503 q^{97} +2.19656 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 2 q^{5} + 3 q^{7} + 3 q^{9} + 2 q^{11} - 2 q^{13} + 2 q^{15} - 3 q^{17} + 4 q^{19} - 3 q^{21} + 2 q^{23} - 5 q^{25} - 3 q^{27} - 6 q^{29} + 10 q^{31} - 2 q^{33} - 2 q^{35} - 4 q^{37} + 2 q^{39}+ \cdots + 2 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 2 * q^5 + 3 * q^7 + 3 * q^9 + 2 * q^11 - 2 * q^13 + 2 * q^15 - 3 * q^17 + 4 * q^19 - 3 * q^21 + 2 * q^23 - 5 * q^25 - 3 * q^27 - 6 * q^29 + 10 * q^31 - 2 * q^33 - 2 * q^35 - 4 * q^37 + 2 * q^39 + 2 * q^41 + 14 * q^43 - 2 * q^45 + 10 * q^47 + 3 * q^49 + 3 * q^51 + 12 * q^55 - 4 * q^57 + 2 * q^59 - 8 * q^61 + 3 * q^63 + 2 * q^65 + 4 * q^67 - 2 * q^69 + 4 * q^71 - 10 * q^73 + 5 * q^75 + 2 * q^77 + 18 * q^79 + 3 * q^81 - 8 * q^83 + 2 * q^85 + 6 * q^87 - 14 * q^89 - 2 * q^91 - 10 * q^93 - 2 * q^95 - 2 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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