LMFDB - Embedded newform 5776.2.a.bt.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5776,2,Mod(1,5776)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5776, 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("5776.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5776 = 2^{4} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5776.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:	$46.1215922075$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 722)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.90211$ of defining polynomial
Character	$\chi$	$=$	5776.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.79360 q^{3} -2.34458 q^{5} +1.28408 q^{7} +4.80423 q^{9} +O(q^{10})$	$q-2.79360 q^{3} -2.34458 q^{5} +1.28408 q^{7} +4.80423 q^{9} -5.75621 q^{11} +0.304282 q^{13} +6.54982 q^{15} +4.18619 q^{17} -3.58721 q^{21} +6.47684 q^{23} +0.497039 q^{25} -5.04029 q^{27} -3.12756 q^{29} -6.44246 q^{31} +16.0806 q^{33} -3.01062 q^{35} -3.97980 q^{37} -0.850045 q^{39} -5.01719 q^{41} +0.989378 q^{43} -11.2639 q^{45} +4.39445 q^{47} -5.35114 q^{49} -11.6946 q^{51} +3.29064 q^{53} +13.4959 q^{55} -3.31375 q^{59} -10.9615 q^{61} +6.16901 q^{63} -0.713414 q^{65} +4.38081 q^{67} -18.0937 q^{69} +4.41570 q^{71} -2.26689 q^{73} -1.38853 q^{75} -7.39144 q^{77} -8.57659 q^{79} -0.332090 q^{81} +9.76464 q^{83} -9.81485 q^{85} +8.73716 q^{87} -15.0765 q^{89} +0.390723 q^{91} +17.9977 q^{93} +17.3522 q^{97} -27.6542 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 4 * q^9	$ 4 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 4 q^{9} - 2 q^{11} + 18 q^{13} - 4 q^{15} + 6 q^{17} + 4 q^{21} + 10 q^{23} + 6 q^{25} + 4 q^{27} - 2 q^{29} - 26 q^{31} + 16 q^{33} - 6 q^{35} + 4 q^{37} + 6 q^{39} - 12 q^{41} + 10 q^{43} - 22 q^{45} + 12 q^{47} - 12 q^{49} + 2 q^{51} + 8 q^{53} + 26 q^{55} + 8 q^{59} + 22 q^{63} - 4 q^{65} - 10 q^{67} - 20 q^{69} - 14 q^{73} - 8 q^{75} + 4 q^{77} - 22 q^{79} - 4 q^{81} + 12 q^{83} - 18 q^{85} + 26 q^{87} - 16 q^{89} + 4 q^{91} + 8 q^{93} + 28 q^{97} - 22 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 4 * q^9 - 2 * q^11 + 18 * q^13 - 4 * q^15 + 6 * q^17 + 4 * q^21 + 10 * q^23 + 6 * q^25 + 4 * q^27 - 2 * q^29 - 26 * q^31 + 16 * q^33 - 6 * q^35 + 4 * q^37 + 6 * q^39 - 12 * q^41 + 10 * q^43 - 22 * q^45 + 12 * q^47 - 12 * q^49 + 2 * q^51 + 8 * q^53 + 26 * q^55 + 8 * q^59 + 22 * q^63 - 4 * q^65 - 10 * q^67 - 20 * q^69 - 14 * q^73 - 8 * q^75 + 4 * q^77 - 22 * q^79 - 4 * q^81 + 12 * q^83 - 18 * q^85 + 26 * q^87 - 16 * q^89 + 4 * q^91 + 8 * q^93 + 28 * q^97 - 22 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.79360	−1.61289	−0.806444	−	0.591310i	$-0.798611\pi$
$3$	−2.79360	−1.61289	−0.806444	+	0.591310i	$0.798611\pi$
$4$	0	0
$5$	−2.34458	−1.04853	−0.524263	−	0.851556i	$-0.675659\pi$
$5$	−2.34458	−1.04853	−0.524263	+	0.851556i	$0.675659\pi$
$6$	0	0
$7$	1.28408	0.485336	0.242668	−	0.970109i	$-0.421977\pi$
$7$	1.28408	0.485336	0.242668	+	0.970109i	$0.421977\pi$
$8$	0	0
$9$	4.80423	1.60141
$10$	0	0
$11$	−5.75621	−1.73556	−0.867782	−	0.496945i	$-0.834455\pi$
$11$	−5.75621	−1.73556	−0.867782	+	0.496945i	$0.834455\pi$
$12$	0	0
$13$	0.304282	0.0843928	0.0421964	−	0.999109i	$-0.486564\pi$
$13$	0.304282	0.0843928	0.0421964	+	0.999109i	$0.486564\pi$
$14$	0	0
$15$	6.54982	1.69116
$16$	0	0
$17$	4.18619	1.01530	0.507650	−	0.861563i	$-0.330514\pi$
$17$	4.18619	1.01530	0.507650	+	0.861563i	$0.330514\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−3.58721	−0.782793
$22$	0	0
$23$	6.47684	1.35051	0.675257	−	0.737583i	$-0.264033\pi$
$23$	6.47684	1.35051	0.675257	+	0.737583i	$0.264033\pi$
$24$	0	0
$25$	0.497039	0.0994078
$26$	0	0
$27$	−5.04029	−0.970005
$28$	0	0
$29$	−3.12756	−0.580773	−0.290387	−	0.956909i	$-0.593784\pi$
$29$	−3.12756	−0.580773	−0.290387	+	0.956909i	$0.593784\pi$
$30$	0	0
$31$	−6.44246	−1.15710	−0.578550	−	0.815647i	$-0.696381\pi$
$31$	−6.44246	−1.15710	−0.578550	+	0.815647i	$0.696381\pi$
$32$	0	0
$33$	16.0806	2.79927
$34$	0	0
$35$	−3.01062	−0.508888
$36$	0	0
$37$	−3.97980	−0.654275	−0.327137	−	0.944977i	$-0.606084\pi$
$37$	−3.97980	−0.654275	−0.327137	+	0.944977i	$0.606084\pi$
$38$	0	0
$39$	−0.850045	−0.136116
$40$	0	0
$41$	−5.01719	−0.783553	−0.391776	−	0.920060i	$-0.628139\pi$
$41$	−5.01719	−0.783553	−0.391776	+	0.920060i	$0.628139\pi$
$42$	0	0
$43$	0.989378	0.150879	0.0754394	−	0.997150i	$-0.475964\pi$
$43$	0.989378	0.150879	0.0754394	+	0.997150i	$0.475964\pi$
$44$	0	0
$45$	−11.2639	−1.67912
$46$	0	0
$47$	4.39445	0.640997	0.320498	−	0.947249i	$-0.396150\pi$
$47$	4.39445	0.640997	0.320498	+	0.947249i	$0.396150\pi$
$48$	0	0
$49$	−5.35114	−0.764449
$50$	0	0
$51$	−11.6946	−1.63757
$52$	0	0
$53$	3.29064	0.452005	0.226002	−	0.974127i	$-0.427434\pi$
$53$	3.29064	0.452005	0.226002	+	0.974127i	$0.427434\pi$
$54$	0	0
$55$	13.4959	1.81978
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.31375	−0.431414	−0.215707	−	0.976458i	$-0.569206\pi$
$59$	−3.31375	−0.431414	−0.215707	+	0.976458i	$0.569206\pi$
$60$	0	0
$61$	−10.9615	−1.40347	−0.701735	−	0.712438i	$-0.747591\pi$
$61$	−10.9615	−1.40347	−0.701735	+	0.712438i	$0.747591\pi$
$62$	0	0
$63$	6.16901	0.777222
$64$	0	0
$65$	−0.713414	−0.0884881
$66$	0	0
$67$	4.38081	0.535202	0.267601	−	0.963530i	$-0.413769\pi$
$67$	4.38081	0.535202	0.267601	+	0.963530i	$0.413769\pi$
$68$	0	0
$69$	−18.0937	−2.17823
$70$	0	0
$71$	4.41570	0.524047	0.262023	−	0.965062i	$-0.415610\pi$
$71$	4.41570	0.524047	0.262023	+	0.965062i	$0.415610\pi$
$72$	0	0
$73$	−2.26689	−0.265320	−0.132660	−	0.991162i	$-0.542352\pi$
$73$	−2.26689	−0.265320	−0.132660	+	0.991162i	$0.542352\pi$
$74$	0	0
$75$	−1.38853	−0.160334
$76$	0	0
$77$	−7.39144	−0.842332
$78$	0	0
$79$	−8.57659	−0.964941	−0.482471	−	0.875912i	$-0.660261\pi$
$79$	−8.57659	−0.964941	−0.482471	+	0.875912i	$0.660261\pi$
$80$	0	0
$81$	−0.332090	−0.0368989
$82$	0	0
$83$	9.76464	1.07181	0.535904	−	0.844279i	$-0.319971\pi$
$83$	9.76464	1.07181	0.535904	+	0.844279i	$0.319971\pi$
$84$	0	0
$85$	−9.81485	−1.06457
$86$	0	0
$87$	8.73716	0.936722
$88$	0	0
$89$	−15.0765	−1.59811	−0.799055	−	0.601259i	$-0.794666\pi$
$89$	−15.0765	−1.59811	−0.799055	+	0.601259i	$0.794666\pi$
$90$	0	0
$91$	0.390723	0.0409589
$92$	0	0
$93$	17.9977	1.86627
$94$	0	0
$95$	0	0
$96$	0	0
$97$	17.3522	1.76185	0.880924	−	0.473259i	$-0.156922\pi$
$97$	17.3522	1.76185	0.880924	+	0.473259i	$0.156922\pi$
$98$	0	0
$99$	−27.6542	−2.77935
$100$	0	0
$101$	3.67667	0.365842	0.182921	−	0.983128i	$-0.441445\pi$
$101$	3.67667	0.365842	0.182921	+	0.983128i	$0.441445\pi$
$102$	0	0
$103$	−19.0135	−1.87346	−0.936729	−	0.350055i	$-0.886163\pi$
$103$	−19.0135	−1.87346	−0.936729	+	0.350055i	$0.886163\pi$
$104$	0	0
$105$	8.41049	0.820779
$106$	0	0
$107$	−10.0373	−0.970340	−0.485170	−	0.874420i	$-0.661242\pi$
$107$	−10.0373	−0.970340	−0.485170	+	0.874420i	$0.661242\pi$
$108$	0	0
$109$	−8.84348	−0.847052	−0.423526	−	0.905884i	$-0.639208\pi$
$109$	−8.84348	−0.847052	−0.423526	+	0.905884i	$0.639208\pi$
$110$	0	0
$111$	11.1180	1.05527
$112$	0	0
$113$	−14.2502	−1.34055	−0.670275	−	0.742113i	$-0.733824\pi$
$113$	−14.2502	−1.34055	−0.670275	+	0.742113i	$0.733824\pi$
$114$	0	0
$115$	−15.1854	−1.41605
$116$	0	0
$117$	1.46184	0.135147
$118$	0	0
$119$	5.37540	0.492762
$120$	0	0
$121$	22.1340	2.01218
$122$	0	0
$123$	14.0160	1.26378
$124$	0	0
$125$	10.5575	0.944295
$126$	0	0
$127$	−2.23015	−0.197893	−0.0989467	−	0.995093i	$-0.531547\pi$
$127$	−2.23015	−0.197893	−0.0989467	+	0.995093i	$0.531547\pi$
$128$	0	0
$129$	−2.76393	−0.243351
$130$	0	0
$131$	−1.52265	−0.133035	−0.0665175	−	0.997785i	$-0.521189\pi$
$131$	−1.52265	−0.133035	−0.0665175	+	0.997785i	$0.521189\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	11.8174	1.01708
$136$	0	0
$137$	10.7118	0.915167	0.457583	−	0.889167i	$-0.348715\pi$
$137$	10.7118	0.915167	0.457583	+	0.889167i	$0.348715\pi$
$138$	0	0
$139$	10.8143	0.917260	0.458630	−	0.888627i	$-0.348340\pi$
$139$	10.8143	0.917260	0.458630	+	0.888627i	$0.348340\pi$
$140$	0	0
$141$	−12.2764	−1.03386
$142$	0	0
$143$	−1.75152	−0.146469
$144$	0	0
$145$	7.33280	0.608956
$146$	0	0
$147$	14.9490	1.23297
$148$	0	0
$149$	−9.33384	−0.764658	−0.382329	−	0.924026i	$-0.624878\pi$
$149$	−9.33384	−0.764658	−0.382329	+	0.924026i	$0.624878\pi$
$150$	0	0
$151$	−19.8232	−1.61319	−0.806593	−	0.591107i	$-0.798691\pi$
$151$	−19.8232	−1.61319	−0.806593	+	0.591107i	$0.798691\pi$
$152$	0	0
$153$	20.1114	1.62591
$154$	0	0
$155$	15.1048	1.21325
$156$	0	0
$157$	13.8676	1.10676	0.553379	−	0.832930i	$-0.313338\pi$
$157$	13.8676	1.10676	0.553379	+	0.832930i	$0.313338\pi$
$158$	0	0
$159$	−9.19276	−0.729033
$160$	0	0
$161$	8.31677	0.655453
$162$	0	0
$163$	−14.2496	−1.11611	−0.558057	−	0.829803i	$-0.688453\pi$
$163$	−14.2496	−1.11611	−0.558057	+	0.829803i	$0.688453\pi$
$164$	0	0
$165$	−37.7022	−2.93511
$166$	0	0
$167$	−11.5901	−0.896870	−0.448435	−	0.893815i	$-0.648018\pi$
$167$	−11.5901	−0.896870	−0.448435	+	0.893815i	$0.648018\pi$
$168$	0	0
$169$	−12.9074	−0.992878
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.98353	0.302862	0.151431	−	0.988468i	$-0.451612\pi$
$173$	3.98353	0.302862	0.151431	+	0.988468i	$0.451612\pi$
$174$	0	0
$175$	0.638237	0.0482462
$176$	0	0
$177$	9.25731	0.695822
$178$	0	0
$179$	8.45089	0.631649	0.315825	−	0.948818i	$-0.397719\pi$
$179$	8.45089	0.631649	0.315825	+	0.948818i	$0.397719\pi$
$180$	0	0
$181$	8.48526	0.630705	0.315352	−	0.948975i	$-0.397877\pi$
$181$	8.48526	0.630705	0.315352	+	0.948975i	$0.397877\pi$
$182$	0	0
$183$	30.6220	2.26364
$184$	0	0
$185$	9.33094	0.686024
$186$	0	0
$187$	−24.0966	−1.76212
$188$	0	0
$189$	−6.47214	−0.470779
$190$	0	0
$191$	22.3790	1.61929	0.809644	−	0.586921i	$-0.199660\pi$
$191$	22.3790	1.61929	0.809644	+	0.586921i	$0.199660\pi$
$192$	0	0
$193$	7.42811	0.534687	0.267344	−	0.963601i	$-0.413854\pi$
$193$	7.42811	0.534687	0.267344	+	0.963601i	$0.413854\pi$
$194$	0	0
$195$	1.99300	0.142721
$196$	0	0
$197$	8.53554	0.608132	0.304066	−	0.952651i	$-0.401656\pi$
$197$	8.53554	0.608132	0.304066	+	0.952651i	$0.401656\pi$
$198$	0	0
$199$	6.35926	0.450796	0.225398	−	0.974267i	$-0.427632\pi$
$199$	6.35926	0.450796	0.225398	+	0.974267i	$0.427632\pi$
$200$	0	0
$201$	−12.2383	−0.863220
$202$	0	0
$203$	−4.01603	−0.281870
$204$	0	0
$205$	11.7632	0.821576
$206$	0	0
$207$	31.1162	2.16272
$208$	0	0
$209$	0	0
$210$	0	0
$211$	13.9578	0.960894	0.480447	−	0.877024i	$-0.340474\pi$
$211$	13.9578	0.960894	0.480447	+	0.877024i	$0.340474\pi$
$212$	0	0
$213$	−12.3357	−0.845229
$214$	0	0
$215$	−2.31967	−0.158200
$216$	0	0
$217$	−8.27263	−0.561583
$218$	0	0
$219$	6.33280	0.427931
$220$	0	0
$221$	1.27378	0.0856840
$222$	0	0
$223$	−8.92400	−0.597595	−0.298798	−	0.954317i	$-0.596586\pi$
$223$	−8.92400	−0.597595	−0.298798	+	0.954317i	$0.596586\pi$
$224$	0	0
$225$	2.38789	0.159193
$226$	0	0
$227$	0.659796	0.0437922	0.0218961	−	0.999760i	$-0.493030\pi$
$227$	0.659796	0.0437922	0.0218961	+	0.999760i	$0.493030\pi$
$228$	0	0
$229$	−6.97208	−0.460728	−0.230364	−	0.973105i	$-0.573992\pi$
$229$	−6.97208	−0.460728	−0.230364	+	0.973105i	$0.573992\pi$
$230$	0	0
$231$	20.6487	1.35859
$232$	0	0
$233$	−29.6097	−1.93980	−0.969898	−	0.243512i	$-0.921700\pi$
$233$	−29.6097	−1.93980	−0.969898	+	0.243512i	$0.921700\pi$
$234$	0	0
$235$	−10.3031	−0.672102
$236$	0	0
$237$	23.9596	1.55634
$238$	0	0
$239$	24.7034	1.59793	0.798965	−	0.601378i	$-0.205381\pi$
$239$	24.7034	1.59793	0.798965	+	0.601378i	$0.205381\pi$
$240$	0	0
$241$	27.5586	1.77521	0.887604	−	0.460607i	$-0.152368\pi$
$241$	27.5586	1.77521	0.887604	+	0.460607i	$0.152368\pi$
$242$	0	0
$243$	16.0486	1.02952
$244$	0	0
$245$	12.5462	0.801545
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−27.2786	−1.72871
$250$	0	0
$251$	2.98855	0.188636	0.0943179	−	0.995542i	$-0.469933\pi$
$251$	2.98855	0.188636	0.0943179	+	0.995542i	$0.469933\pi$
$252$	0	0
$253$	−37.2821	−2.34390
$254$	0	0
$255$	27.4188	1.71703
$256$	0	0
$257$	14.1323	0.881546	0.440773	−	0.897619i	$-0.354704\pi$
$257$	14.1323	0.881546	0.440773	+	0.897619i	$0.354704\pi$
$258$	0	0
$259$	−5.11037	−0.317543
$260$	0	0
$261$	−15.0255	−0.930055
$262$	0	0
$263$	−7.56887	−0.466717	−0.233358	−	0.972391i	$-0.574971\pi$
$263$	−7.56887	−0.466717	−0.233358	+	0.972391i	$0.574971\pi$
$264$	0	0
$265$	−7.71517	−0.473939
$266$	0	0
$267$	42.1179	2.57757
$268$	0	0
$269$	−7.67074	−0.467694	−0.233847	−	0.972273i	$-0.575131\pi$
$269$	−7.67074	−0.467694	−0.233847	+	0.972273i	$0.575131\pi$
$270$	0	0
$271$	−11.6649	−0.708592	−0.354296	−	0.935133i	$-0.615279\pi$
$271$	−11.6649	−0.708592	−0.354296	+	0.935133i	$0.615279\pi$
$272$	0	0
$273$	−1.09152	−0.0660621
$274$	0	0
$275$	−2.86106	−0.172529
$276$	0	0
$277$	27.4405	1.64874	0.824370	−	0.566052i	$-0.191530\pi$
$277$	27.4405	1.64874	0.824370	+	0.566052i	$0.191530\pi$
$278$	0	0
$279$	−30.9511	−1.85299
$280$	0	0
$281$	16.2715	0.970675	0.485338	−	0.874327i	$-0.338697\pi$
$281$	16.2715	0.970675	0.485338	+	0.874327i	$0.338697\pi$
$282$	0	0
$283$	−3.39374	−0.201737	−0.100868	−	0.994900i	$-0.532162\pi$
$283$	−3.39374	−0.201737	−0.100868	+	0.994900i	$0.532162\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.44246	−0.380287
$288$	0	0
$289$	0.524204	0.0308355
$290$	0	0
$291$	−48.4751	−2.84166
$292$	0	0
$293$	23.3783	1.36578	0.682888	−	0.730523i	$-0.260724\pi$
$293$	23.3783	1.36578	0.682888	+	0.730523i	$0.260724\pi$
$294$	0	0
$295$	7.76934	0.452349
$296$	0	0
$297$	29.0130	1.68351
$298$	0	0
$299$	1.97079	0.113974
$300$	0	0
$301$	1.27044	0.0732270
$302$	0	0
$303$	−10.2712	−0.590062
$304$	0	0
$305$	25.7000	1.47158
$306$	0	0
$307$	1.38802	0.0792185	0.0396093	−	0.999215i	$-0.487389\pi$
$307$	1.38802	0.0792185	0.0396093	+	0.999215i	$0.487389\pi$
$308$	0	0
$309$	53.1163	3.02168
$310$	0	0
$311$	12.1902	0.691246	0.345623	−	0.938374i	$-0.387668\pi$
$311$	12.1902	0.691246	0.345623	+	0.938374i	$0.387668\pi$
$312$	0	0
$313$	16.5314	0.934411	0.467205	−	0.884149i	$-0.345261\pi$
$313$	16.5314	0.934411	0.467205	+	0.884149i	$0.345261\pi$
$314$	0	0
$315$	−14.4637	−0.814938
$316$	0	0
$317$	2.68811	0.150979	0.0754897	−	0.997147i	$-0.475948\pi$
$317$	2.68811	0.150979	0.0754897	+	0.997147i	$0.475948\pi$
$318$	0	0
$319$	18.0029	1.00797
$320$	0	0
$321$	28.0402	1.56505
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0.151240	0.00838930
$326$	0	0
$327$	24.7052	1.36620
$328$	0	0
$329$	5.64282	0.311099
$330$	0	0
$331$	−5.21303	−0.286534	−0.143267	−	0.989684i	$-0.545761\pi$
$331$	−5.21303	−0.286534	−0.143267	+	0.989684i	$0.545761\pi$
$332$	0	0
$333$	−19.1198	−1.04776
$334$	0	0
$335$	−10.2712	−0.561173
$336$	0	0
$337$	−26.7275	−1.45594	−0.727969	−	0.685610i	$-0.759536\pi$
$337$	−26.7275	−1.45594	−0.727969	+	0.685610i	$0.759536\pi$
$338$	0	0
$339$	39.8095	2.16216
$340$	0	0
$341$	37.0842	2.00822
$342$	0	0
$343$	−15.8598	−0.856351
$344$	0	0
$345$	42.4221	2.28393
$346$	0	0
$347$	32.6682	1.75372	0.876860	−	0.480745i	$-0.159634\pi$
$347$	32.6682	1.75372	0.876860	+	0.480745i	$0.159634\pi$
$348$	0	0
$349$	−29.1058	−1.55800	−0.778998	−	0.627026i	$-0.784272\pi$
$349$	−29.1058	−1.55800	−0.778998	+	0.627026i	$0.784272\pi$
$350$	0	0
$351$	−1.53367	−0.0818614
$352$	0	0
$353$	−14.3049	−0.761372	−0.380686	−	0.924704i	$-0.624312\pi$
$353$	−14.3049	−0.761372	−0.380686	+	0.924704i	$0.624312\pi$
$354$	0	0
$355$	−10.3529	−0.549477
$356$	0	0
$357$	−15.0167	−0.794770
$358$	0	0
$359$	−19.6707	−1.03818	−0.519090	−	0.854720i	$-0.673729\pi$
$359$	−19.6707	−1.03818	−0.519090	+	0.854720i	$0.673729\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−61.8337	−3.24543
$364$	0	0
$365$	5.31490	0.278195
$366$	0	0
$367$	−9.55053	−0.498534	−0.249267	−	0.968435i	$-0.580190\pi$
$367$	−9.55053	−0.498534	−0.249267	+	0.968435i	$0.580190\pi$
$368$	0	0
$369$	−24.1037	−1.25479
$370$	0	0
$371$	4.22545	0.219374
$372$	0	0
$373$	3.64010	0.188477	0.0942387	−	0.995550i	$-0.469958\pi$
$373$	3.64010	0.188477	0.0942387	+	0.995550i	$0.469958\pi$
$374$	0	0
$375$	−29.4936	−1.52304
$376$	0	0
$377$	−0.951662	−0.0490131
$378$	0	0
$379$	−8.34741	−0.428778	−0.214389	−	0.976748i	$-0.568776\pi$
$379$	−8.34741	−0.428778	−0.214389	+	0.976748i	$0.568776\pi$
$380$	0	0
$381$	6.23015	0.319180
$382$	0	0
$383$	9.48566	0.484695	0.242347	−	0.970190i	$-0.422083\pi$
$383$	9.48566	0.484695	0.242347	+	0.970190i	$0.422083\pi$
$384$	0	0
$385$	17.3298	0.883208
$386$	0	0
$387$	4.75320	0.241619
$388$	0	0
$389$	36.0480	1.82770	0.913852	−	0.406047i	$-0.133093\pi$
$389$	36.0480	1.82770	0.913852	+	0.406047i	$0.133093\pi$
$390$	0	0
$391$	27.1133	1.37118
$392$	0	0
$393$	4.25369	0.214570
$394$	0	0
$395$	20.1085	1.01177
$396$	0	0
$397$	−6.40906	−0.321662	−0.160831	−	0.986982i	$-0.551417\pi$
$397$	−6.40906	−0.321662	−0.160831	+	0.986982i	$0.551417\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0.187660	0.00937128	0.00468564	−	0.999989i	$-0.498509\pi$
$401$	0.187660	0.00937128	0.00468564	+	0.999989i	$0.498509\pi$
$402$	0	0
$403$	−1.96033	−0.0976509
$404$	0	0
$405$	0.778611	0.0386895
$406$	0	0
$407$	22.9086	1.13554
$408$	0	0
$409$	17.2663	0.853762	0.426881	−	0.904308i	$-0.359612\pi$
$409$	17.2663	0.853762	0.426881	+	0.904308i	$0.359612\pi$
$410$	0	0
$411$	−29.9244	−1.47606
$412$	0	0
$413$	−4.25512	−0.209381
$414$	0	0
$415$	−22.8940	−1.12382
$416$	0	0
$417$	−30.2110	−1.47944
$418$	0	0
$419$	1.72572	0.0843068	0.0421534	−	0.999111i	$-0.486578\pi$
$419$	1.72572	0.0843068	0.0421534	+	0.999111i	$0.486578\pi$
$420$	0	0
$421$	32.7546	1.59636	0.798180	−	0.602420i	$-0.205797\pi$
$421$	32.7546	1.59636	0.798180	+	0.602420i	$0.205797\pi$
$422$	0	0
$423$	21.1119	1.02650
$424$	0	0
$425$	2.08070	0.100929
$426$	0	0
$427$	−14.0754	−0.681155
$428$	0	0
$429$	4.89304	0.236238
$430$	0	0
$431$	25.5119	1.22887	0.614433	−	0.788969i	$-0.289385\pi$
$431$	25.5119	1.22887	0.614433	+	0.788969i	$0.289385\pi$
$432$	0	0
$433$	24.4834	1.17660	0.588299	−	0.808644i	$-0.299798\pi$
$433$	24.4834	1.17660	0.588299	+	0.808644i	$0.299798\pi$
$434$	0	0
$435$	−20.4849	−0.982178
$436$	0	0
$437$	0	0
$438$	0	0
$439$	30.8388	1.47186	0.735928	−	0.677060i	$-0.236746\pi$
$439$	30.8388	1.47186	0.735928	+	0.677060i	$0.236746\pi$
$440$	0	0
$441$	−25.7081	−1.22419
$442$	0	0
$443$	−10.7047	−0.508595	−0.254297	−	0.967126i	$-0.581844\pi$
$443$	−10.7047	−0.508595	−0.254297	+	0.967126i	$0.581844\pi$
$444$	0	0
$445$	35.3481	1.67566
$446$	0	0
$447$	26.0751	1.23331
$448$	0	0
$449$	2.31562	0.109281	0.0546403	−	0.998506i	$-0.482599\pi$
$449$	2.31562	0.109281	0.0546403	+	0.998506i	$0.482599\pi$
$450$	0	0
$451$	28.8800	1.35991
$452$	0	0
$453$	55.3781	2.60189
$454$	0	0
$455$	−0.916079	−0.0429465
$456$	0	0
$457$	−0.0521810	−0.00244092	−0.00122046	−	0.999999i	$-0.500388\pi$
$457$	−0.0521810	−0.00244092	−0.00122046	+	0.999999i	$0.500388\pi$
$458$	0	0
$459$	−21.0996	−0.984847
$460$	0	0
$461$	39.4720	1.83840	0.919198	−	0.393796i	$-0.128838\pi$
$461$	39.4720	1.83840	0.919198	+	0.393796i	$0.128838\pi$
$462$	0	0
$463$	6.19056	0.287700	0.143850	−	0.989600i	$-0.454052\pi$
$463$	6.19056	0.287700	0.143850	+	0.989600i	$0.454052\pi$
$464$	0	0
$465$	−42.1970	−1.95684
$466$	0	0
$467$	20.9962	0.971586	0.485793	−	0.874074i	$-0.338531\pi$
$467$	20.9962	0.971586	0.485793	+	0.874074i	$0.338531\pi$
$468$	0	0
$469$	5.62531	0.259753
$470$	0	0
$471$	−38.7407	−1.78508
$472$	0	0
$473$	−5.69507	−0.261860
$474$	0	0
$475$	0	0
$476$	0	0
$477$	15.8090	0.723844
$478$	0	0
$479$	−15.5229	−0.709261	−0.354631	−	0.935006i	$-0.615393\pi$
$479$	−15.5229	−0.709261	−0.354631	+	0.935006i	$0.615393\pi$
$480$	0	0
$481$	−1.21098	−0.0552160
$482$	0	0
$483$	−23.2338	−1.05717
$484$	0	0
$485$	−40.6835	−1.84734
$486$	0	0
$487$	28.6065	1.29629	0.648143	−	0.761519i	$-0.275546\pi$
$487$	28.6065	1.29629	0.648143	+	0.761519i	$0.275546\pi$
$488$	0	0
$489$	39.8077	1.80017
$490$	0	0
$491$	30.3488	1.36962	0.684812	−	0.728720i	$-0.259884\pi$
$491$	30.3488	1.36962	0.684812	+	0.728720i	$0.259884\pi$
$492$	0	0
$493$	−13.0926	−0.589659
$494$	0	0
$495$	64.8373	2.91422
$496$	0	0
$497$	5.67010	0.254339
$498$	0	0
$499$	−8.77624	−0.392878	−0.196439	−	0.980516i	$-0.562938\pi$
$499$	−8.77624	−0.392878	−0.196439	+	0.980516i	$0.562938\pi$
$500$	0	0
$501$	32.3782	1.44655
$502$	0	0
$503$	21.1043	0.940996	0.470498	−	0.882401i	$-0.344074\pi$
$503$	21.1043	0.940996	0.470498	+	0.882401i	$0.344074\pi$
$504$	0	0
$505$	−8.62023	−0.383595
$506$	0	0
$507$	36.0582	1.60140
$508$	0	0
$509$	22.2212	0.984936	0.492468	−	0.870331i	$-0.336095\pi$
$509$	22.2212	0.984936	0.492468	+	0.870331i	$0.336095\pi$
$510$	0	0
$511$	−2.91087	−0.128769
$512$	0	0
$513$	0	0
$514$	0	0
$515$	44.5787	1.96437
$516$	0	0
$517$	−25.2954	−1.11249
$518$	0	0
$519$	−11.1284	−0.488482
$520$	0	0
$521$	10.0332	0.439563	0.219782	−	0.975549i	$-0.429466\pi$
$521$	10.0332	0.439563	0.219782	+	0.975549i	$0.429466\pi$
$522$	0	0
$523$	−29.2823	−1.28042	−0.640212	−	0.768198i	$-0.721154\pi$
$523$	−29.2823	−1.28042	−0.640212	+	0.768198i	$0.721154\pi$
$524$	0	0
$525$	−1.78298	−0.0778158
$526$	0	0
$527$	−26.9694	−1.17481
$528$	0	0
$529$	18.9494	0.823887
$530$	0	0
$531$	−15.9200	−0.690870
$532$	0	0
$533$	−1.52664	−0.0661262
$534$	0	0
$535$	23.5332	1.01743
$536$	0	0
$537$	−23.6085	−1.01878
$538$	0	0
$539$	30.8023	1.32675
$540$	0	0
$541$	−37.8404	−1.62688	−0.813442	−	0.581646i	$-0.802409\pi$
$541$	−37.8404	−1.62688	−0.813442	+	0.581646i	$0.802409\pi$
$542$	0	0
$543$	−23.7045	−1.01726
$544$	0	0
$545$	20.7342	0.888156
$546$	0	0
$547$	14.4976	0.619871	0.309936	−	0.950758i	$-0.399692\pi$
$547$	14.4976	0.619871	0.309936	+	0.950758i	$0.399692\pi$
$548$	0	0
$549$	−52.6613	−2.24753
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−11.0130	−0.468321
$554$	0	0
$555$	−26.0669	−1.10648
$556$	0	0
$557$	−4.33801	−0.183807	−0.0919037	−	0.995768i	$-0.529295\pi$
$557$	−4.33801	−0.183807	−0.0919037	+	0.995768i	$0.529295\pi$
$558$	0	0
$559$	0.301051	0.0127331
$560$	0	0
$561$	67.3164	2.84210
$562$	0	0
$563$	27.2936	1.15029	0.575144	−	0.818052i	$-0.304946\pi$
$563$	27.2936	1.15029	0.575144	+	0.818052i	$0.304946\pi$
$564$	0	0
$565$	33.4108	1.40560
$566$	0	0
$567$	−0.426430	−0.0179084
$568$	0	0
$569$	−7.19043	−0.301439	−0.150719	−	0.988577i	$-0.548159\pi$
$569$	−7.19043	−0.301439	−0.150719	+	0.988577i	$0.548159\pi$
$570$	0	0
$571$	−42.0134	−1.75821	−0.879103	−	0.476631i	$-0.841858\pi$
$571$	−42.0134	−1.75821	−0.879103	+	0.476631i	$0.841858\pi$
$572$	0	0
$573$	−62.5181	−2.61173
$574$	0	0
$575$	3.21924	0.134252
$576$	0	0
$577$	21.6713	0.902190	0.451095	−	0.892476i	$-0.351034\pi$
$577$	21.6713	0.902190	0.451095	+	0.892476i	$0.351034\pi$
$578$	0	0
$579$	−20.7512	−0.862391
$580$	0	0
$581$	12.5386	0.520188
$582$	0	0
$583$	−18.9417	−0.784483
$584$	0	0
$585$	−3.42740	−0.141706
$586$	0	0
$587$	1.26864	0.0523626	0.0261813	−	0.999657i	$-0.491665\pi$
$587$	1.26864	0.0523626	0.0261813	+	0.999657i	$0.491665\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−23.8449	−0.980849
$592$	0	0
$593$	21.5600	0.885363	0.442682	−	0.896679i	$-0.354027\pi$
$593$	21.5600	0.885363	0.442682	+	0.896679i	$0.354027\pi$
$594$	0	0
$595$	−12.6030	−0.516674
$596$	0	0
$597$	−17.7652	−0.727083
$598$	0	0
$599$	28.6417	1.17027	0.585135	−	0.810936i	$-0.301042\pi$
$599$	28.6417	1.17027	0.585135	+	0.810936i	$0.301042\pi$
$600$	0	0
$601$	7.55865	0.308324	0.154162	−	0.988046i	$-0.450732\pi$
$601$	7.55865	0.308324	0.154162	+	0.988046i	$0.450732\pi$
$602$	0	0
$603$	21.0464	0.857076
$604$	0	0
$605$	−51.8949	−2.10983
$606$	0	0
$607$	−1.72737	−0.0701117	−0.0350558	−	0.999385i	$-0.511161\pi$
$607$	−1.72737	−0.0701117	−0.0350558	+	0.999385i	$0.511161\pi$
$608$	0	0
$609$	11.2192	0.454625
$610$	0	0
$611$	1.33715	0.0540955
$612$	0	0
$613$	−27.4169	−1.10736	−0.553680	−	0.832730i	$-0.686777\pi$
$613$	−27.4169	−1.10736	−0.553680	+	0.832730i	$0.686777\pi$
$614$	0	0
$615$	−32.8617	−1.32511
$616$	0	0
$617$	10.6275	0.427847	0.213924	−	0.976850i	$-0.431376\pi$
$617$	10.6275	0.427847	0.213924	+	0.976850i	$0.431376\pi$
$618$	0	0
$619$	−8.36008	−0.336020	−0.168010	−	0.985785i	$-0.553734\pi$
$619$	−8.36008	−0.336020	−0.168010	+	0.985785i	$0.553734\pi$
$620$	0	0
$621$	−32.6452	−1.31000
$622$	0	0
$623$	−19.3595	−0.775620
$624$	0	0
$625$	−27.2381	−1.08953
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−16.6602	−0.664285
$630$	0	0
$631$	−24.5301	−0.976530	−0.488265	−	0.872696i	$-0.662370\pi$
$631$	−24.5301	−0.976530	−0.488265	+	0.872696i	$0.662370\pi$
$632$	0	0
$633$	−38.9926	−1.54982
$634$	0	0
$635$	5.22875	0.207497
$636$	0	0
$637$	−1.62826	−0.0645139
$638$	0	0
$639$	21.2140	0.839213
$640$	0	0
$641$	−6.97270	−0.275405	−0.137703	−	0.990474i	$-0.543972\pi$
$641$	−6.97270	−0.275405	−0.137703	+	0.990474i	$0.543972\pi$
$642$	0	0
$643$	9.09024	0.358484	0.179242	−	0.983805i	$-0.442635\pi$
$643$	9.09024	0.358484	0.179242	+	0.983805i	$0.442635\pi$
$644$	0	0
$645$	6.48025	0.255160
$646$	0	0
$647$	−30.0074	−1.17971	−0.589856	−	0.807509i	$-0.700815\pi$
$647$	−30.0074	−1.17971	−0.589856	+	0.807509i	$0.700815\pi$
$648$	0	0
$649$	19.0747	0.748746
$650$	0	0
$651$	23.1105	0.905770
$652$	0	0
$653$	10.4191	0.407732	0.203866	−	0.978999i	$-0.434649\pi$
$653$	10.4191	0.407732	0.203866	+	0.978999i	$0.434649\pi$
$654$	0	0
$655$	3.56998	0.139491
$656$	0	0
$657$	−10.8907	−0.424885
$658$	0	0
$659$	26.3756	1.02745	0.513724	−	0.857956i	$-0.328266\pi$
$659$	26.3756	1.02745	0.513724	+	0.857956i	$0.328266\pi$
$660$	0	0
$661$	−24.6939	−0.960483	−0.480242	−	0.877136i	$-0.659451\pi$
$661$	−24.6939	−0.960483	−0.480242	+	0.877136i	$0.659451\pi$
$662$	0	0
$663$	−3.55845	−0.138199
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−20.2567	−0.784342
$668$	0	0
$669$	24.9301	0.963854
$670$	0	0
$671$	63.0965	2.43581
$672$	0	0
$673$	−9.77380	−0.376752	−0.188376	−	0.982097i	$-0.560322\pi$
$673$	−9.77380	−0.376752	−0.188376	+	0.982097i	$0.560322\pi$
$674$	0	0
$675$	−2.50522	−0.0964261
$676$	0	0
$677$	−21.5747	−0.829184	−0.414592	−	0.910007i	$-0.636076\pi$
$677$	−21.5747	−0.829184	−0.414592	+	0.910007i	$0.636076\pi$
$678$	0	0
$679$	22.2816	0.855088
$680$	0	0
$681$	−1.84321	−0.0706319
$682$	0	0
$683$	−33.2509	−1.27231	−0.636155	−	0.771561i	$-0.719476\pi$
$683$	−33.2509	−1.27231	−0.636155	+	0.771561i	$0.719476\pi$
$684$	0	0
$685$	−25.1145	−0.959576
$686$	0	0
$687$	19.4772	0.743103
$688$	0	0
$689$	1.00129	0.0381459
$690$	0	0
$691$	−4.80932	−0.182955	−0.0914776	−	0.995807i	$-0.529159\pi$
$691$	−4.80932	−0.182955	−0.0914776	+	0.995807i	$0.529159\pi$
$692$	0	0
$693$	−35.5101	−1.34892
$694$	0	0
$695$	−25.3550	−0.961772
$696$	0	0
$697$	−21.0029	−0.795542
$698$	0	0
$699$	82.7178	3.12867
$700$	0	0
$701$	3.46466	0.130859	0.0654293	−	0.997857i	$-0.479158\pi$
$701$	3.46466	0.130859	0.0654293	+	0.997857i	$0.479158\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	28.7829	1.08403
$706$	0	0
$707$	4.72113	0.177556
$708$	0	0
$709$	20.5743	0.772684	0.386342	−	0.922356i	$-0.373738\pi$
$709$	20.5743	0.772684	0.386342	+	0.922356i	$0.373738\pi$
$710$	0	0
$711$	−41.2039	−1.54527
$712$	0	0
$713$	−41.7268	−1.56268
$714$	0	0
$715$	4.10656	0.153577
$716$	0	0
$717$	−69.0115	−2.57728
$718$	0	0
$719$	−24.1885	−0.902080	−0.451040	−	0.892504i	$-0.648947\pi$
$719$	−24.1885	−0.902080	−0.451040	+	0.892504i	$0.648947\pi$
$720$	0	0
$721$	−24.4149	−0.909257
$722$	0	0
$723$	−76.9880	−2.86321
$724$	0	0
$725$	−1.55452	−0.0577334
$726$	0	0
$727$	−18.1276	−0.672317	−0.336158	−	0.941806i	$-0.609128\pi$
$727$	−18.1276	−0.672317	−0.336158	+	0.941806i	$0.609128\pi$
$728$	0	0
$729$	−43.8372	−1.62360
$730$	0	0
$731$	4.14173	0.153187
$732$	0	0
$733$	22.7363	0.839785	0.419893	−	0.907574i	$-0.362068\pi$
$733$	22.7363	0.839785	0.419893	+	0.907574i	$0.362068\pi$
$734$	0	0
$735$	−35.0490	−1.29280
$736$	0	0
$737$	−25.2169	−0.928877
$738$	0	0
$739$	−15.1787	−0.558356	−0.279178	−	0.960239i	$-0.590062\pi$
$739$	−15.1787	−0.558356	−0.279178	+	0.960239i	$0.590062\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	3.27625	0.120194	0.0600970	−	0.998193i	$-0.480859\pi$
$743$	3.27625	0.120194	0.0600970	+	0.998193i	$0.480859\pi$
$744$	0	0
$745$	21.8839	0.801764
$746$	0	0
$747$	46.9116	1.71640
$748$	0	0
$749$	−12.8887	−0.470941
$750$	0	0
$751$	−52.2323	−1.90598	−0.952992	−	0.302995i	$-0.902013\pi$
$751$	−52.2323	−1.90598	−0.952992	+	0.302995i	$0.902013\pi$
$752$	0	0
$753$	−8.34884	−0.304248
$754$	0	0
$755$	46.4769	1.69147
$756$	0	0
$757$	−24.2342	−0.880807	−0.440404	−	0.897800i	$-0.645165\pi$
$757$	−24.2342	−0.880807	−0.440404	+	0.897800i	$0.645165\pi$
$758$	0	0
$759$	104.151	3.78045
$760$	0	0
$761$	−24.8054	−0.899195	−0.449597	−	0.893231i	$-0.648433\pi$
$761$	−24.8054	−0.899195	−0.449597	+	0.893231i	$0.648433\pi$
$762$	0	0
$763$	−11.3557	−0.411105
$764$	0	0
$765$	−47.1527	−1.70481
$766$	0	0
$767$	−1.00832	−0.0364082
$768$	0	0
$769$	9.64289	0.347732	0.173866	−	0.984769i	$-0.444374\pi$
$769$	9.64289	0.347732	0.173866	+	0.984769i	$0.444374\pi$
$770$	0	0
$771$	−39.4799	−1.42184
$772$	0	0
$773$	1.28731	0.0463014	0.0231507	−	0.999732i	$-0.492630\pi$
$773$	1.28731	0.0463014	0.0231507	+	0.999732i	$0.492630\pi$
$774$	0	0
$775$	−3.20216	−0.115025
$776$	0	0
$777$	14.2764	0.512162
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−25.4177	−0.909517
$782$	0	0
$783$	15.7638	0.563353
$784$	0	0
$785$	−32.5137	−1.16046
$786$	0	0
$787$	16.3446	0.582624	0.291312	−	0.956628i	$-0.405908\pi$
$787$	16.3446	0.582624	0.291312	+	0.956628i	$0.405908\pi$
$788$	0	0
$789$	21.1444	0.752762
$790$	0	0
$791$	−18.2984	−0.650617
$792$	0	0
$793$	−3.33538	−0.118443
$794$	0	0
$795$	21.5531	0.764410
$796$	0	0
$797$	−15.6884	−0.555712	−0.277856	−	0.960623i	$-0.589624\pi$
$797$	−15.6884	−0.555712	−0.277856	+	0.960623i	$0.589624\pi$
$798$	0	0
$799$	18.3960	0.650804
$800$	0	0
$801$	−72.4311	−2.55923
$802$	0	0
$803$	13.0487	0.460479
$804$	0	0
$805$	−19.4993	−0.687260
$806$	0	0
$807$	21.4290	0.754337
$808$	0	0
$809$	−11.5248	−0.405192	−0.202596	−	0.979262i	$-0.564938\pi$
$809$	−11.5248	−0.405192	−0.202596	+	0.979262i	$0.564938\pi$
$810$	0	0
$811$	8.30102	0.291488	0.145744	−	0.989322i	$-0.453442\pi$
$811$	8.30102	0.291488	0.145744	+	0.989322i	$0.453442\pi$
$812$	0	0
$813$	32.5871	1.14288
$814$	0	0
$815$	33.4093	1.17028
$816$	0	0
$817$	0	0
$818$	0	0
$819$	1.87712	0.0655919
$820$	0	0
$821$	−33.7112	−1.17653	−0.588265	−	0.808668i	$-0.700189\pi$
$821$	−33.7112	−1.17653	−0.588265	+	0.808668i	$0.700189\pi$
$822$	0	0
$823$	−21.8678	−0.762262	−0.381131	−	0.924521i	$-0.624465\pi$
$823$	−21.8678	−0.762262	−0.381131	+	0.924521i	$0.624465\pi$
$824$	0	0
$825$	7.99268	0.278269
$826$	0	0
$827$	−39.1695	−1.36206	−0.681028	−	0.732257i	$-0.738467\pi$
$827$	−39.1695	−1.36206	−0.681028	+	0.732257i	$0.738467\pi$
$828$	0	0
$829$	−14.4176	−0.500743	−0.250371	−	0.968150i	$-0.580553\pi$
$829$	−14.4176	−0.500743	−0.250371	+	0.968150i	$0.580553\pi$
$830$	0	0
$831$	−76.6579	−2.65923
$832$	0	0
$833$	−22.4009	−0.776145
$834$	0	0
$835$	27.1739	0.940392
$836$	0	0
$837$	32.4719	1.12239
$838$	0	0
$839$	−20.9532	−0.723385	−0.361693	−	0.932297i	$-0.617801\pi$
$839$	−20.9532	−0.723385	−0.361693	+	0.932297i	$0.617801\pi$
$840$	0	0
$841$	−19.2184	−0.662702
$842$	0	0
$843$	−45.4561	−1.56559
$844$	0	0
$845$	30.2624	1.04106
$846$	0	0
$847$	28.4218	0.976585
$848$	0	0
$849$	9.48077	0.325379
$850$	0	0
$851$	−25.7765	−0.883607
$852$	0	0
$853$	51.9553	1.77892	0.889458	−	0.457017i	$-0.151082\pi$
$853$	51.9553	1.77892	0.889458	+	0.457017i	$0.151082\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	29.1835	0.996888	0.498444	−	0.866922i	$-0.333905\pi$
$857$	29.1835	0.996888	0.498444	+	0.866922i	$0.333905\pi$
$858$	0	0
$859$	−21.9913	−0.750335	−0.375168	−	0.926957i	$-0.622415\pi$
$859$	−21.9913	−0.750335	−0.375168	+	0.926957i	$0.622415\pi$
$860$	0	0
$861$	17.9977	0.613360
$862$	0	0
$863$	35.8157	1.21918	0.609590	−	0.792717i	$-0.291334\pi$
$863$	35.8157	1.21918	0.609590	+	0.792717i	$0.291334\pi$
$864$	0	0
$865$	−9.33968	−0.317559
$866$	0	0
$867$	−1.46442	−0.0497343
$868$	0	0
$869$	49.3687	1.67472
$870$	0	0
$871$	1.33300	0.0451671
$872$	0	0
$873$	83.3638	2.82144
$874$	0	0
$875$	13.5567	0.458300
$876$	0	0
$877$	7.82483	0.264226	0.132113	−	0.991235i	$-0.457824\pi$
$877$	7.82483	0.264226	0.132113	+	0.991235i	$0.457824\pi$
$878$	0	0
$879$	−65.3097	−2.20284
$880$	0	0
$881$	−14.4829	−0.487941	−0.243970	−	0.969783i	$-0.578450\pi$
$881$	−14.4829	−0.487941	−0.243970	+	0.969783i	$0.578450\pi$
$882$	0	0
$883$	23.5529	0.792619	0.396310	−	0.918117i	$-0.370291\pi$
$883$	23.5529	0.792619	0.396310	+	0.918117i	$0.370291\pi$
$884$	0	0
$885$	−21.7045	−0.729588
$886$	0	0
$887$	15.8736	0.532982	0.266491	−	0.963837i	$-0.414136\pi$
$887$	15.8736	0.532982	0.266491	+	0.963837i	$0.414136\pi$
$888$	0	0
$889$	−2.86368	−0.0960449
$890$	0	0
$891$	1.91158	0.0640404
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−19.8138	−0.662301
$896$	0	0
$897$	−5.50560	−0.183827
$898$	0	0
$899$	20.1492	0.672013
$900$	0	0
$901$	13.7753	0.458921
$902$	0	0
$903$	−3.54911	−0.118107
$904$	0	0
$905$	−19.8944	−0.661311
$906$	0	0
$907$	−45.0127	−1.49462	−0.747311	−	0.664474i	$-0.768656\pi$
$907$	−45.0127	−1.49462	−0.747311	+	0.664474i	$0.768656\pi$
$908$	0	0
$909$	17.6635	0.585863
$910$	0	0
$911$	2.75901	0.0914100	0.0457050	−	0.998955i	$-0.485447\pi$
$911$	2.75901	0.0914100	0.0457050	+	0.998955i	$0.485447\pi$
$912$	0	0
$913$	−56.2074	−1.86019
$914$	0	0
$915$	−71.7956	−2.37349
$916$	0	0
$917$	−1.95521	−0.0645667
$918$	0	0
$919$	39.3083	1.29666	0.648331	−	0.761359i	$-0.275467\pi$
$919$	39.3083	1.29666	0.648331	+	0.761359i	$0.275467\pi$
$920$	0	0
$921$	−3.87758	−0.127771
$922$	0	0
$923$	1.34362	0.0442258
$924$	0	0
$925$	−1.97811	−0.0650400
$926$	0	0
$927$	−91.3453	−3.00017
$928$	0	0
$929$	−29.7417	−0.975795	−0.487897	−	0.872901i	$-0.662236\pi$
$929$	−29.7417	−0.975795	−0.487897	+	0.872901i	$0.662236\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−34.0547	−1.11490
$934$	0	0
$935$	56.4964	1.84763
$936$	0	0
$937$	37.7211	1.23229	0.616147	−	0.787631i	$-0.288693\pi$
$937$	37.7211	1.23229	0.616147	+	0.787631i	$0.288693\pi$
$938$	0	0
$939$	−46.1822	−1.50710
$940$	0	0
$941$	−6.47652	−0.211129	−0.105564	−	0.994412i	$-0.533665\pi$
$941$	−6.47652	−0.211129	−0.105564	+	0.994412i	$0.533665\pi$
$942$	0	0
$943$	−32.4955	−1.05820
$944$	0	0
$945$	15.1744	0.493624
$946$	0	0
$947$	−16.2913	−0.529396	−0.264698	−	0.964331i	$-0.585272\pi$
$947$	−16.2913	−0.529396	−0.264698	+	0.964331i	$0.585272\pi$
$948$	0	0
$949$	−0.689776	−0.0223911
$950$	0	0
$951$	−7.50953	−0.243513
$952$	0	0
$953$	−34.6213	−1.12150	−0.560748	−	0.827987i	$-0.689486\pi$
$953$	−34.6213	−1.12150	−0.560748	+	0.827987i	$0.689486\pi$
$954$	0	0
$955$	−52.4693	−1.69787
$956$	0	0
$957$	−50.2930	−1.62574
$958$	0	0
$959$	13.7547	0.444164
$960$	0	0
$961$	10.5053	0.338882
$962$	0	0
$963$	−48.2213	−1.55391
$964$	0	0
$965$	−17.4158	−0.560634
$966$	0	0
$967$	48.9105	1.57286	0.786428	−	0.617682i	$-0.211928\pi$
$967$	48.9105	1.57286	0.786428	+	0.617682i	$0.211928\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−6.07592	−0.194986	−0.0974928	−	0.995236i	$-0.531082\pi$
$971$	−6.07592	−0.194986	−0.0974928	+	0.995236i	$0.531082\pi$
$972$	0	0
$973$	13.8865	0.445180
$974$	0	0
$975$	−0.422505	−0.0135310
$976$	0	0
$977$	57.0477	1.82512	0.912559	−	0.408944i	$-0.134103\pi$
$977$	57.0477	1.82512	0.912559	+	0.408944i	$0.134103\pi$
$978$	0	0
$979$	86.7838	2.77362
$980$	0	0
$981$	−42.4861	−1.35648
$982$	0	0
$983$	−0.186519	−0.00594905	−0.00297452	−	0.999996i	$-0.500947\pi$
$983$	−0.186519	−0.00594905	−0.00297452	+	0.999996i	$0.500947\pi$
$984$	0	0
$985$	−20.0122	−0.637642
$986$	0	0
$987$	−15.7638	−0.501768
$988$	0	0
$989$	6.40804	0.203764
$990$	0	0
$991$	46.2398	1.46886	0.734428	−	0.678686i	$-0.237450\pi$
$991$	46.2398	1.46886	0.734428	+	0.678686i	$0.237450\pi$
$992$	0	0
$993$	14.5631	0.462147
$994$	0	0
$995$	−14.9098	−0.472671
$996$	0	0
$997$	−20.8490	−0.660295	−0.330147	−	0.943929i	$-0.607098\pi$
$997$	−20.8490	−0.660295	−0.330147	+	0.943929i	$0.607098\pi$
$998$	0	0
$999$	20.0593	0.634650

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5776.2.a.bt.1.1		4
4.3	odd	2		722.2.a.n.1.4	yes	4
12.11	even	2		6498.2.a.bx.1.3		4
19.18	odd	2		5776.2.a.bv.1.4		4
76.3	even	18		722.2.e.s.389.1		24
76.7	odd	6		722.2.c.m.429.1		8
76.11	odd	6		722.2.c.m.653.1		8
76.15	even	18		722.2.e.s.415.4		24
76.23	odd	18		722.2.e.r.415.1		24
76.27	even	6		722.2.c.n.653.4		8
76.31	even	6		722.2.c.n.429.4		8
76.35	odd	18		722.2.e.r.389.4		24
76.43	odd	18		722.2.e.r.595.1		24
76.47	odd	18		722.2.e.r.423.4		24
76.51	even	18		722.2.e.s.245.1		24
76.55	odd	18		722.2.e.r.99.4		24
76.59	even	18		722.2.e.s.99.1		24
76.63	odd	18		722.2.e.r.245.4		24
76.67	even	18		722.2.e.s.423.1		24
76.71	even	18		722.2.e.s.595.4		24
76.75	even	2		722.2.a.m.1.1	✓	4
228.227	odd	2		6498.2.a.ca.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
722.2.a.m.1.1	✓	4	76.75	even	2
722.2.a.n.1.4	yes	4	4.3	odd	2
722.2.c.m.429.1		8	76.7	odd	6
722.2.c.m.653.1		8	76.11	odd	6
722.2.c.n.429.4		8	76.31	even	6
722.2.c.n.653.4		8	76.27	even	6
722.2.e.r.99.4		24	76.55	odd	18
722.2.e.r.245.4		24	76.63	odd	18
722.2.e.r.389.4		24	76.35	odd	18
722.2.e.r.415.1		24	76.23	odd	18
722.2.e.r.423.4		24	76.47	odd	18
722.2.e.r.595.1		24	76.43	odd	18
722.2.e.s.99.1		24	76.59	even	18
722.2.e.s.245.1		24	76.51	even	18
722.2.e.s.389.1		24	76.3	even	18
722.2.e.s.415.4		24	76.15	even	18
722.2.e.s.423.1		24	76.67	even	18
722.2.e.s.595.4		24	76.71	even	18
5776.2.a.bt.1.1		4	1.1	even	1	trivial
5776.2.a.bv.1.4		4	19.18	odd	2
6498.2.a.bx.1.3		4	12.11	even	2
6498.2.a.ca.1.3		4	228.227	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5776 = 2^{4} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5776.a (trivial)

\(f(q)\)	\(=\)	\(q-2.79360 q^{3} -2.34458 q^{5} +1.28408 q^{7} +4.80423 q^{9} +O(q^{10})\)	\(q-2.79360 q^{3} -2.34458 q^{5} +1.28408 q^{7} +4.80423 q^{9} -5.75621 q^{11} +0.304282 q^{13} +6.54982 q^{15} +4.18619 q^{17} -3.58721 q^{21} +6.47684 q^{23} +0.497039 q^{25} -5.04029 q^{27} -3.12756 q^{29} -6.44246 q^{31} +16.0806 q^{33} -3.01062 q^{35} -3.97980 q^{37} -0.850045 q^{39} -5.01719 q^{41} +0.989378 q^{43} -11.2639 q^{45} +4.39445 q^{47} -5.35114 q^{49} -11.6946 q^{51} +3.29064 q^{53} +13.4959 q^{55} -3.31375 q^{59} -10.9615 q^{61} +6.16901 q^{63} -0.713414 q^{65} +4.38081 q^{67} -18.0937 q^{69} +4.41570 q^{71} -2.26689 q^{73} -1.38853 q^{75} -7.39144 q^{77} -8.57659 q^{79} -0.332090 q^{81} +9.76464 q^{83} -9.81485 q^{85} +8.73716 q^{87} -15.0765 q^{89} +0.390723 q^{91} +17.9977 q^{93} +17.3522 q^{97} -27.6542 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 4 * q^9	\( 4 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 4 q^{9} - 2 q^{11} + 18 q^{13} - 4 q^{15} + 6 q^{17} + 4 q^{21} + 10 q^{23} + 6 q^{25} + 4 q^{27} - 2 q^{29} - 26 q^{31} + 16 q^{33} - 6 q^{35} + 4 q^{37} + 6 q^{39} - 12 q^{41} + 10 q^{43} - 22 q^{45} + 12 q^{47} - 12 q^{49} + 2 q^{51} + 8 q^{53} + 26 q^{55} + 8 q^{59} + 22 q^{63} - 4 q^{65} - 10 q^{67} - 20 q^{69} - 14 q^{73} - 8 q^{75} + 4 q^{77} - 22 q^{79} - 4 q^{81} + 12 q^{83} - 18 q^{85} + 26 q^{87} - 16 q^{89} + 4 q^{91} + 8 q^{93} + 28 q^{97} - 22 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 4 * q^9 - 2 * q^11 + 18 * q^13 - 4 * q^15 + 6 * q^17 + 4 * q^21 + 10 * q^23 + 6 * q^25 + 4 * q^27 - 2 * q^29 - 26 * q^31 + 16 * q^33 - 6 * q^35 + 4 * q^37 + 6 * q^39 - 12 * q^41 + 10 * q^43 - 22 * q^45 + 12 * q^47 - 12 * q^49 + 2 * q^51 + 8 * q^53 + 26 * q^55 + 8 * q^59 + 22 * q^63 - 4 * q^65 - 10 * q^67 - 20 * q^69 - 14 * q^73 - 8 * q^75 + 4 * q^77 - 22 * q^79 - 4 * q^81 + 12 * q^83 - 18 * q^85 + 26 * q^87 - 16 * q^89 + 4 * q^91 + 8 * q^93 + 28 * q^97 - 22 * q^99

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