LMFDB - Embedded newform 2816.2.c.t.1409.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2816,2,Mod(1409,2816)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2816, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2816.1409");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2816 = 2^{8} \cdot 11$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2816.c (of order $2$ , degree $1$ , not minimal)

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:	no
Analytic conductor:	$22.4858732092$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 9x^{2} + 16$ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 352)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1409.3
Root			$1.56155i$ of defining polynomial
Character	$\chi$	$=$	2816.1409
Dual form			2816.2.c.t.1409.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155i q^{3} -3.56155i q^{5} +0.561553 q^{9} +1.00000i q^{11} +2.00000i q^{13} +5.56155 q^{15} -1.12311 q^{17} -7.12311i q^{19} +4.68466 q^{23} -7.68466 q^{25} +5.56155i q^{27} -1.12311i q^{29} +9.56155 q^{31} -1.56155 q^{33} -6.68466i q^{37} -3.12311 q^{39} -8.24621 q^{41} +7.12311i q^{43} -2.00000i q^{45} +4.00000 q^{47} -7.00000 q^{49} -1.75379i q^{51} +8.24621i q^{53} +3.56155 q^{55} +11.1231 q^{57} -12.6847i q^{59} -15.3693i q^{61} +7.12311 q^{65} +4.68466i q^{67} +7.31534i q^{69} +3.31534 q^{71} +6.00000 q^{73} -12.0000i q^{75} +4.87689 q^{79} -7.00000 q^{81} -13.3693i q^{83} +4.00000i q^{85} +1.75379 q^{87} +3.56155 q^{89} +14.9309i q^{93} -25.3693 q^{95} -6.68466 q^{97} +0.561553i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 6 q^{9} + 14 q^{15} + 12 q^{17} - 6 q^{23} - 6 q^{25} + 30 q^{31} + 2 q^{33} + 4 q^{39} + 16 q^{47} - 28 q^{49} + 6 q^{55} + 28 q^{57} + 12 q^{65} + 38 q^{71} + 24 q^{73} + 36 q^{79} - 28 q^{81} + 40 q^{87}+ \cdots - 2 q^{97}+O(q^{100})$ 4 * q - 6 * q^9 + 14 * q^15 + 12 * q^17 - 6 * q^23 - 6 * q^25 + 30 * q^31 + 2 * q^33 + 4 * q^39 + 16 * q^47 - 28 * q^49 + 6 * q^55 + 28 * q^57 + 12 * q^65 + 38 * q^71 + 24 * q^73 + 36 * q^79 - 28 * q^81 + 40 * q^87 + 6 * q^89 - 52 * q^95 - 2 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2816\mathbb{Z}\right)^\times$ .

$n$	$1025$	$1541$	$2047$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	1.56155i	0.901563i	0.892634	+	0.450781i	$0.148855\pi$
$3$	1.56155i	0.901563i	−0.892634	+	0.450781i	$0.851145\pi$
$4$	0	0
$5$	− 3.56155i	− 1.59277i	−0.604787	−	0.796387i	$-0.706742\pi$
$5$	− 3.56155i	− 1.59277i	0.604787	−	0.796387i	$-0.293258\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	0.561553	0.187184
$10$	0	0
$11$	1.00000i	0.301511i
$11$	1.00000i	0.301511i
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	0	0
$15$	5.56155	1.43599
$16$	0	0
$17$	−1.12311	−0.272393	−0.136197	−	0.990682i	$-0.543488\pi$
$17$	−1.12311	−0.272393	−0.136197	+	0.990682i	$0.543488\pi$
$18$	0	0
$19$	− 7.12311i	− 1.63415i	−0.576530	−	0.817076i	$-0.695593\pi$
$19$	− 7.12311i	− 1.63415i	0.576530	−	0.817076i	$-0.304407\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.68466	0.976819	0.488409	−	0.872615i	$-0.337577\pi$
$23$	4.68466	0.976819	0.488409	+	0.872615i	$0.337577\pi$
$24$	0	0
$25$	−7.68466	−1.53693
$26$	0	0
$27$	5.56155i	1.07032i
$28$	0	0
$29$	− 1.12311i	− 0.208555i	−0.994548	−	0.104278i	$-0.966747\pi$
$29$	− 1.12311i	− 0.208555i	0.994548	−	0.104278i	$-0.0332531\pi$
$30$	0	0
$31$	9.56155	1.71731	0.858653	−	0.512558i	$-0.171302\pi$
$31$	9.56155	1.71731	0.858653	+	0.512558i	$0.171302\pi$
$32$	0	0
$33$	−1.56155	−0.271831
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.68466i	− 1.09895i	−0.835510	−	0.549476i	$-0.814828\pi$
$37$	− 6.68466i	− 1.09895i	0.835510	−	0.549476i	$-0.185172\pi$
$38$	0	0
$39$	−3.12311	−0.500097
$40$	0	0
$41$	−8.24621	−1.28784	−0.643921	−	0.765092i	$-0.722693\pi$
$41$	−8.24621	−1.28784	−0.643921	+	0.765092i	$0.722693\pi$
$42$	0	0
$43$	7.12311i	1.08626i	0.839648	+	0.543132i	$0.182762\pi$
$43$	7.12311i	1.08626i	−0.839648	+	0.543132i	$0.817238\pi$
$44$	0	0
$45$	− 2.00000i	− 0.298142i
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	− 1.75379i	− 0.245580i
$52$	0	0
$53$	8.24621i	1.13270i	0.824163	+	0.566352i	$0.191646\pi$
$53$	8.24621i	1.13270i	−0.824163	+	0.566352i	$0.808354\pi$
$54$	0	0
$55$	3.56155	0.480240
$56$	0	0
$57$	11.1231	1.47329
$58$	0	0
$59$	− 12.6847i	− 1.65140i	−0.564108	−	0.825701i	$-0.690780\pi$
$59$	− 12.6847i	− 1.65140i	0.564108	−	0.825701i	$-0.309220\pi$
$60$	0	0
$61$	− 15.3693i	− 1.96784i	−0.178611	−	0.983920i	$-0.557161\pi$
$61$	− 15.3693i	− 1.96784i	0.178611	−	0.983920i	$-0.442839\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	7.12311	0.883513
$66$	0	0
$67$	4.68466i	0.572322i	0.958182	+	0.286161i	$0.0923792\pi$
$67$	4.68466i	0.572322i	−0.958182	+	0.286161i	$0.907621\pi$
$68$	0	0
$69$	7.31534i	0.880664i
$70$	0	0
$71$	3.31534	0.393459	0.196729	−	0.980458i	$-0.436968\pi$
$71$	3.31534	0.393459	0.196729	+	0.980458i	$0.436968\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	− 12.0000i	− 1.38564i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	4.87689	0.548693	0.274347	−	0.961631i	$-0.411538\pi$
$79$	4.87689	0.548693	0.274347	+	0.961631i	$0.411538\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	− 13.3693i	− 1.46747i	−0.679434	−	0.733737i	$-0.737775\pi$
$83$	− 13.3693i	− 1.46747i	0.679434	−	0.733737i	$-0.262225\pi$
$84$	0	0
$85$	4.00000i	0.433861i
$86$	0	0
$87$	1.75379	0.188026
$88$	0	0
$89$	3.56155	0.377524	0.188762	−	0.982023i	$-0.439553\pi$
$89$	3.56155	0.377524	0.188762	+	0.982023i	$0.439553\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	14.9309i	1.54826i
$94$	0	0
$95$	−25.3693	−2.60284
$96$	0	0
$97$	−6.68466	−0.678724	−0.339362	−	0.940656i	$-0.610211\pi$
$97$	−6.68466	−0.678724	−0.339362	+	0.940656i	$0.610211\pi$
$98$	0	0
$99$	0.561553i	0.0564382i
$100$	0	0
$101$	− 13.1231i	− 1.30580i	−0.757445	−	0.652899i	$-0.773553\pi$
$101$	− 13.1231i	− 1.30580i	0.757445	−	0.652899i	$-0.226447\pi$
$102$	0	0
$103$	2.24621	0.221326	0.110663	−	0.993858i	$-0.464703\pi$
$103$	2.24621	0.221326	0.110663	+	0.993858i	$0.464703\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	0	0
$109$	− 14.0000i	− 1.34096i	−0.741929	−	0.670478i	$-0.766089\pi$
$109$	− 14.0000i	− 1.34096i	0.741929	−	0.670478i	$-0.233911\pi$
$110$	0	0
$111$	10.4384	0.990774
$112$	0	0
$113$	15.5616	1.46391	0.731954	−	0.681354i	$-0.238609\pi$
$113$	15.5616	1.46391	0.731954	+	0.681354i	$0.238609\pi$
$114$	0	0
$115$	− 16.6847i	− 1.55585i
$116$	0	0
$117$	1.12311i	0.103831i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	− 12.8769i	− 1.16107i
$124$	0	0
$125$	9.56155i	0.855211i
$126$	0	0
$127$	9.36932	0.831392	0.415696	−	0.909504i	$-0.363538\pi$
$127$	9.36932	0.831392	0.415696	+	0.909504i	$0.363538\pi$
$128$	0	0
$129$	−11.1231	−0.979335
$130$	0	0
$131$	− 4.00000i	− 0.349482i	−0.984614	−	0.174741i	$-0.944091\pi$
$131$	− 4.00000i	− 0.349482i	0.984614	−	0.174741i	$-0.0559088\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	19.8078	1.70478
$136$	0	0
$137$	−1.31534	−0.112377	−0.0561886	−	0.998420i	$-0.517895\pi$
$137$	−1.31534	−0.112377	−0.0561886	+	0.998420i	$0.517895\pi$
$138$	0	0
$139$	12.0000i	1.01783i	0.860818	+	0.508913i	$0.169953\pi$
$139$	12.0000i	1.01783i	−0.860818	+	0.508913i	$0.830047\pi$
$140$	0	0
$141$	6.24621i	0.526026i
$142$	0	0
$143$	−2.00000	−0.167248
$144$	0	0
$145$	−4.00000	−0.332182
$146$	0	0
$147$	− 10.9309i	− 0.901563i
$148$	0	0
$149$	− 13.1231i	− 1.07509i	−0.843236	−	0.537543i	$-0.819352\pi$
$149$	− 13.1231i	− 1.07509i	0.843236	−	0.537543i	$-0.180648\pi$
$150$	0	0
$151$	−9.36932	−0.762464	−0.381232	−	0.924479i	$-0.624500\pi$
$151$	−9.36932	−0.762464	−0.381232	+	0.924479i	$0.624500\pi$
$152$	0	0
$153$	−0.630683	−0.0509877
$154$	0	0
$155$	− 34.0540i	− 2.73528i
$156$	0	0
$157$	− 10.6847i	− 0.852729i	−0.904552	−	0.426364i	$-0.859794\pi$
$157$	− 10.6847i	− 0.852729i	0.904552	−	0.426364i	$-0.140206\pi$
$158$	0	0
$159$	−12.8769	−1.02120
$160$	0	0
$161$	0	0
$162$	0	0
$163$	12.0000i	0.939913i	0.882690	+	0.469956i	$0.155730\pi$
$163$	12.0000i	0.939913i	−0.882690	+	0.469956i	$0.844270\pi$
$164$	0	0
$165$	5.56155i	0.432966i
$166$	0	0
$167$	−17.3693	−1.34408	−0.672039	−	0.740516i	$-0.734581\pi$
$167$	−17.3693	−1.34408	−0.672039	+	0.740516i	$0.734581\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	− 4.00000i	− 0.305888i
$172$	0	0
$173$	18.0000i	1.36851i	0.729241	+	0.684257i	$0.239873\pi$
$173$	18.0000i	1.36851i	−0.729241	+	0.684257i	$0.760127\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	19.8078	1.48884
$178$	0	0
$179$	4.68466i	0.350148i	0.984555	+	0.175074i	$0.0560164\pi$
$179$	4.68466i	0.350148i	−0.984555	+	0.175074i	$0.943984\pi$
$180$	0	0
$181$	12.0540i	0.895965i	0.894042	+	0.447982i	$0.147857\pi$
$181$	12.0540i	0.895965i	−0.894042	+	0.447982i	$0.852143\pi$
$182$	0	0
$183$	24.0000	1.77413
$184$	0	0
$185$	−23.8078	−1.75038
$186$	0	0
$187$	− 1.12311i	− 0.0821296i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.68466	0.338970	0.169485	−	0.985533i	$-0.445790\pi$
$191$	4.68466	0.338970	0.169485	+	0.985533i	$0.445790\pi$
$192$	0	0
$193$	11.3693	0.818381	0.409191	−	0.912449i	$-0.365811\pi$
$193$	11.3693	0.818381	0.409191	+	0.912449i	$0.365811\pi$
$194$	0	0
$195$	11.1231i	0.796542i
$196$	0	0
$197$	− 8.24621i	− 0.587518i	−0.955879	−	0.293759i	$-0.905094\pi$
$197$	− 8.24621i	− 0.587518i	0.955879	−	0.293759i	$-0.0949064\pi$
$198$	0	0
$199$	2.24621	0.159230	0.0796148	−	0.996826i	$-0.474631\pi$
$199$	2.24621	0.159230	0.0796148	+	0.996826i	$0.474631\pi$
$200$	0	0
$201$	−7.31534	−0.515984
$202$	0	0
$203$	0	0
$204$	0	0
$205$	29.3693i	2.05124i
$206$	0	0
$207$	2.63068	0.182845
$208$	0	0
$209$	7.12311	0.492716
$210$	0	0
$211$	16.4924i	1.13539i	0.823241	+	0.567693i	$0.192164\pi$
$211$	16.4924i	1.13539i	−0.823241	+	0.567693i	$0.807836\pi$
$212$	0	0
$213$	5.17708i	0.354728i
$214$	0	0
$215$	25.3693	1.73017
$216$	0	0
$217$	0	0
$218$	0	0
$219$	9.36932i	0.633120i
$220$	0	0
$221$	− 2.24621i	− 0.151097i
$222$	0	0
$223$	−4.68466	−0.313708	−0.156854	−	0.987622i	$-0.550135\pi$
$223$	−4.68466	−0.313708	−0.156854	+	0.987622i	$0.550135\pi$
$224$	0	0
$225$	−4.31534	−0.287689
$226$	0	0
$227$	− 20.0000i	− 1.32745i	−0.747978	−	0.663723i	$-0.768975\pi$
$227$	− 20.0000i	− 1.32745i	0.747978	−	0.663723i	$-0.231025\pi$
$228$	0	0
$229$	9.31534i	0.615575i	0.951455	+	0.307788i	$0.0995886\pi$
$229$	9.31534i	0.615575i	−0.951455	+	0.307788i	$0.900411\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.12311	0.0735771	0.0367885	−	0.999323i	$-0.488287\pi$
$233$	1.12311	0.0735771	0.0367885	+	0.999323i	$0.488287\pi$
$234$	0	0
$235$	− 14.2462i	− 0.929320i
$236$	0	0
$237$	7.61553i	0.494682i
$238$	0	0
$239$	25.3693	1.64100	0.820502	−	0.571643i	$-0.193694\pi$
$239$	25.3693	1.64100	0.820502	+	0.571643i	$0.193694\pi$
$240$	0	0
$241$	27.3693	1.76301	0.881506	−	0.472172i	$-0.156530\pi$
$241$	27.3693	1.76301	0.881506	+	0.472172i	$0.156530\pi$
$242$	0	0
$243$	5.75379i	0.369106i
$244$	0	0
$245$	24.9309i	1.59277i
$246$	0	0
$247$	14.2462	0.906465
$248$	0	0
$249$	20.8769	1.32302
$250$	0	0
$251$	14.0540i	0.887079i	0.896255	+	0.443540i	$0.146277\pi$
$251$	14.0540i	0.887079i	−0.896255	+	0.443540i	$0.853723\pi$
$252$	0	0
$253$	4.68466i	0.294522i
$254$	0	0
$255$	−6.24621	−0.391153
$256$	0	0
$257$	8.24621	0.514385	0.257192	−	0.966360i	$-0.417203\pi$
$257$	8.24621	0.514385	0.257192	+	0.966360i	$0.417203\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	− 0.630683i	− 0.0390383i
$262$	0	0
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	0	0
$265$	29.3693	1.80414
$266$	0	0
$267$	5.56155i	0.340362i
$268$	0	0
$269$	− 22.4924i	− 1.37139i	−0.727890	−	0.685694i	$-0.759499\pi$
$269$	− 22.4924i	− 1.37139i	0.727890	−	0.685694i	$-0.240501\pi$
$270$	0	0
$271$	4.87689	0.296250	0.148125	−	0.988969i	$-0.452676\pi$
$271$	4.87689	0.296250	0.148125	+	0.988969i	$0.452676\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 7.68466i	− 0.463402i
$276$	0	0
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	0	0
$279$	5.36932	0.321453
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	26.2462i	1.56018i	0.625670	+	0.780088i	$0.284826\pi$
$283$	26.2462i	1.56018i	−0.625670	+	0.780088i	$0.715174\pi$
$284$	0	0
$285$	− 39.6155i	− 2.34662i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−15.7386	−0.925802
$290$	0	0
$291$	− 10.4384i	− 0.611913i
$292$	0	0
$293$	− 22.4924i	− 1.31402i	−0.753881	−	0.657011i	$-0.771821\pi$
$293$	− 22.4924i	− 1.31402i	0.753881	−	0.657011i	$-0.228179\pi$
$294$	0	0
$295$	−45.1771	−2.63031
$296$	0	0
$297$	−5.56155	−0.322714
$298$	0	0
$299$	9.36932i	0.541842i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	20.4924	1.17726
$304$	0	0
$305$	−54.7386	−3.13433
$306$	0	0
$307$	− 2.63068i	− 0.150141i	−0.997178	−	0.0750705i	$-0.976082\pi$
$307$	− 2.63068i	− 0.150141i	0.997178	−	0.0750705i	$-0.0239182\pi$
$308$	0	0
$309$	3.50758i	0.199539i
$310$	0	0
$311$	−30.7386	−1.74303	−0.871514	−	0.490371i	$-0.836861\pi$
$311$	−30.7386	−1.74303	−0.871514	+	0.490371i	$0.836861\pi$
$312$	0	0
$313$	−18.6847	−1.05612	−0.528060	−	0.849207i	$-0.677080\pi$
$313$	−18.6847	−1.05612	−0.528060	+	0.849207i	$0.677080\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.05398i	0.452356i	0.974086	+	0.226178i	$0.0726232\pi$
$317$	8.05398i	0.452356i	−0.974086	+	0.226178i	$0.927377\pi$
$318$	0	0
$319$	1.12311	0.0628818
$320$	0	0
$321$	18.7386	1.04589
$322$	0	0
$323$	8.00000i	0.445132i
$324$	0	0
$325$	− 15.3693i	− 0.852536i
$326$	0	0
$327$	21.8617	1.20896
$328$	0	0
$329$	0	0
$330$	0	0
$331$	28.6847i	1.57665i	0.615258	+	0.788326i	$0.289052\pi$
$331$	28.6847i	1.57665i	−0.615258	+	0.788326i	$0.710948\pi$
$332$	0	0
$333$	− 3.75379i	− 0.205706i
$334$	0	0
$335$	16.6847	0.911580
$336$	0	0
$337$	−31.3693	−1.70880	−0.854398	−	0.519619i	$-0.826074\pi$
$337$	−31.3693	−1.70880	−0.854398	+	0.519619i	$0.826074\pi$
$338$	0	0
$339$	24.3002i	1.31980i
$340$	0	0
$341$	9.56155i	0.517787i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	26.0540	1.40270
$346$	0	0
$347$	2.63068i	0.141222i	0.997504	+	0.0706112i	$0.0224950\pi$
$347$	2.63068i	0.141222i	−0.997504	+	0.0706112i	$0.977505\pi$
$348$	0	0
$349$	11.3693i	0.608586i	0.952579	+	0.304293i	$0.0984201\pi$
$349$	11.3693i	0.608586i	−0.952579	+	0.304293i	$0.901580\pi$
$350$	0	0
$351$	−11.1231	−0.593707
$352$	0	0
$353$	20.4384	1.08783	0.543914	−	0.839141i	$-0.316942\pi$
$353$	20.4384	1.08783	0.543914	+	0.839141i	$0.316942\pi$
$354$	0	0
$355$	− 11.8078i	− 0.626691i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−9.36932	−0.494494	−0.247247	−	0.968953i	$-0.579526\pi$
$359$	−9.36932	−0.494494	−0.247247	+	0.968953i	$0.579526\pi$
$360$	0	0
$361$	−31.7386	−1.67045
$362$	0	0
$363$	− 1.56155i	− 0.0819603i
$364$	0	0
$365$	− 21.3693i	− 1.11852i
$366$	0	0
$367$	38.0540	1.98640	0.993201	−	0.116415i	$-0.0371402\pi$
$367$	38.0540	1.98640	0.993201	+	0.116415i	$0.0371402\pi$
$368$	0	0
$369$	−4.63068	−0.241064
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 3.36932i	− 0.174457i	−0.996188	−	0.0872283i	$-0.972199\pi$
$373$	− 3.36932i	− 0.174457i	0.996188	−	0.0872283i	$-0.0278010\pi$
$374$	0	0
$375$	−14.9309	−0.771027
$376$	0	0
$377$	2.24621	0.115686
$378$	0	0
$379$	14.4384i	0.741653i	0.928702	+	0.370827i	$0.120926\pi$
$379$	14.4384i	0.741653i	−0.928702	+	0.370827i	$0.879074\pi$
$380$	0	0
$381$	14.6307i	0.749553i
$382$	0	0
$383$	−20.6847	−1.05694	−0.528468	−	0.848953i	$-0.677233\pi$
$383$	−20.6847	−1.05694	−0.528468	+	0.848953i	$0.677233\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.00000i	0.203331i
$388$	0	0
$389$	6.19224i	0.313959i	0.987602	+	0.156979i	$0.0501756\pi$
$389$	6.19224i	0.313959i	−0.987602	+	0.156979i	$0.949824\pi$
$390$	0	0
$391$	−5.26137	−0.266079
$392$	0	0
$393$	6.24621	0.315080
$394$	0	0
$395$	− 17.3693i	− 0.873945i
$396$	0	0
$397$	14.0000i	0.702640i	0.936255	+	0.351320i	$0.114267\pi$
$397$	14.0000i	0.702640i	−0.936255	+	0.351320i	$0.885733\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−10.4924	−0.523967	−0.261983	−	0.965072i	$-0.584377\pi$
$401$	−10.4924	−0.523967	−0.261983	+	0.965072i	$0.584377\pi$
$402$	0	0
$403$	19.1231i	0.952590i
$404$	0	0
$405$	24.9309i	1.23882i
$406$	0	0
$407$	6.68466	0.331346
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	− 2.05398i	− 0.101315i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−47.6155	−2.33735
$416$	0	0
$417$	−18.7386	−0.917635
$418$	0	0
$419$	− 30.7386i	− 1.50168i	−0.660484	−	0.750840i	$-0.729649\pi$
$419$	− 30.7386i	− 1.50168i	0.660484	−	0.750840i	$-0.270351\pi$
$420$	0	0
$421$	10.0000i	0.487370i	0.969854	+	0.243685i	$0.0783563\pi$
$421$	10.0000i	0.487370i	−0.969854	+	0.243685i	$0.921644\pi$
$422$	0	0
$423$	2.24621	0.109215
$424$	0	0
$425$	8.63068	0.418650
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 3.12311i	− 0.150785i
$430$	0	0
$431$	−10.7386	−0.517262	−0.258631	−	0.965976i	$-0.583271\pi$
$431$	−10.7386	−0.517262	−0.258631	+	0.965976i	$0.583271\pi$
$432$	0	0
$433$	−6.68466	−0.321244	−0.160622	−	0.987016i	$-0.551350\pi$
$433$	−6.68466	−0.321244	−0.160622	+	0.987016i	$0.551350\pi$
$434$	0	0
$435$	− 6.24621i	− 0.299483i
$436$	0	0
$437$	− 33.3693i	− 1.59627i
$438$	0	0
$439$	−9.75379	−0.465523	−0.232761	−	0.972534i	$-0.574776\pi$
$439$	−9.75379	−0.465523	−0.232761	+	0.972534i	$0.574776\pi$
$440$	0	0
$441$	−3.93087	−0.187184
$442$	0	0
$443$	− 20.6847i	− 0.982758i	−0.870946	−	0.491379i	$-0.836493\pi$
$443$	− 20.6847i	− 0.982758i	0.870946	−	0.491379i	$-0.163507\pi$
$444$	0	0
$445$	− 12.6847i	− 0.601310i
$446$	0	0
$447$	20.4924	0.969258
$448$	0	0
$449$	−17.8078	−0.840400	−0.420200	−	0.907431i	$-0.638040\pi$
$449$	−17.8078	−0.840400	−0.420200	+	0.907431i	$0.638040\pi$
$450$	0	0
$451$	− 8.24621i	− 0.388299i
$452$	0	0
$453$	− 14.6307i	− 0.687409i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−16.6307	−0.777951	−0.388975	−	0.921248i	$-0.627171\pi$
$457$	−16.6307	−0.777951	−0.388975	+	0.921248i	$0.627171\pi$
$458$	0	0
$459$	− 6.24621i	− 0.291548i
$460$	0	0
$461$	− 10.8769i	− 0.506587i	−0.967389	−	0.253294i	$-0.918486\pi$
$461$	− 10.8769i	− 0.506587i	0.967389	−	0.253294i	$-0.0815139\pi$
$462$	0	0
$463$	−5.06913	−0.235582	−0.117791	−	0.993038i	$-0.537581\pi$
$463$	−5.06913	−0.235582	−0.117791	+	0.993038i	$0.537581\pi$
$464$	0	0
$465$	53.1771	2.46603
$466$	0	0
$467$	20.6847i	0.957172i	0.878041	+	0.478586i	$0.158850\pi$
$467$	20.6847i	0.957172i	−0.878041	+	0.478586i	$0.841150\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	16.6847	0.768788
$472$	0	0
$473$	−7.12311	−0.327521
$474$	0	0
$475$	54.7386i	2.51158i
$476$	0	0
$477$	4.63068i	0.212024i
$478$	0	0
$479$	−25.3693	−1.15915	−0.579577	−	0.814918i	$-0.696782\pi$
$479$	−25.3693	−1.15915	−0.579577	+	0.814918i	$0.696782\pi$
$480$	0	0
$481$	13.3693	0.609588
$482$	0	0
$483$	0	0
$484$	0	0
$485$	23.8078i	1.08105i
$486$	0	0
$487$	−19.3153	−0.875262	−0.437631	−	0.899155i	$-0.644182\pi$
$487$	−19.3153	−0.875262	−0.437631	+	0.899155i	$0.644182\pi$
$488$	0	0
$489$	−18.7386	−0.847390
$490$	0	0
$491$	22.7386i	1.02618i	0.858335	+	0.513090i	$0.171499\pi$
$491$	22.7386i	1.02618i	−0.858335	+	0.513090i	$0.828501\pi$
$492$	0	0
$493$	1.26137i	0.0568091i
$494$	0	0
$495$	2.00000	0.0898933
$496$	0	0
$497$	0	0
$498$	0	0
$499$	7.50758i	0.336085i	0.985780	+	0.168043i	$0.0537446\pi$
$499$	7.50758i	0.336085i	−0.985780	+	0.168043i	$0.946255\pi$
$500$	0	0
$501$	− 27.1231i	− 1.21177i
$502$	0	0
$503$	−9.36932	−0.417757	−0.208879	−	0.977942i	$-0.566981\pi$
$503$	−9.36932	−0.417757	−0.208879	+	0.977942i	$0.566981\pi$
$504$	0	0
$505$	−46.7386	−2.07984
$506$	0	0
$507$	14.0540i	0.624159i
$508$	0	0
$509$	− 1.31534i	− 0.0583015i	−0.999575	−	0.0291507i	$-0.990720\pi$
$509$	− 1.31534i	− 0.0583015i	0.999575	−	0.0291507i	$-0.00928028\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	39.6155	1.74907
$514$	0	0
$515$	− 8.00000i	− 0.352522i
$516$	0	0
$517$	4.00000i	0.175920i
$518$	0	0
$519$	−28.1080	−1.23380
$520$	0	0
$521$	−20.0540	−0.878581	−0.439290	−	0.898345i	$-0.644770\pi$
$521$	−20.0540	−0.878581	−0.439290	+	0.898345i	$0.644770\pi$
$522$	0	0
$523$	− 40.1080i	− 1.75380i	−0.480674	−	0.876899i	$-0.659608\pi$
$523$	− 40.1080i	− 1.75380i	0.480674	−	0.876899i	$-0.340392\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−10.7386	−0.467782
$528$	0	0
$529$	−1.05398	−0.0458250
$530$	0	0
$531$	− 7.12311i	− 0.309116i
$532$	0	0
$533$	− 16.4924i	− 0.714366i
$534$	0	0
$535$	−42.7386	−1.84775
$536$	0	0
$537$	−7.31534	−0.315680
$538$	0	0
$539$	− 7.00000i	− 0.301511i
$540$	0	0
$541$	18.0000i	0.773880i	0.922105	+	0.386940i	$0.126468\pi$
$541$	18.0000i	0.773880i	−0.922105	+	0.386940i	$0.873532\pi$
$542$	0	0
$543$	−18.8229	−0.807769
$544$	0	0
$545$	−49.8617	−2.13584
$546$	0	0
$547$	− 12.0000i	− 0.513083i	−0.966533	−	0.256541i	$-0.917417\pi$
$547$	− 12.0000i	− 0.513083i	0.966533	−	0.256541i	$-0.0825830\pi$
$548$	0	0
$549$	− 8.63068i	− 0.368349i
$550$	0	0
$551$	−8.00000	−0.340811
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 37.1771i	− 1.57808i
$556$	0	0
$557$	8.63068i	0.365694i	0.983141	+	0.182847i	$0.0585313\pi$
$557$	8.63068i	0.365694i	−0.983141	+	0.182847i	$0.941469\pi$
$558$	0	0
$559$	−14.2462	−0.602551
$560$	0	0
$561$	1.75379	0.0740450
$562$	0	0
$563$	30.7386i	1.29548i	0.761862	+	0.647739i	$0.224285\pi$
$563$	30.7386i	1.29548i	−0.761862	+	0.647739i	$0.775715\pi$
$564$	0	0
$565$	− 55.4233i	− 2.33168i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−3.36932	−0.141249	−0.0706246	−	0.997503i	$-0.522499\pi$
$569$	−3.36932	−0.141249	−0.0706246	+	0.997503i	$0.522499\pi$
$570$	0	0
$571$	12.0000i	0.502184i	0.967963	+	0.251092i	$0.0807897\pi$
$571$	12.0000i	0.502184i	−0.967963	+	0.251092i	$0.919210\pi$
$572$	0	0
$573$	7.31534i	0.305603i
$574$	0	0
$575$	−36.0000	−1.50130
$576$	0	0
$577$	−6.68466	−0.278286	−0.139143	−	0.990272i	$-0.544435\pi$
$577$	−6.68466	−0.278286	−0.139143	+	0.990272i	$0.544435\pi$
$578$	0	0
$579$	17.7538i	0.737822i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−8.24621	−0.341523
$584$	0	0
$585$	4.00000	0.165380
$586$	0	0
$587$	28.0000i	1.15568i	0.816149	+	0.577842i	$0.196105\pi$
$587$	28.0000i	1.15568i	−0.816149	+	0.577842i	$0.803895\pi$
$588$	0	0
$589$	− 68.1080i	− 2.80634i
$590$	0	0
$591$	12.8769	0.529685
$592$	0	0
$593$	31.8617	1.30840	0.654202	−	0.756320i	$-0.273004\pi$
$593$	31.8617	1.30840	0.654202	+	0.756320i	$0.273004\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3.50758i	0.143556i
$598$	0	0
$599$	14.7386	0.602204	0.301102	−	0.953592i	$-0.402645\pi$
$599$	14.7386	0.602204	0.301102	+	0.953592i	$0.402645\pi$
$600$	0	0
$601$	42.1080	1.71762	0.858810	−	0.512295i	$-0.171205\pi$
$601$	42.1080	1.71762	0.858810	+	0.512295i	$0.171205\pi$
$602$	0	0
$603$	2.63068i	0.107130i
$604$	0	0
$605$	3.56155i	0.144798i
$606$	0	0
$607$	−28.8769	−1.17208	−0.586038	−	0.810283i	$-0.699313\pi$
$607$	−28.8769	−1.17208	−0.586038	+	0.810283i	$0.699313\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8.00000i	0.323645i
$612$	0	0
$613$	7.36932i	0.297644i	0.988864	+	0.148822i	$0.0475481\pi$
$613$	7.36932i	0.297644i	−0.988864	+	0.148822i	$0.952452\pi$
$614$	0	0
$615$	−45.8617	−1.84932
$616$	0	0
$617$	15.7538	0.634224	0.317112	−	0.948388i	$-0.397287\pi$
$617$	15.7538	0.634224	0.317112	+	0.948388i	$0.397287\pi$
$618$	0	0
$619$	4.68466i	0.188292i	0.995558	+	0.0941462i	$0.0300121\pi$
$619$	4.68466i	0.188292i	−0.995558	+	0.0941462i	$0.969988\pi$
$620$	0	0
$621$	26.0540i	1.04551i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−4.36932	−0.174773
$626$	0	0
$627$	11.1231i	0.444214i
$628$	0	0
$629$	7.50758i	0.299347i
$630$	0	0
$631$	23.4233	0.932467	0.466233	−	0.884662i	$-0.345611\pi$
$631$	23.4233	0.932467	0.466233	+	0.884662i	$0.345611\pi$
$632$	0	0
$633$	−25.7538	−1.02362
$634$	0	0
$635$	− 33.3693i	− 1.32422i
$636$	0	0
$637$	− 14.0000i	− 0.554700i
$638$	0	0
$639$	1.86174	0.0736493
$640$	0	0
$641$	0.930870	0.0367671	0.0183836	−	0.999831i	$-0.494148\pi$
$641$	0.930870	0.0367671	0.0183836	+	0.999831i	$0.494148\pi$
$642$	0	0
$643$	38.4384i	1.51586i	0.652333	+	0.757932i	$0.273790\pi$
$643$	38.4384i	1.51586i	−0.652333	+	0.757932i	$0.726210\pi$
$644$	0	0
$645$	39.6155i	1.55986i
$646$	0	0
$647$	19.3153	0.759364	0.379682	−	0.925117i	$-0.376033\pi$
$647$	19.3153	0.759364	0.379682	+	0.925117i	$0.376033\pi$
$648$	0	0
$649$	12.6847	0.497916
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 1.31534i	− 0.0514733i	−0.999669	−	0.0257366i	$-0.991807\pi$
$653$	− 1.31534i	− 0.0514733i	0.999669	−	0.0257366i	$-0.00819313\pi$
$654$	0	0
$655$	−14.2462	−0.556646
$656$	0	0
$657$	3.36932	0.131450
$658$	0	0
$659$	37.3693i	1.45570i	0.685735	+	0.727851i	$0.259481\pi$
$659$	37.3693i	1.45570i	−0.685735	+	0.727851i	$0.740519\pi$
$660$	0	0
$661$	− 16.0540i	− 0.624427i	−0.950012	−	0.312214i	$-0.898930\pi$
$661$	− 16.0540i	− 0.624427i	0.950012	−	0.312214i	$-0.101070\pi$
$662$	0	0
$663$	3.50758	0.136223
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 5.26137i	− 0.203721i
$668$	0	0
$669$	− 7.31534i	− 0.282827i
$670$	0	0
$671$	15.3693	0.593326
$672$	0	0
$673$	−24.7386	−0.953604	−0.476802	−	0.879011i	$-0.658204\pi$
$673$	−24.7386	−0.953604	−0.476802	+	0.879011i	$0.658204\pi$
$674$	0	0
$675$	− 42.7386i	− 1.64501i
$676$	0	0
$677$	19.8617i	0.763349i	0.924297	+	0.381674i	$0.124652\pi$
$677$	19.8617i	0.763349i	−0.924297	+	0.381674i	$0.875348\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	31.2311	1.19678
$682$	0	0
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	0	0
$685$	4.68466i	0.178992i
$686$	0	0
$687$	−14.5464	−0.554980
$688$	0	0
$689$	−16.4924	−0.628311
$690$	0	0
$691$	− 28.3002i	− 1.07659i	−0.842757	−	0.538295i	$-0.819069\pi$
$691$	− 28.3002i	− 1.07659i	0.842757	−	0.538295i	$-0.180931\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	42.7386	1.62117
$696$	0	0
$697$	9.26137	0.350799
$698$	0	0
$699$	1.75379i	0.0663344i
$700$	0	0
$701$	− 6.38447i	− 0.241138i	−0.992705	−	0.120569i	$-0.961528\pi$
$701$	− 6.38447i	− 0.241138i	0.992705	−	0.120569i	$-0.0384719\pi$
$702$	0	0
$703$	−47.6155	−1.79585
$704$	0	0
$705$	22.2462	0.837841
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 22.6847i	− 0.851940i	−0.904737	−	0.425970i	$-0.859933\pi$
$709$	− 22.6847i	− 0.851940i	0.904737	−	0.425970i	$-0.140067\pi$
$710$	0	0
$711$	2.73863	0.102707
$712$	0	0
$713$	44.7926	1.67750
$714$	0	0
$715$	7.12311i	0.266389i
$716$	0	0
$717$	39.6155i	1.47947i
$718$	0	0
$719$	−28.6847	−1.06976	−0.534879	−	0.844929i	$-0.679643\pi$
$719$	−28.6847	−1.06976	−0.534879	+	0.844929i	$0.679643\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	42.7386i	1.58947i
$724$	0	0
$725$	8.63068i	0.320536i
$726$	0	0
$727$	28.6847	1.06386	0.531928	−	0.846790i	$-0.321468\pi$
$727$	28.6847	1.06386	0.531928	+	0.846790i	$0.321468\pi$
$728$	0	0
$729$	−29.9848	−1.11055
$730$	0	0
$731$	− 8.00000i	− 0.295891i
$732$	0	0
$733$	38.1080i	1.40755i	0.710423	+	0.703775i	$0.248504\pi$
$733$	38.1080i	1.40755i	−0.710423	+	0.703775i	$0.751496\pi$
$734$	0	0
$735$	−38.9309	−1.43599
$736$	0	0
$737$	−4.68466	−0.172562
$738$	0	0
$739$	11.6155i	0.427284i	0.976912	+	0.213642i	$0.0685326\pi$
$739$	11.6155i	0.427284i	−0.976912	+	0.213642i	$0.931467\pi$
$740$	0	0
$741$	22.2462i	0.817235i
$742$	0	0
$743$	18.7386	0.687454	0.343727	−	0.939070i	$-0.388311\pi$
$743$	18.7386	0.687454	0.343727	+	0.939070i	$0.388311\pi$
$744$	0	0
$745$	−46.7386	−1.71237
$746$	0	0
$747$	− 7.50758i	− 0.274688i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	33.1771	1.21065	0.605324	−	0.795979i	$-0.293043\pi$
$751$	33.1771	1.21065	0.605324	+	0.795979i	$0.293043\pi$
$752$	0	0
$753$	−21.9460	−0.799758
$754$	0	0
$755$	33.3693i	1.21443i
$756$	0	0
$757$	10.0000i	0.363456i	0.983349	+	0.181728i	$0.0581691\pi$
$757$	10.0000i	0.363456i	−0.983349	+	0.181728i	$0.941831\pi$
$758$	0	0
$759$	−7.31534	−0.265530
$760$	0	0
$761$	20.2462	0.733925	0.366962	−	0.930236i	$-0.380398\pi$
$761$	20.2462	0.733925	0.366962	+	0.930236i	$0.380398\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	2.24621i	0.0812119i
$766$	0	0
$767$	25.3693	0.916033
$768$	0	0
$769$	−38.0000	−1.37032	−0.685158	−	0.728395i	$-0.740267\pi$
$769$	−38.0000	−1.37032	−0.685158	+	0.728395i	$0.740267\pi$
$770$	0	0
$771$	12.8769i	0.463750i
$772$	0	0
$773$	8.24621i	0.296596i	0.988943	+	0.148298i	$0.0473794\pi$
$773$	8.24621i	0.296596i	−0.988943	+	0.148298i	$0.952621\pi$
$774$	0	0
$775$	−73.4773	−2.63938
$776$	0	0
$777$	0	0
$778$	0	0
$779$	58.7386i	2.10453i
$780$	0	0
$781$	3.31534i	0.118632i
$782$	0	0
$783$	6.24621	0.223221
$784$	0	0
$785$	−38.0540	−1.35820
$786$	0	0
$787$	12.0000i	0.427754i	0.976861	+	0.213877i	$0.0686091\pi$
$787$	12.0000i	0.427754i	−0.976861	+	0.213877i	$0.931391\pi$
$788$	0	0
$789$	12.4924i	0.444742i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	30.7386	1.09156
$794$	0	0
$795$	45.8617i	1.62655i
$796$	0	0
$797$	− 20.4384i	− 0.723967i	−0.932185	−	0.361983i	$-0.882100\pi$
$797$	− 20.4384i	− 0.723967i	0.932185	−	0.361983i	$-0.117900\pi$
$798$	0	0
$799$	−4.49242	−0.158930
$800$	0	0
$801$	2.00000	0.0706665
$802$	0	0
$803$	6.00000i	0.211735i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	35.1231	1.23639
$808$	0	0
$809$	−8.24621	−0.289921	−0.144961	−	0.989437i	$-0.546306\pi$
$809$	−8.24621	−0.289921	−0.144961	+	0.989437i	$0.546306\pi$
$810$	0	0
$811$	26.6307i	0.935130i	0.883959	+	0.467565i	$0.154869\pi$
$811$	26.6307i	0.935130i	−0.883959	+	0.467565i	$0.845131\pi$
$812$	0	0
$813$	7.61553i	0.267088i
$814$	0	0
$815$	42.7386	1.49707
$816$	0	0
$817$	50.7386	1.77512
$818$	0	0
$819$	0	0
$820$	0	0
$821$	48.7386i	1.70099i	0.525983	+	0.850495i	$0.323698\pi$
$821$	48.7386i	1.70099i	−0.525983	+	0.850495i	$0.676302\pi$
$822$	0	0
$823$	14.4384	0.503293	0.251646	−	0.967819i	$-0.419028\pi$
$823$	14.4384	0.503293	0.251646	+	0.967819i	$0.419028\pi$
$824$	0	0
$825$	12.0000	0.417786
$826$	0	0
$827$	− 21.3693i	− 0.743084i	−0.928416	−	0.371542i	$-0.878829\pi$
$827$	− 21.3693i	− 0.743084i	0.928416	−	0.371542i	$-0.121171\pi$
$828$	0	0
$829$	5.31534i	0.184609i	0.995731	+	0.0923047i	$0.0294234\pi$
$829$	5.31534i	0.184609i	−0.995731	+	0.0923047i	$0.970577\pi$
$830$	0	0
$831$	3.12311	0.108339
$832$	0	0
$833$	7.86174	0.272393
$834$	0	0
$835$	61.8617i	2.14081i
$836$	0	0
$837$	53.1771i	1.83807i
$838$	0	0
$839$	31.4233	1.08485	0.542426	−	0.840103i	$-0.317506\pi$
$839$	31.4233	1.08485	0.542426	+	0.840103i	$0.317506\pi$
$840$	0	0
$841$	27.7386	0.956505
$842$	0	0
$843$	9.36932i	0.322696i
$844$	0	0
$845$	− 32.0540i	− 1.10269i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−40.9848	−1.40660
$850$	0	0
$851$	− 31.3153i	− 1.07348i
$852$	0	0
$853$	8.73863i	0.299205i	0.988746	+	0.149603i	$0.0477994\pi$
$853$	8.73863i	0.299205i	−0.988746	+	0.149603i	$0.952201\pi$
$854$	0	0
$855$	−14.2462	−0.487210
$856$	0	0
$857$	−13.1231	−0.448277	−0.224138	−	0.974557i	$-0.571957\pi$
$857$	−13.1231	−0.448277	−0.224138	+	0.974557i	$0.571957\pi$
$858$	0	0
$859$	29.0691i	0.991826i	0.868372	+	0.495913i	$0.165167\pi$
$859$	29.0691i	0.991826i	−0.868372	+	0.495913i	$0.834833\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	30.7386	1.04636	0.523178	−	0.852224i	$-0.324746\pi$
$863$	30.7386	1.04636	0.523178	+	0.852224i	$0.324746\pi$
$864$	0	0
$865$	64.1080	2.17974
$866$	0	0
$867$	− 24.5767i	− 0.834669i
$868$	0	0
$869$	4.87689i	0.165437i
$870$	0	0
$871$	−9.36932	−0.317467
$872$	0	0
$873$	−3.75379	−0.127047
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 56.7386i	− 1.91593i	−0.286889	−	0.957964i	$-0.592621\pi$
$877$	− 56.7386i	− 1.91593i	0.286889	−	0.957964i	$-0.407379\pi$
$878$	0	0
$879$	35.1231	1.18467
$880$	0	0
$881$	34.3002	1.15560	0.577801	−	0.816177i	$-0.303911\pi$
$881$	34.3002	1.15560	0.577801	+	0.816177i	$0.303911\pi$
$882$	0	0
$883$	36.0000i	1.21150i	0.795656	+	0.605748i	$0.207126\pi$
$883$	36.0000i	1.21150i	−0.795656	+	0.605748i	$0.792874\pi$
$884$	0	0
$885$	− 70.5464i	− 2.37139i
$886$	0	0
$887$	−52.1080	−1.74961	−0.874807	−	0.484472i	$-0.839012\pi$
$887$	−52.1080	−1.74961	−0.874807	+	0.484472i	$0.839012\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 7.00000i	− 0.234509i
$892$	0	0
$893$	− 28.4924i	− 0.953463i
$894$	0	0
$895$	16.6847	0.557707
$896$	0	0
$897$	−14.6307	−0.488504
$898$	0	0
$899$	− 10.7386i	− 0.358153i
$900$	0	0
$901$	− 9.26137i	− 0.308541i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	42.9309	1.42707
$906$	0	0
$907$	16.4924i	0.547622i	0.961784	+	0.273811i	$0.0882843\pi$
$907$	16.4924i	0.547622i	−0.961784	+	0.273811i	$0.911716\pi$
$908$	0	0
$909$	− 7.36932i	− 0.244425i
$910$	0	0
$911$	22.7386	0.753365	0.376682	−	0.926343i	$-0.377065\pi$
$911$	22.7386	0.753365	0.376682	+	0.926343i	$0.377065\pi$
$912$	0	0
$913$	13.3693	0.442460
$914$	0	0
$915$	− 85.4773i	− 2.82579i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−23.6155	−0.779004	−0.389502	−	0.921026i	$-0.627353\pi$
$919$	−23.6155	−0.779004	−0.389502	+	0.921026i	$0.627353\pi$
$920$	0	0
$921$	4.10795	0.135362
$922$	0	0
$923$	6.63068i	0.218252i
$924$	0	0
$925$	51.3693i	1.68901i
$926$	0	0
$927$	1.26137	0.0414287
$928$	0	0
$929$	−10.4924	−0.344245	−0.172123	−	0.985076i	$-0.555063\pi$
$929$	−10.4924	−0.344245	−0.172123	+	0.985076i	$0.555063\pi$
$930$	0	0
$931$	49.8617i	1.63415i
$932$	0	0
$933$	− 48.0000i	− 1.57145i
$934$	0	0
$935$	−4.00000	−0.130814
$936$	0	0
$937$	−28.7386	−0.938850	−0.469425	−	0.882972i	$-0.655539\pi$
$937$	−28.7386	−0.938850	−0.469425	+	0.882972i	$0.655539\pi$
$938$	0	0
$939$	− 29.1771i	− 0.952158i
$940$	0	0
$941$	8.24621i	0.268819i	0.990926	+	0.134409i	$0.0429137\pi$
$941$	8.24621i	0.268819i	−0.990926	+	0.134409i	$0.957086\pi$
$942$	0	0
$943$	−38.6307	−1.25799
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 19.3153i	− 0.627664i	−0.949478	−	0.313832i	$-0.898387\pi$
$947$	− 19.3153i	− 0.627664i	0.949478	−	0.313832i	$-0.101613\pi$
$948$	0	0
$949$	12.0000i	0.389536i
$950$	0	0
$951$	−12.5767	−0.407828
$952$	0	0
$953$	6.38447	0.206813	0.103407	−	0.994639i	$-0.467026\pi$
$953$	6.38447	0.206813	0.103407	+	0.994639i	$0.467026\pi$
$954$	0	0
$955$	− 16.6847i	− 0.539903i
$956$	0	0
$957$	1.75379i	0.0566919i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	60.4233	1.94914
$962$	0	0
$963$	− 6.73863i	− 0.217149i
$964$	0	0
$965$	− 40.4924i	− 1.30350i
$966$	0	0
$967$	18.7386	0.602594	0.301297	−	0.953530i	$-0.402580\pi$
$967$	18.7386	0.602594	0.301297	+	0.953530i	$0.402580\pi$
$968$	0	0
$969$	−12.4924	−0.401314
$970$	0	0
$971$	28.6847i	0.920534i	0.887780	+	0.460267i	$0.152246\pi$
$971$	28.6847i	0.920534i	−0.887780	+	0.460267i	$0.847754\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	24.0000	0.768615
$976$	0	0
$977$	53.0388	1.69686	0.848431	−	0.529306i	$-0.177548\pi$
$977$	53.0388	1.69686	0.848431	+	0.529306i	$0.177548\pi$
$978$	0	0
$979$	3.56155i	0.113828i
$980$	0	0
$981$	− 7.86174i	− 0.251006i
$982$	0	0
$983$	28.6847	0.914899	0.457449	−	0.889236i	$-0.348763\pi$
$983$	28.6847	0.914899	0.457449	+	0.889236i	$0.348763\pi$
$984$	0	0
$985$	−29.3693	−0.935784
$986$	0	0
$987$	0	0
$988$	0	0
$989$	33.3693i	1.06108i
$990$	0	0
$991$	−54.7386	−1.73883	−0.869415	−	0.494083i	$-0.835504\pi$
$991$	−54.7386	−1.73883	−0.869415	+	0.494083i	$0.835504\pi$
$992$	0	0
$993$	−44.7926	−1.42145
$994$	0	0
$995$	− 8.00000i	− 0.253617i
$996$	0	0
$997$	− 0.630683i	− 0.0199739i	−0.999950	−	0.00998697i	$-0.996821\pi$
$997$	− 0.630683i	− 0.0199739i	0.999950	−	0.00998697i	$-0.00317900\pi$
$998$	0	0
$999$	37.1771	1.17623

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2816.2.c.t.1409.3		4
4.3	odd	2		2816.2.c.s.1409.2		4
8.3	odd	2		2816.2.c.s.1409.3		4
8.5	even	2	inner	2816.2.c.t.1409.2		4
16.3	odd	4		704.2.a.n.1.2		2
16.5	even	4		352.2.a.g.1.2	✓	2
16.11	odd	4		352.2.a.h.1.1	yes	2
16.13	even	4		704.2.a.o.1.1		2
48.5	odd	4		3168.2.a.bd.1.1		2
48.11	even	4		3168.2.a.bc.1.1		2
48.29	odd	4		6336.2.a.cv.1.2		2
48.35	even	4		6336.2.a.cw.1.2		2
80.59	odd	4		8800.2.a.bd.1.2		2
80.69	even	4		8800.2.a.be.1.1		2
176.21	odd	4		3872.2.a.p.1.2		2
176.43	even	4		3872.2.a.ba.1.1		2
176.109	odd	4		7744.2.a.cm.1.1		2
176.131	even	4		7744.2.a.bw.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
352.2.a.g.1.2	✓	2	16.5	even	4
352.2.a.h.1.1	yes	2	16.11	odd	4
704.2.a.n.1.2		2	16.3	odd	4
704.2.a.o.1.1		2	16.13	even	4
2816.2.c.s.1409.2		4	4.3	odd	2
2816.2.c.s.1409.3		4	8.3	odd	2
2816.2.c.t.1409.2		4	8.5	even	2	inner
2816.2.c.t.1409.3		4	1.1	even	1	trivial
3168.2.a.bc.1.1		2	48.11	even	4
3168.2.a.bd.1.1		2	48.5	odd	4
3872.2.a.p.1.2		2	176.21	odd	4
3872.2.a.ba.1.1		2	176.43	even	4
6336.2.a.cv.1.2		2	48.29	odd	4
6336.2.a.cw.1.2		2	48.35	even	4
7744.2.a.bw.1.2		2	176.131	even	4
7744.2.a.cm.1.1		2	176.109	odd	4
8800.2.a.bd.1.2		2	80.59	odd	4
8800.2.a.be.1.1		2	80.69	even	4

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

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2816.2.c.t.1409.3

Level	$2816$
Weight	$2$
Character	2816.1409
Analytic conductor	$22.486$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$