LMFDB - Embedded newform 6292.2.a.v.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6292,2,Mod(1,6292)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6292, 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("6292.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6292 = 2^{2} \cdot 11^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6292.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:	$50.2418729518$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 19x^{8} + 40x^{7} + 106x^{6} - 244x^{5} - 154x^{4} + 488x^{3} - 107x^{2} - 138x + 33 $ x^10 - 2x^9 - 19x^8 + 40x^7 + 106x^6 - 244x^5 - 154x^4 + 488x^3 - 107x^2 - 138*x + 33
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.99408$ of defining polynomial
Character	$\chi$	$=$	6292.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.99408 q^{3} -2.78548 q^{5} +3.21617 q^{7} +0.976354 q^{9} +O(q^{10})$	$q-1.99408 q^{3} -2.78548 q^{5} +3.21617 q^{7} +0.976354 q^{9} -1.00000 q^{13} +5.55447 q^{15} -2.38564 q^{17} +7.07321 q^{19} -6.41330 q^{21} +5.82556 q^{23} +2.75889 q^{25} +4.03531 q^{27} +0.273393 q^{29} -8.15791 q^{31} -8.95857 q^{35} -7.71874 q^{37} +1.99408 q^{39} +5.36338 q^{41} -4.83138 q^{43} -2.71961 q^{45} +10.9497 q^{47} +3.34374 q^{49} +4.75715 q^{51} -2.27559 q^{53} -14.1046 q^{57} +11.1250 q^{59} -10.6235 q^{61} +3.14012 q^{63} +2.78548 q^{65} +11.7704 q^{67} -11.6166 q^{69} +4.13323 q^{71} -9.94948 q^{73} -5.50145 q^{75} -5.18790 q^{79} -10.9758 q^{81} -8.83200 q^{83} +6.64514 q^{85} -0.545168 q^{87} -5.10075 q^{89} -3.21617 q^{91} +16.2675 q^{93} -19.7023 q^{95} +8.71314 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{3} - 6 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) $ 10 * q - 2 * q^3 - 6 * q^5 + 4 * q^7 + 12 * q^9	$ 10 q - 2 q^{3} - 6 q^{5} + 4 q^{7} + 12 q^{9} - 10 q^{13} - 4 q^{15} + 20 q^{17} + 14 q^{19} + 2 q^{21} - 2 q^{23} + 2 q^{25} + 10 q^{27} + 16 q^{29} - 2 q^{31} - 20 q^{35} + 2 q^{39} + 22 q^{41} - 12 q^{43} - 8 q^{45} + 6 q^{47} - 2 q^{49} - 8 q^{51} + 6 q^{53} + 12 q^{57} - 4 q^{59} + 24 q^{61} + 68 q^{63} + 6 q^{65} - 6 q^{67} - 2 q^{69} - 8 q^{71} - 6 q^{73} - 18 q^{75} - 24 q^{79} + 10 q^{81} + 22 q^{83} + 34 q^{85} - 4 q^{87} - 42 q^{89} - 4 q^{91} - 38 q^{93} + 14 q^{97}+O(q^{100}) $ 10 * q - 2 * q^3 - 6 * q^5 + 4 * q^7 + 12 * q^9 - 10 * q^13 - 4 * q^15 + 20 * q^17 + 14 * q^19 + 2 * q^21 - 2 * q^23 + 2 * q^25 + 10 * q^27 + 16 * q^29 - 2 * q^31 - 20 * q^35 + 2 * q^39 + 22 * q^41 - 12 * q^43 - 8 * q^45 + 6 * q^47 - 2 * q^49 - 8 * q^51 + 6 * q^53 + 12 * q^57 - 4 * q^59 + 24 * q^61 + 68 * q^63 + 6 * q^65 - 6 * q^67 - 2 * q^69 - 8 * q^71 - 6 * q^73 - 18 * q^75 - 24 * q^79 + 10 * q^81 + 22 * q^83 + 34 * q^85 - 4 * q^87 - 42 * q^89 - 4 * q^91 - 38 * q^93 + 14 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.99408	−1.15128	−0.575641	−	0.817702i	$-0.695247\pi$
$3$	−1.99408	−1.15128	−0.575641	+	0.817702i	$0.695247\pi$
$4$	0	0
$5$	−2.78548	−1.24570	−0.622852	−	0.782340i	$-0.714026\pi$
$5$	−2.78548	−1.24570	−0.622852	+	0.782340i	$0.714026\pi$
$6$	0	0
$7$	3.21617	1.21560	0.607799	−	0.794091i	$-0.292053\pi$
$7$	3.21617	1.21560	0.607799	+	0.794091i	$0.292053\pi$
$8$	0	0
$9$	0.976354	0.325451
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	5.55447	1.43416
$16$	0	0
$17$	−2.38564	−0.578602	−0.289301	−	0.957238i	$-0.593423\pi$
$17$	−2.38564	−0.578602	−0.289301	+	0.957238i	$0.593423\pi$
$18$	0	0
$19$	7.07321	1.62271	0.811353	−	0.584556i	$-0.198731\pi$
$19$	7.07321	1.62271	0.811353	+	0.584556i	$0.198731\pi$
$20$	0	0
$21$	−6.41330	−1.39950
$22$	0	0
$23$	5.82556	1.21471	0.607356	−	0.794430i	$-0.292230\pi$
$23$	5.82556	1.21471	0.607356	+	0.794430i	$0.292230\pi$
$24$	0	0
$25$	2.75889	0.551779
$26$	0	0
$27$	4.03531	0.776596
$28$	0	0
$29$	0.273393	0.0507679	0.0253839	−	0.999678i	$-0.491919\pi$
$29$	0.273393	0.0507679	0.0253839	+	0.999678i	$0.491919\pi$
$30$	0	0
$31$	−8.15791	−1.46520	−0.732602	−	0.680657i	$-0.761694\pi$
$31$	−8.15791	−1.46520	−0.732602	+	0.680657i	$0.761694\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−8.95857	−1.51427
$36$	0	0
$37$	−7.71874	−1.26895	−0.634477	−	0.772942i	$-0.718784\pi$
$37$	−7.71874	−1.26895	−0.634477	+	0.772942i	$0.718784\pi$
$38$	0	0
$39$	1.99408	0.319308
$40$	0	0
$41$	5.36338	0.837620	0.418810	−	0.908074i	$-0.362447\pi$
$41$	5.36338	0.837620	0.418810	+	0.908074i	$0.362447\pi$
$42$	0	0
$43$	−4.83138	−0.736779	−0.368390	−	0.929672i	$-0.620091\pi$
$43$	−4.83138	−0.736779	−0.368390	+	0.929672i	$0.620091\pi$
$44$	0	0
$45$	−2.71961	−0.405416
$46$	0	0
$47$	10.9497	1.59717	0.798586	−	0.601881i	$-0.205582\pi$
$47$	10.9497	1.59717	0.798586	+	0.601881i	$0.205582\pi$
$48$	0	0
$49$	3.34374	0.477677
$50$	0	0
$51$	4.75715	0.666134
$52$	0	0
$53$	−2.27559	−0.312577	−0.156288	−	0.987711i	$-0.549953\pi$
$53$	−2.27559	−0.312577	−0.156288	+	0.987711i	$0.549953\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−14.1046	−1.86819
$58$	0	0
$59$	11.1250	1.44836	0.724178	−	0.689613i	$-0.242219\pi$
$59$	11.1250	1.44836	0.724178	+	0.689613i	$0.242219\pi$
$60$	0	0
$61$	−10.6235	−1.36020	−0.680101	−	0.733118i	$-0.738064\pi$
$61$	−10.6235	−1.36020	−0.680101	+	0.733118i	$0.738064\pi$
$62$	0	0
$63$	3.14012	0.395618
$64$	0	0
$65$	2.78548	0.345496
$66$	0	0
$67$	11.7704	1.43799	0.718993	−	0.695017i	$-0.244603\pi$
$67$	11.7704	1.43799	0.718993	+	0.695017i	$0.244603\pi$
$68$	0	0
$69$	−11.6166	−1.39848
$70$	0	0
$71$	4.13323	0.490524	0.245262	−	0.969457i	$-0.421126\pi$
$71$	4.13323	0.490524	0.245262	+	0.969457i	$0.421126\pi$
$72$	0	0
$73$	−9.94948	−1.16450	−0.582249	−	0.813010i	$-0.697827\pi$
$73$	−9.94948	−1.16450	−0.582249	+	0.813010i	$0.697827\pi$
$74$	0	0
$75$	−5.50145	−0.635253
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−5.18790	−0.583684	−0.291842	−	0.956467i	$-0.594268\pi$
$79$	−5.18790	−0.583684	−0.291842	+	0.956467i	$0.594268\pi$
$80$	0	0
$81$	−10.9758	−1.21953
$82$	0	0
$83$	−8.83200	−0.969438	−0.484719	−	0.874670i	$-0.661078\pi$
$83$	−8.83200	−0.969438	−0.484719	+	0.874670i	$0.661078\pi$
$84$	0	0
$85$	6.64514	0.720766
$86$	0	0
$87$	−0.545168	−0.0584481
$88$	0	0
$89$	−5.10075	−0.540679	−0.270339	−	0.962765i	$-0.587136\pi$
$89$	−5.10075	−0.540679	−0.270339	+	0.962765i	$0.587136\pi$
$90$	0	0
$91$	−3.21617	−0.337146
$92$	0	0
$93$	16.2675	1.68686
$94$	0	0
$95$	−19.7023	−2.02141
$96$	0	0
$97$	8.71314	0.884686	0.442343	−	0.896846i	$-0.354147\pi$
$97$	8.71314	0.884686	0.442343	+	0.896846i	$0.354147\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−3.02355	−0.300855	−0.150427	−	0.988621i	$-0.548065\pi$
$101$	−3.02355	−0.300855	−0.150427	+	0.988621i	$0.548065\pi$
$102$	0	0
$103$	−4.05855	−0.399900	−0.199950	−	0.979806i	$-0.564078\pi$
$103$	−4.05855	−0.399900	−0.199950	+	0.979806i	$0.564078\pi$
$104$	0	0
$105$	17.8641	1.74336
$106$	0	0
$107$	−12.7447	−1.23207	−0.616036	−	0.787718i	$-0.711262\pi$
$107$	−12.7447	−1.23207	−0.616036	+	0.787718i	$0.711262\pi$
$108$	0	0
$109$	−1.43206	−0.137166	−0.0685830	−	0.997645i	$-0.521848\pi$
$109$	−1.43206	−0.137166	−0.0685830	+	0.997645i	$0.521848\pi$
$110$	0	0
$111$	15.3918	1.46092
$112$	0	0
$113$	−9.73205	−0.915514	−0.457757	−	0.889077i	$-0.651347\pi$
$113$	−9.73205	−0.915514	−0.457757	+	0.889077i	$0.651347\pi$
$114$	0	0
$115$	−16.2270	−1.51317
$116$	0	0
$117$	−0.976354	−0.0902639
$118$	0	0
$119$	−7.67261	−0.703347
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−10.6950	−0.964337
$124$	0	0
$125$	6.24256	0.558351
$126$	0	0
$127$	19.0657	1.69180	0.845902	−	0.533339i	$-0.179063\pi$
$127$	19.0657	1.69180	0.845902	+	0.533339i	$0.179063\pi$
$128$	0	0
$129$	9.63416	0.848241
$130$	0	0
$131$	−8.96242	−0.783050	−0.391525	−	0.920168i	$-0.628052\pi$
$131$	−8.96242	−0.783050	−0.391525	+	0.920168i	$0.628052\pi$
$132$	0	0
$133$	22.7487	1.97256
$134$	0	0
$135$	−11.2403	−0.967409
$136$	0	0
$137$	−10.5768	−0.903637	−0.451819	−	0.892110i	$-0.649225\pi$
$137$	−10.5768	−0.903637	−0.451819	+	0.892110i	$0.649225\pi$
$138$	0	0
$139$	−4.20402	−0.356580	−0.178290	−	0.983978i	$-0.557057\pi$
$139$	−4.20402	−0.356580	−0.178290	+	0.983978i	$0.557057\pi$
$140$	0	0
$141$	−21.8345	−1.83880
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−0.761531	−0.0632417
$146$	0	0
$147$	−6.66769	−0.549942
$148$	0	0
$149$	8.32334	0.681875	0.340937	−	0.940086i	$-0.389256\pi$
$149$	8.32334	0.681875	0.340937	+	0.940086i	$0.389256\pi$
$150$	0	0
$151$	12.7350	1.03636	0.518179	−	0.855272i	$-0.326610\pi$
$151$	12.7350	1.03636	0.518179	+	0.855272i	$0.326610\pi$
$152$	0	0
$153$	−2.32922	−0.188307
$154$	0	0
$155$	22.7237	1.82521
$156$	0	0
$157$	−4.72894	−0.377410	−0.188705	−	0.982034i	$-0.560429\pi$
$157$	−4.72894	−0.377410	−0.188705	+	0.982034i	$0.560429\pi$
$158$	0	0
$159$	4.53772	0.359864
$160$	0	0
$161$	18.7360	1.47660
$162$	0	0
$163$	−20.1040	−1.57467	−0.787335	−	0.616525i	$-0.788540\pi$
$163$	−20.1040	−1.57467	−0.787335	+	0.616525i	$0.788540\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	22.2950	1.72524	0.862618	−	0.505856i	$-0.168823\pi$
$167$	22.2950	1.72524	0.862618	+	0.505856i	$0.168823\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	6.90596	0.528112
$172$	0	0
$173$	18.3676	1.39646	0.698232	−	0.715871i	$-0.253970\pi$
$173$	18.3676	1.39646	0.698232	+	0.715871i	$0.253970\pi$
$174$	0	0
$175$	8.87306	0.670741
$176$	0	0
$177$	−22.1842	−1.66747
$178$	0	0
$179$	19.7199	1.47393	0.736967	−	0.675929i	$-0.236258\pi$
$179$	19.7199	1.47393	0.736967	+	0.675929i	$0.236258\pi$
$180$	0	0
$181$	−11.8838	−0.883313	−0.441657	−	0.897184i	$-0.645609\pi$
$181$	−11.8838	−0.883313	−0.441657	+	0.897184i	$0.645609\pi$
$182$	0	0
$183$	21.1842	1.56598
$184$	0	0
$185$	21.5004	1.58074
$186$	0	0
$187$	0	0
$188$	0	0
$189$	12.9782	0.944028
$190$	0	0
$191$	8.01125	0.579674	0.289837	−	0.957076i	$-0.406399\pi$
$191$	8.01125	0.579674	0.289837	+	0.957076i	$0.406399\pi$
$192$	0	0
$193$	19.9487	1.43594	0.717968	−	0.696076i	$-0.245072\pi$
$193$	19.9487	1.43594	0.717968	+	0.696076i	$0.245072\pi$
$194$	0	0
$195$	−5.55447	−0.397764
$196$	0	0
$197$	−12.7780	−0.910392	−0.455196	−	0.890391i	$-0.650431\pi$
$197$	−12.7780	−0.910392	−0.455196	+	0.890391i	$0.650431\pi$
$198$	0	0
$199$	12.1966	0.864594	0.432297	−	0.901731i	$-0.357703\pi$
$199$	12.1966	0.864594	0.432297	+	0.901731i	$0.357703\pi$
$200$	0	0
$201$	−23.4712	−1.65553
$202$	0	0
$203$	0.879279	0.0617133
$204$	0	0
$205$	−14.9396	−1.04343
$206$	0	0
$207$	5.68780	0.395330
$208$	0	0
$209$	0	0
$210$	0	0
$211$	1.58721	0.109268	0.0546341	−	0.998506i	$-0.482601\pi$
$211$	1.58721	0.109268	0.0546341	+	0.998506i	$0.482601\pi$
$212$	0	0
$213$	−8.24199	−0.564732
$214$	0	0
$215$	13.4577	0.917809
$216$	0	0
$217$	−26.2372	−1.78110
$218$	0	0
$219$	19.8401	1.34067
$220$	0	0
$221$	2.38564	0.160475
$222$	0	0
$223$	−1.65393	−0.110755	−0.0553775	−	0.998465i	$-0.517636\pi$
$223$	−1.65393	−0.110755	−0.0553775	+	0.998465i	$0.517636\pi$
$224$	0	0
$225$	2.69365	0.179577
$226$	0	0
$227$	−4.04640	−0.268569	−0.134284	−	0.990943i	$-0.542874\pi$
$227$	−4.04640	−0.268569	−0.134284	+	0.990943i	$0.542874\pi$
$228$	0	0
$229$	−9.55806	−0.631614	−0.315807	−	0.948823i	$-0.602275\pi$
$229$	−9.55806	−0.631614	−0.315807	+	0.948823i	$0.602275\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.1840	0.798197	0.399099	−	0.916908i	$-0.369323\pi$
$233$	12.1840	0.798197	0.399099	+	0.916908i	$0.369323\pi$
$234$	0	0
$235$	−30.5000	−1.98960
$236$	0	0
$237$	10.3451	0.671985
$238$	0	0
$239$	19.1386	1.23797	0.618986	−	0.785402i	$-0.287544\pi$
$239$	19.1386	1.23797	0.618986	+	0.785402i	$0.287544\pi$
$240$	0	0
$241$	−11.4947	−0.740441	−0.370220	−	0.928944i	$-0.620718\pi$
$241$	−11.4947	−0.740441	−0.370220	+	0.928944i	$0.620718\pi$
$242$	0	0
$243$	9.78067	0.627430
$244$	0	0
$245$	−9.31392	−0.595045
$246$	0	0
$247$	−7.07321	−0.450058
$248$	0	0
$249$	17.6117	1.11610
$250$	0	0
$251$	−8.34770	−0.526902	−0.263451	−	0.964673i	$-0.584861\pi$
$251$	−8.34770	−0.526902	−0.263451	+	0.964673i	$0.584861\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−13.2509	−0.829806
$256$	0	0
$257$	26.9863	1.68336	0.841680	−	0.539977i	$-0.181567\pi$
$257$	26.9863	1.68336	0.841680	+	0.539977i	$0.181567\pi$
$258$	0	0
$259$	−24.8248	−1.54254
$260$	0	0
$261$	0.266928	0.0165225
$262$	0	0
$263$	13.1466	0.810656	0.405328	−	0.914171i	$-0.367157\pi$
$263$	13.1466	0.810656	0.405328	+	0.914171i	$0.367157\pi$
$264$	0	0
$265$	6.33862	0.389378
$266$	0	0
$267$	10.1713	0.622474
$268$	0	0
$269$	15.9683	0.973606	0.486803	−	0.873512i	$-0.338163\pi$
$269$	15.9683	0.973606	0.486803	+	0.873512i	$0.338163\pi$
$270$	0	0
$271$	−14.3387	−0.871014	−0.435507	−	0.900185i	$-0.643431\pi$
$271$	−14.3387	−0.871014	−0.435507	+	0.900185i	$0.643431\pi$
$272$	0	0
$273$	6.41330	0.388150
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−11.6004	−0.697000	−0.348500	−	0.937309i	$-0.613309\pi$
$277$	−11.6004	−0.697000	−0.348500	+	0.937309i	$0.613309\pi$
$278$	0	0
$279$	−7.96500	−0.476852
$280$	0	0
$281$	29.7706	1.77597	0.887984	−	0.459875i	$-0.152106\pi$
$281$	29.7706	1.77597	0.887984	+	0.459875i	$0.152106\pi$
$282$	0	0
$283$	−17.1133	−1.01728	−0.508640	−	0.860979i	$-0.669852\pi$
$283$	−17.1133	−1.01728	−0.508640	+	0.860979i	$0.669852\pi$
$284$	0	0
$285$	39.2879	2.32722
$286$	0	0
$287$	17.2495	1.01821
$288$	0	0
$289$	−11.3087	−0.665220
$290$	0	0
$291$	−17.3747	−1.01852
$292$	0	0
$293$	−15.6658	−0.915206	−0.457603	−	0.889157i	$-0.651292\pi$
$293$	−15.6658	−0.915206	−0.457603	+	0.889157i	$0.651292\pi$
$294$	0	0
$295$	−30.9885	−1.80422
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−5.82556	−0.336901
$300$	0	0
$301$	−15.5385	−0.895627
$302$	0	0
$303$	6.02921	0.346369
$304$	0	0
$305$	29.5916	1.69441
$306$	0	0
$307$	28.3490	1.61796	0.808980	−	0.587836i	$-0.200020\pi$
$307$	28.3490	1.61796	0.808980	+	0.587836i	$0.200020\pi$
$308$	0	0
$309$	8.09306	0.460398
$310$	0	0
$311$	6.15835	0.349208	0.174604	−	0.984639i	$-0.444135\pi$
$311$	6.15835	0.349208	0.174604	+	0.984639i	$0.444135\pi$
$312$	0	0
$313$	16.0151	0.905229	0.452615	−	0.891706i	$-0.350491\pi$
$313$	16.0151	0.905229	0.452615	+	0.891706i	$0.350491\pi$
$314$	0	0
$315$	−8.74673	−0.492823
$316$	0	0
$317$	−7.56967	−0.425155	−0.212577	−	0.977144i	$-0.568186\pi$
$317$	−7.56967	−0.425155	−0.212577	+	0.977144i	$0.568186\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	25.4139	1.41846
$322$	0	0
$323$	−16.8741	−0.938901
$324$	0	0
$325$	−2.75889	−0.153036
$326$	0	0
$327$	2.85563	0.157917
$328$	0	0
$329$	35.2159	1.94152
$330$	0	0
$331$	−15.1766	−0.834184	−0.417092	−	0.908864i	$-0.636951\pi$
$331$	−15.1766	−0.834184	−0.417092	+	0.908864i	$0.636951\pi$
$332$	0	0
$333$	−7.53622	−0.412982
$334$	0	0
$335$	−32.7863	−1.79130
$336$	0	0
$337$	29.7659	1.62145	0.810725	−	0.585428i	$-0.199073\pi$
$337$	29.7659	1.62145	0.810725	+	0.585428i	$0.199073\pi$
$338$	0	0
$339$	19.4065	1.05401
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−11.7591	−0.634934
$344$	0	0
$345$	32.3579	1.74209
$346$	0	0
$347$	19.8297	1.06452	0.532258	−	0.846582i	$-0.321344\pi$
$347$	19.8297	1.06452	0.532258	+	0.846582i	$0.321344\pi$
$348$	0	0
$349$	15.5683	0.833355	0.416677	−	0.909054i	$-0.363195\pi$
$349$	15.5683	0.833355	0.416677	+	0.909054i	$0.363195\pi$
$350$	0	0
$351$	−4.03531	−0.215389
$352$	0	0
$353$	−17.1009	−0.910189	−0.455095	−	0.890443i	$-0.650395\pi$
$353$	−17.1009	−0.910189	−0.455095	+	0.890443i	$0.650395\pi$
$354$	0	0
$355$	−11.5130	−0.611048
$356$	0	0
$357$	15.2998	0.809751
$358$	0	0
$359$	11.0586	0.583651	0.291826	−	0.956472i	$-0.405737\pi$
$359$	11.0586	0.583651	0.291826	+	0.956472i	$0.405737\pi$
$360$	0	0
$361$	31.0304	1.63318
$362$	0	0
$363$	0	0
$364$	0	0
$365$	27.7141	1.45062
$366$	0	0
$367$	31.9017	1.66525	0.832627	−	0.553834i	$-0.186836\pi$
$367$	31.9017	1.66525	0.832627	+	0.553834i	$0.186836\pi$
$368$	0	0
$369$	5.23656	0.272604
$370$	0	0
$371$	−7.31869	−0.379968
$372$	0	0
$373$	11.9127	0.616816	0.308408	−	0.951254i	$-0.400204\pi$
$373$	11.9127	0.616816	0.308408	+	0.951254i	$0.400204\pi$
$374$	0	0
$375$	−12.4482	−0.642820
$376$	0	0
$377$	−0.273393	−0.0140805
$378$	0	0
$379$	13.8454	0.711192	0.355596	−	0.934640i	$-0.384278\pi$
$379$	13.8454	0.711192	0.355596	+	0.934640i	$0.384278\pi$
$380$	0	0
$381$	−38.0184	−1.94774
$382$	0	0
$383$	3.97149	0.202934	0.101467	−	0.994839i	$-0.467646\pi$
$383$	3.97149	0.202934	0.101467	+	0.994839i	$0.467646\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.71714	−0.239786
$388$	0	0
$389$	24.6753	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$389$	24.6753	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$390$	0	0
$391$	−13.8977	−0.702835
$392$	0	0
$393$	17.8718	0.901512
$394$	0	0
$395$	14.4508	0.727097
$396$	0	0
$397$	4.36074	0.218859	0.109430	−	0.993995i	$-0.465098\pi$
$397$	4.36074	0.218859	0.109430	+	0.993995i	$0.465098\pi$
$398$	0	0
$399$	−45.3626	−2.27097
$400$	0	0
$401$	15.9645	0.797231	0.398616	−	0.917118i	$-0.369491\pi$
$401$	15.9645	0.797231	0.398616	+	0.917118i	$0.369491\pi$
$402$	0	0
$403$	8.15791	0.406374
$404$	0	0
$405$	30.5728	1.51918
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−10.3161	−0.510100	−0.255050	−	0.966928i	$-0.582092\pi$
$409$	−10.3161	−0.510100	−0.255050	+	0.966928i	$0.582092\pi$
$410$	0	0
$411$	21.0910	1.04034
$412$	0	0
$413$	35.7800	1.76062
$414$	0	0
$415$	24.6013	1.20763
$416$	0	0
$417$	8.38315	0.410525
$418$	0	0
$419$	−6.09348	−0.297686	−0.148843	−	0.988861i	$-0.547555\pi$
$419$	−6.09348	−0.297686	−0.148843	+	0.988861i	$0.547555\pi$
$420$	0	0
$421$	15.3584	0.748524	0.374262	−	0.927323i	$-0.377896\pi$
$421$	15.3584	0.748524	0.374262	+	0.927323i	$0.377896\pi$
$422$	0	0
$423$	10.6907	0.519801
$424$	0	0
$425$	−6.58171	−0.319260
$426$	0	0
$427$	−34.1670	−1.65346
$428$	0	0
$429$	0	0
$430$	0	0
$431$	20.7710	1.00050	0.500252	−	0.865880i	$-0.333241\pi$
$431$	20.7710	1.00050	0.500252	+	0.865880i	$0.333241\pi$
$432$	0	0
$433$	3.12766	0.150306	0.0751528	−	0.997172i	$-0.476056\pi$
$433$	3.12766	0.150306	0.0751528	+	0.997172i	$0.476056\pi$
$434$	0	0
$435$	1.51855	0.0728091
$436$	0	0
$437$	41.2054	1.97112
$438$	0	0
$439$	−39.1047	−1.86637	−0.933183	−	0.359403i	$-0.882981\pi$
$439$	−39.1047	−1.86637	−0.933183	+	0.359403i	$0.882981\pi$
$440$	0	0
$441$	3.26467	0.155461
$442$	0	0
$443$	−35.9676	−1.70887	−0.854437	−	0.519555i	$-0.826098\pi$
$443$	−35.9676	−1.70887	−0.854437	+	0.519555i	$0.826098\pi$
$444$	0	0
$445$	14.2080	0.673525
$446$	0	0
$447$	−16.5974	−0.785031
$448$	0	0
$449$	27.2073	1.28399	0.641995	−	0.766709i	$-0.278107\pi$
$449$	27.2073	1.28399	0.641995	+	0.766709i	$0.278107\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−25.3946	−1.19314
$454$	0	0
$455$	8.95857	0.419984
$456$	0	0
$457$	18.8174	0.880243	0.440121	−	0.897938i	$-0.354935\pi$
$457$	18.8174	0.880243	0.440121	+	0.897938i	$0.354935\pi$
$458$	0	0
$459$	−9.62678	−0.449340
$460$	0	0
$461$	3.98667	0.185678	0.0928390	−	0.995681i	$-0.470406\pi$
$461$	3.98667	0.185678	0.0928390	+	0.995681i	$0.470406\pi$
$462$	0	0
$463$	29.9656	1.39262	0.696310	−	0.717741i	$-0.254824\pi$
$463$	29.9656	1.39262	0.696310	+	0.717741i	$0.254824\pi$
$464$	0	0
$465$	−45.3128	−2.10133
$466$	0	0
$467$	−4.72745	−0.218761	−0.109380	−	0.994000i	$-0.534887\pi$
$467$	−4.72745	−0.218761	−0.109380	+	0.994000i	$0.534887\pi$
$468$	0	0
$469$	37.8557	1.74801
$470$	0	0
$471$	9.42988	0.434506
$472$	0	0
$473$	0	0
$474$	0	0
$475$	19.5142	0.895375
$476$	0	0
$477$	−2.22178	−0.101729
$478$	0	0
$479$	9.97452	0.455747	0.227874	−	0.973691i	$-0.426823\pi$
$479$	9.97452	0.455747	0.227874	+	0.973691i	$0.426823\pi$
$480$	0	0
$481$	7.71874	0.351944
$482$	0	0
$483$	−37.3610	−1.69999
$484$	0	0
$485$	−24.2703	−1.10206
$486$	0	0
$487$	40.5055	1.83548	0.917740	−	0.397182i	$-0.130012\pi$
$487$	40.5055	1.83548	0.917740	+	0.397182i	$0.130012\pi$
$488$	0	0
$489$	40.0891	1.81289
$490$	0	0
$491$	13.3224	0.601233	0.300617	−	0.953745i	$-0.402808\pi$
$491$	13.3224	0.601233	0.300617	+	0.953745i	$0.402808\pi$
$492$	0	0
$493$	−0.652217	−0.0293744
$494$	0	0
$495$	0	0
$496$	0	0
$497$	13.2932	0.596280
$498$	0	0
$499$	32.4353	1.45200	0.726002	−	0.687692i	$-0.241376\pi$
$499$	32.4353	1.45200	0.726002	+	0.687692i	$0.241376\pi$
$500$	0	0
$501$	−44.4579	−1.98623
$502$	0	0
$503$	−15.9976	−0.713299	−0.356649	−	0.934238i	$-0.616081\pi$
$503$	−15.9976	−0.713299	−0.356649	+	0.934238i	$0.616081\pi$
$504$	0	0
$505$	8.42205	0.374776
$506$	0	0
$507$	−1.99408	−0.0885602
$508$	0	0
$509$	−17.5423	−0.777551	−0.388775	−	0.921333i	$-0.627102\pi$
$509$	−17.5423	−0.777551	−0.388775	+	0.921333i	$0.627102\pi$
$510$	0	0
$511$	−31.9992	−1.41556
$512$	0	0
$513$	28.5426	1.26019
$514$	0	0
$515$	11.3050	0.498158
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−36.6265	−1.60773
$520$	0	0
$521$	−25.1789	−1.10311	−0.551555	−	0.834139i	$-0.685965\pi$
$521$	−25.1789	−1.10311	−0.551555	+	0.834139i	$0.685965\pi$
$522$	0	0
$523$	−1.58676	−0.0693843	−0.0346922	−	0.999398i	$-0.511045\pi$
$523$	−1.58676	−0.0693843	−0.0346922	+	0.999398i	$0.511045\pi$
$524$	0	0
$525$	−17.6936	−0.772212
$526$	0	0
$527$	19.4618	0.847769
$528$	0	0
$529$	10.9371	0.475527
$530$	0	0
$531$	10.8620	0.471369
$532$	0	0
$533$	−5.36338	−0.232314
$534$	0	0
$535$	35.5000	1.53480
$536$	0	0
$537$	−39.3230	−1.69691
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−29.9806	−1.28897	−0.644484	−	0.764618i	$-0.722928\pi$
$541$	−29.9806	−1.28897	−0.644484	+	0.764618i	$0.722928\pi$
$542$	0	0
$543$	23.6972	1.01694
$544$	0	0
$545$	3.98896	0.170868
$546$	0	0
$547$	7.62460	0.326005	0.163002	−	0.986626i	$-0.447882\pi$
$547$	7.62460	0.326005	0.163002	+	0.986626i	$0.447882\pi$
$548$	0	0
$549$	−10.3723	−0.442680
$550$	0	0
$551$	1.93377	0.0823813
$552$	0	0
$553$	−16.6852	−0.709525
$554$	0	0
$555$	−42.8735	−1.81988
$556$	0	0
$557$	21.9848	0.931525	0.465763	−	0.884910i	$-0.345780\pi$
$557$	21.9848	0.931525	0.465763	+	0.884910i	$0.345780\pi$
$558$	0	0
$559$	4.83138	0.204346
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.23799	0.136465	0.0682325	−	0.997669i	$-0.478264\pi$
$563$	3.23799	0.136465	0.0682325	+	0.997669i	$0.478264\pi$
$564$	0	0
$565$	27.1084	1.14046
$566$	0	0
$567$	−35.3000	−1.48246
$568$	0	0
$569$	−34.8088	−1.45926	−0.729630	−	0.683842i	$-0.760308\pi$
$569$	−34.8088	−1.45926	−0.729630	+	0.683842i	$0.760308\pi$
$570$	0	0
$571$	−23.9418	−1.00193	−0.500966	−	0.865467i	$-0.667022\pi$
$571$	−23.9418	−1.00193	−0.500966	+	0.865467i	$0.667022\pi$
$572$	0	0
$573$	−15.9751	−0.667368
$574$	0	0
$575$	16.0721	0.670252
$576$	0	0
$577$	1.65654	0.0689627	0.0344814	−	0.999405i	$-0.489022\pi$
$577$	1.65654	0.0689627	0.0344814	+	0.999405i	$0.489022\pi$
$578$	0	0
$579$	−39.7792	−1.65317
$580$	0	0
$581$	−28.4052	−1.17845
$582$	0	0
$583$	0	0
$584$	0	0
$585$	2.71961	0.112442
$586$	0	0
$587$	32.5869	1.34500	0.672502	−	0.740095i	$-0.265220\pi$
$587$	32.5869	1.34500	0.672502	+	0.740095i	$0.265220\pi$
$588$	0	0
$589$	−57.7026	−2.37760
$590$	0	0
$591$	25.4803	1.04812
$592$	0	0
$593$	30.6545	1.25883	0.629414	−	0.777070i	$-0.283295\pi$
$593$	30.6545	1.25883	0.629414	+	0.777070i	$0.283295\pi$
$594$	0	0
$595$	21.3719	0.876162
$596$	0	0
$597$	−24.3210	−0.995392
$598$	0	0
$599$	31.8235	1.30027	0.650136	−	0.759818i	$-0.274712\pi$
$599$	31.8235	1.30027	0.650136	+	0.759818i	$0.274712\pi$
$600$	0	0
$601$	14.9627	0.610341	0.305170	−	0.952298i	$-0.401287\pi$
$601$	14.9627	0.610341	0.305170	+	0.952298i	$0.401287\pi$
$602$	0	0
$603$	11.4921	0.467994
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−27.9902	−1.13609	−0.568044	−	0.822998i	$-0.692300\pi$
$607$	−27.9902	−1.13609	−0.568044	+	0.822998i	$0.692300\pi$
$608$	0	0
$609$	−1.75335	−0.0710494
$610$	0	0
$611$	−10.9497	−0.442976
$612$	0	0
$613$	2.02951	0.0819713	0.0409857	−	0.999160i	$-0.486950\pi$
$613$	2.02951	0.0819713	0.0409857	+	0.999160i	$0.486950\pi$
$614$	0	0
$615$	29.7907	1.20128
$616$	0	0
$617$	−37.9480	−1.52773	−0.763864	−	0.645378i	$-0.776700\pi$
$617$	−37.9480	−1.52773	−0.763864	+	0.645378i	$0.776700\pi$
$618$	0	0
$619$	27.8987	1.12134	0.560672	−	0.828038i	$-0.310543\pi$
$619$	27.8987	1.12134	0.560672	+	0.828038i	$0.310543\pi$
$620$	0	0
$621$	23.5079	0.943341
$622$	0	0
$623$	−16.4049	−0.657247
$624$	0	0
$625$	−31.1830	−1.24732
$626$	0	0
$627$	0	0
$628$	0	0
$629$	18.4141	0.734219
$630$	0	0
$631$	−13.5282	−0.538548	−0.269274	−	0.963064i	$-0.586784\pi$
$631$	−13.5282	−0.538548	−0.269274	+	0.963064i	$0.586784\pi$
$632$	0	0
$633$	−3.16503	−0.125799
$634$	0	0
$635$	−53.1070	−2.10749
$636$	0	0
$637$	−3.34374	−0.132484
$638$	0	0
$639$	4.03549	0.159642
$640$	0	0
$641$	46.6627	1.84306	0.921532	−	0.388302i	$-0.126938\pi$
$641$	46.6627	1.84306	0.921532	+	0.388302i	$0.126938\pi$
$642$	0	0
$643$	29.1332	1.14890	0.574450	−	0.818540i	$-0.305216\pi$
$643$	29.1332	1.14890	0.574450	+	0.818540i	$0.305216\pi$
$644$	0	0
$645$	−26.8358	−1.05666
$646$	0	0
$647$	−41.8786	−1.64642	−0.823208	−	0.567740i	$-0.807818\pi$
$647$	−41.8786	−1.64642	−0.823208	+	0.567740i	$0.807818\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	52.3191	2.05055
$652$	0	0
$653$	15.6038	0.610622	0.305311	−	0.952253i	$-0.401240\pi$
$653$	15.6038	0.610622	0.305311	+	0.952253i	$0.401240\pi$
$654$	0	0
$655$	24.9646	0.975448
$656$	0	0
$657$	−9.71421	−0.378987
$658$	0	0
$659$	17.8596	0.695711	0.347856	−	0.937548i	$-0.386910\pi$
$659$	17.8596	0.695711	0.347856	+	0.937548i	$0.386910\pi$
$660$	0	0
$661$	−40.9352	−1.59220	−0.796098	−	0.605168i	$-0.793106\pi$
$661$	−40.9352	−1.59220	−0.796098	+	0.605168i	$0.793106\pi$
$662$	0	0
$663$	−4.75715	−0.184752
$664$	0	0
$665$	−63.3659	−2.45722
$666$	0	0
$667$	1.59267	0.0616683
$668$	0	0
$669$	3.29806	0.127510
$670$	0	0
$671$	0	0
$672$	0	0
$673$	39.9346	1.53937	0.769683	−	0.638426i	$-0.220414\pi$
$673$	39.9346	1.53937	0.769683	+	0.638426i	$0.220414\pi$
$674$	0	0
$675$	11.1330	0.428509
$676$	0	0
$677$	41.2733	1.58626	0.793132	−	0.609050i	$-0.208449\pi$
$677$	41.2733	1.58626	0.793132	+	0.609050i	$0.208449\pi$
$678$	0	0
$679$	28.0229	1.07542
$680$	0	0
$681$	8.06883	0.309198
$682$	0	0
$683$	27.2392	1.04228	0.521140	−	0.853471i	$-0.325507\pi$
$683$	27.2392	1.04228	0.521140	+	0.853471i	$0.325507\pi$
$684$	0	0
$685$	29.4615	1.12566
$686$	0	0
$687$	19.0595	0.727166
$688$	0	0
$689$	2.27559	0.0866932
$690$	0	0
$691$	15.2481	0.580067	0.290033	−	0.957017i	$-0.406334\pi$
$691$	15.2481	0.580067	0.290033	+	0.957017i	$0.406334\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	11.7102	0.444194
$696$	0	0
$697$	−12.7951	−0.484648
$698$	0	0
$699$	−24.2958	−0.918951
$700$	0	0
$701$	9.61407	0.363118	0.181559	−	0.983380i	$-0.441886\pi$
$701$	9.61407	0.363118	0.181559	+	0.983380i	$0.441886\pi$
$702$	0	0
$703$	−54.5963	−2.05914
$704$	0	0
$705$	60.8195	2.29059
$706$	0	0
$707$	−9.72426	−0.365718
$708$	0	0
$709$	14.3299	0.538169	0.269085	−	0.963117i	$-0.413279\pi$
$709$	14.3299	0.538169	0.269085	+	0.963117i	$0.413279\pi$
$710$	0	0
$711$	−5.06522	−0.189961
$712$	0	0
$713$	−47.5244	−1.77980
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−38.1638	−1.42525
$718$	0	0
$719$	−39.5105	−1.47349	−0.736746	−	0.676169i	$-0.763639\pi$
$719$	−39.5105	−1.47349	−0.736746	+	0.676169i	$0.763639\pi$
$720$	0	0
$721$	−13.0530	−0.486118
$722$	0	0
$723$	22.9214	0.852457
$724$	0	0
$725$	0.754263	0.0280126
$726$	0	0
$727$	−25.9008	−0.960607	−0.480303	−	0.877102i	$-0.659473\pi$
$727$	−25.9008	−0.960607	−0.480303	+	0.877102i	$0.659473\pi$
$728$	0	0
$729$	13.4239	0.497183
$730$	0	0
$731$	11.5259	0.426302
$732$	0	0
$733$	28.0318	1.03538	0.517689	−	0.855569i	$-0.326792\pi$
$733$	28.0318	1.03538	0.517689	+	0.855569i	$0.326792\pi$
$734$	0	0
$735$	18.5727	0.685064
$736$	0	0
$737$	0	0
$738$	0	0
$739$	4.57164	0.168170	0.0840852	−	0.996459i	$-0.473203\pi$
$739$	4.57164	0.168170	0.0840852	+	0.996459i	$0.473203\pi$
$740$	0	0
$741$	14.1046	0.518144
$742$	0	0
$743$	46.0115	1.68800	0.843998	−	0.536346i	$-0.180196\pi$
$743$	46.0115	1.68800	0.843998	+	0.536346i	$0.180196\pi$
$744$	0	0
$745$	−23.1845	−0.849414
$746$	0	0
$747$	−8.62315	−0.315505
$748$	0	0
$749$	−40.9890	−1.49770
$750$	0	0
$751$	−37.0459	−1.35183	−0.675913	−	0.736982i	$-0.736250\pi$
$751$	−37.0459	−1.35183	−0.675913	+	0.736982i	$0.736250\pi$
$752$	0	0
$753$	16.6460	0.606613
$754$	0	0
$755$	−35.4730	−1.29100
$756$	0	0
$757$	−43.0924	−1.56622	−0.783110	−	0.621883i	$-0.786368\pi$
$757$	−43.0924	−1.56622	−0.783110	+	0.621883i	$0.786368\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	11.3831	0.412637	0.206319	−	0.978485i	$-0.433852\pi$
$761$	11.3831	0.412637	0.206319	+	0.978485i	$0.433852\pi$
$762$	0	0
$763$	−4.60573	−0.166739
$764$	0	0
$765$	6.48800	0.234574
$766$	0	0
$767$	−11.1250	−0.401702
$768$	0	0
$769$	5.70904	0.205873	0.102937	−	0.994688i	$-0.467176\pi$
$769$	5.70904	0.205873	0.102937	+	0.994688i	$0.467176\pi$
$770$	0	0
$771$	−53.8128	−1.93802
$772$	0	0
$773$	19.9294	0.716812	0.358406	−	0.933566i	$-0.383320\pi$
$773$	19.9294	0.716812	0.358406	+	0.933566i	$0.383320\pi$
$774$	0	0
$775$	−22.5068	−0.808468
$776$	0	0
$777$	49.5026	1.77590
$778$	0	0
$779$	37.9364	1.35921
$780$	0	0
$781$	0	0
$782$	0	0
$783$	1.10323	0.0394261
$784$	0	0
$785$	13.1724	0.470142
$786$	0	0
$787$	−52.7378	−1.87990	−0.939950	−	0.341312i	$-0.889129\pi$
$787$	−52.7378	−1.87990	−0.939950	+	0.341312i	$0.889129\pi$
$788$	0	0
$789$	−26.2154	−0.933294
$790$	0	0
$791$	−31.2999	−1.11290
$792$	0	0
$793$	10.6235	0.377252
$794$	0	0
$795$	−12.6397	−0.448284
$796$	0	0
$797$	46.8506	1.65953	0.829767	−	0.558110i	$-0.188473\pi$
$797$	46.8506	1.65953	0.829767	+	0.558110i	$0.188473\pi$
$798$	0	0
$799$	−26.1219	−0.924126
$800$	0	0
$801$	−4.98014	−0.175964
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−52.1887	−1.83941
$806$	0	0
$807$	−31.8421	−1.12090
$808$	0	0
$809$	−39.7750	−1.39841	−0.699207	−	0.714919i	$-0.746463\pi$
$809$	−39.7750	−1.39841	−0.699207	+	0.714919i	$0.746463\pi$
$810$	0	0
$811$	27.6442	0.970720	0.485360	−	0.874314i	$-0.338689\pi$
$811$	27.6442	0.970720	0.485360	+	0.874314i	$0.338689\pi$
$812$	0	0
$813$	28.5925	1.00278
$814$	0	0
$815$	55.9994	1.96157
$816$	0	0
$817$	−34.1734	−1.19558
$818$	0	0
$819$	−3.14012	−0.109725
$820$	0	0
$821$	37.3212	1.30252	0.651260	−	0.758855i	$-0.274241\pi$
$821$	37.3212	1.30252	0.651260	+	0.758855i	$0.274241\pi$
$822$	0	0
$823$	−37.8634	−1.31984	−0.659918	−	0.751338i	$-0.729409\pi$
$823$	−37.8634	−1.31984	−0.659918	+	0.751338i	$0.729409\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−37.8621	−1.31660	−0.658298	−	0.752758i	$-0.728723\pi$
$827$	−37.8621	−1.31660	−0.658298	+	0.752758i	$0.728723\pi$
$828$	0	0
$829$	−41.5938	−1.44461	−0.722306	−	0.691573i	$-0.756918\pi$
$829$	−41.5938	−1.44461	−0.722306	+	0.691573i	$0.756918\pi$
$830$	0	0
$831$	23.1321	0.802444
$832$	0	0
$833$	−7.97695	−0.276385
$834$	0	0
$835$	−62.1021	−2.14913
$836$	0	0
$837$	−32.9197	−1.13787
$838$	0	0
$839$	−25.4803	−0.879677	−0.439838	−	0.898077i	$-0.644964\pi$
$839$	−25.4803	−0.879677	−0.439838	+	0.898077i	$0.644964\pi$
$840$	0	0
$841$	−28.9253	−0.997423
$842$	0	0
$843$	−59.3650	−2.04464
$844$	0	0
$845$	−2.78548	−0.0958234
$846$	0	0
$847$	0	0
$848$	0	0
$849$	34.1253	1.17118
$850$	0	0
$851$	−44.9660	−1.54141
$852$	0	0
$853$	−45.8903	−1.57125	−0.785626	−	0.618701i	$-0.787659\pi$
$853$	−45.8903	−1.57125	−0.785626	+	0.618701i	$0.787659\pi$
$854$	0	0
$855$	−19.2364	−0.657871
$856$	0	0
$857$	−41.4563	−1.41612	−0.708060	−	0.706152i	$-0.750430\pi$
$857$	−41.4563	−1.41612	−0.708060	+	0.706152i	$0.750430\pi$
$858$	0	0
$859$	35.4784	1.21051	0.605254	−	0.796032i	$-0.293071\pi$
$859$	35.4784	1.21051	0.605254	+	0.796032i	$0.293071\pi$
$860$	0	0
$861$	−34.3970	−1.17225
$862$	0	0
$863$	−0.462173	−0.0157325	−0.00786627	−	0.999969i	$-0.502504\pi$
$863$	−0.462173	−0.0157325	−0.00786627	+	0.999969i	$0.502504\pi$
$864$	0	0
$865$	−51.1626	−1.73958
$866$	0	0
$867$	22.5505	0.765856
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−11.7704	−0.398825
$872$	0	0
$873$	8.50711	0.287922
$874$	0	0
$875$	20.0771	0.678730
$876$	0	0
$877$	18.3884	0.620933	0.310466	−	0.950584i	$-0.399515\pi$
$877$	18.3884	0.620933	0.310466	+	0.950584i	$0.399515\pi$
$878$	0	0
$879$	31.2389	1.05366
$880$	0	0
$881$	7.32658	0.246839	0.123419	−	0.992355i	$-0.460614\pi$
$881$	7.32658	0.246839	0.123419	+	0.992355i	$0.460614\pi$
$882$	0	0
$883$	−8.97409	−0.302002	−0.151001	−	0.988534i	$-0.548250\pi$
$883$	−8.97409	−0.302002	−0.151001	+	0.988534i	$0.548250\pi$
$884$	0	0
$885$	61.7936	2.07717
$886$	0	0
$887$	−11.7206	−0.393540	−0.196770	−	0.980450i	$-0.563045\pi$
$887$	−11.7206	−0.393540	−0.196770	+	0.980450i	$0.563045\pi$
$888$	0	0
$889$	61.3184	2.05655
$890$	0	0
$891$	0	0
$892$	0	0
$893$	77.4493	2.59174
$894$	0	0
$895$	−54.9293	−1.83608
$896$	0	0
$897$	11.6166	0.387868
$898$	0	0
$899$	−2.23032	−0.0743852
$900$	0	0
$901$	5.42874	0.180857
$902$	0	0
$903$	30.9851	1.03112
$904$	0	0
$905$	33.1020	1.10035
$906$	0	0
$907$	−35.1896	−1.16845	−0.584226	−	0.811591i	$-0.698602\pi$
$907$	−35.1896	−1.16845	−0.584226	+	0.811591i	$0.698602\pi$
$908$	0	0
$909$	−2.95206	−0.0979136
$910$	0	0
$911$	23.0597	0.764003	0.382001	−	0.924162i	$-0.375235\pi$
$911$	23.0597	0.764003	0.382001	+	0.924162i	$0.375235\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−59.0080	−1.95074
$916$	0	0
$917$	−28.8246	−0.951873
$918$	0	0
$919$	39.0812	1.28917	0.644584	−	0.764533i	$-0.277030\pi$
$919$	39.0812	1.28917	0.644584	+	0.764533i	$0.277030\pi$
$920$	0	0
$921$	−56.5301	−1.86273
$922$	0	0
$923$	−4.13323	−0.136047
$924$	0	0
$925$	−21.2952	−0.700181
$926$	0	0
$927$	−3.96258	−0.130148
$928$	0	0
$929$	−38.6549	−1.26823	−0.634113	−	0.773240i	$-0.718635\pi$
$929$	−38.6549	−1.26823	−0.634113	+	0.773240i	$0.718635\pi$
$930$	0	0
$931$	23.6510	0.775130
$932$	0	0
$933$	−12.2802	−0.402037
$934$	0	0
$935$	0	0
$936$	0	0
$937$	30.4326	0.994188	0.497094	−	0.867697i	$-0.334400\pi$
$937$	30.4326	0.994188	0.497094	+	0.867697i	$0.334400\pi$
$938$	0	0
$939$	−31.9355	−1.04217
$940$	0	0
$941$	−39.1831	−1.27733	−0.638666	−	0.769484i	$-0.720514\pi$
$941$	−39.1831	−1.27733	−0.638666	+	0.769484i	$0.720514\pi$
$942$	0	0
$943$	31.2447	1.01747
$944$	0	0
$945$	−36.1506	−1.17598
$946$	0	0
$947$	−51.0655	−1.65941	−0.829703	−	0.558205i	$-0.811490\pi$
$947$	−51.0655	−1.65941	−0.829703	+	0.558205i	$0.811490\pi$
$948$	0	0
$949$	9.94948	0.322974
$950$	0	0
$951$	15.0945	0.489473
$952$	0	0
$953$	5.44898	0.176510	0.0882549	−	0.996098i	$-0.471871\pi$
$953$	5.44898	0.176510	0.0882549	+	0.996098i	$0.471871\pi$
$954$	0	0
$955$	−22.3152	−0.722102
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−34.0168	−1.09846
$960$	0	0
$961$	35.5515	1.14682
$962$	0	0
$963$	−12.4433	−0.400979
$964$	0	0
$965$	−55.5666	−1.78875
$966$	0	0
$967$	−16.8874	−0.543063	−0.271531	−	0.962430i	$-0.587530\pi$
$967$	−16.8874	−0.543063	−0.271531	+	0.962430i	$0.587530\pi$
$968$	0	0
$969$	33.6483	1.08094
$970$	0	0
$971$	52.4609	1.68355	0.841775	−	0.539828i	$-0.181511\pi$
$971$	52.4609	1.68355	0.841775	+	0.539828i	$0.181511\pi$
$972$	0	0
$973$	−13.5208	−0.433458
$974$	0	0
$975$	5.50145	0.176187
$976$	0	0
$977$	60.1999	1.92597	0.962983	−	0.269563i	$-0.0868792\pi$
$977$	60.1999	1.92597	0.962983	+	0.269563i	$0.0868792\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−1.39819	−0.0446408
$982$	0	0
$983$	−24.0360	−0.766630	−0.383315	−	0.923618i	$-0.625218\pi$
$983$	−24.0360	−0.766630	−0.383315	+	0.923618i	$0.625218\pi$
$984$	0	0
$985$	35.5928	1.13408
$986$	0	0
$987$	−70.2234	−2.23524
$988$	0	0
$989$	−28.1455	−0.894975
$990$	0	0
$991$	3.16297	0.100475	0.0502376	−	0.998737i	$-0.484002\pi$
$991$	3.16297	0.100475	0.0502376	+	0.998737i	$0.484002\pi$
$992$	0	0
$993$	30.2634	0.960381
$994$	0	0
$995$	−33.9734	−1.07703
$996$	0	0
$997$	−28.8770	−0.914545	−0.457272	−	0.889327i	$-0.651174\pi$
$997$	−28.8770	−0.914545	−0.457272	+	0.889327i	$0.651174\pi$
$998$	0	0
$999$	−31.1475	−0.985464

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6292.2.a.v.1.3	yes	10
11.10	odd	2		6292.2.a.u.1.3	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6292.2.a.u.1.3	✓	10	11.10	odd	2
6292.2.a.v.1.3	yes	10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6292 = 2^{2} \cdot 11^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6292.a (trivial)

\(f(q)\)	\(=\)	\(q-1.99408 q^{3} -2.78548 q^{5} +3.21617 q^{7} +0.976354 q^{9} +O(q^{10})\)	\(q-1.99408 q^{3} -2.78548 q^{5} +3.21617 q^{7} +0.976354 q^{9} -1.00000 q^{13} +5.55447 q^{15} -2.38564 q^{17} +7.07321 q^{19} -6.41330 q^{21} +5.82556 q^{23} +2.75889 q^{25} +4.03531 q^{27} +0.273393 q^{29} -8.15791 q^{31} -8.95857 q^{35} -7.71874 q^{37} +1.99408 q^{39} +5.36338 q^{41} -4.83138 q^{43} -2.71961 q^{45} +10.9497 q^{47} +3.34374 q^{49} +4.75715 q^{51} -2.27559 q^{53} -14.1046 q^{57} +11.1250 q^{59} -10.6235 q^{61} +3.14012 q^{63} +2.78548 q^{65} +11.7704 q^{67} -11.6166 q^{69} +4.13323 q^{71} -9.94948 q^{73} -5.50145 q^{75} -5.18790 q^{79} -10.9758 q^{81} -8.83200 q^{83} +6.64514 q^{85} -0.545168 q^{87} -5.10075 q^{89} -3.21617 q^{91} +16.2675 q^{93} -19.7023 q^{95} +8.71314 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{3} - 6 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) \) 10 * q - 2 * q^3 - 6 * q^5 + 4 * q^7 + 12 * q^9	\( 10 q - 2 q^{3} - 6 q^{5} + 4 q^{7} + 12 q^{9} - 10 q^{13} - 4 q^{15} + 20 q^{17} + 14 q^{19} + 2 q^{21} - 2 q^{23} + 2 q^{25} + 10 q^{27} + 16 q^{29} - 2 q^{31} - 20 q^{35} + 2 q^{39} + 22 q^{41} - 12 q^{43} - 8 q^{45} + 6 q^{47} - 2 q^{49} - 8 q^{51} + 6 q^{53} + 12 q^{57} - 4 q^{59} + 24 q^{61} + 68 q^{63} + 6 q^{65} - 6 q^{67} - 2 q^{69} - 8 q^{71} - 6 q^{73} - 18 q^{75} - 24 q^{79} + 10 q^{81} + 22 q^{83} + 34 q^{85} - 4 q^{87} - 42 q^{89} - 4 q^{91} - 38 q^{93} + 14 q^{97}+O(q^{100}) \) 10 * q - 2 * q^3 - 6 * q^5 + 4 * q^7 + 12 * q^9 - 10 * q^13 - 4 * q^15 + 20 * q^17 + 14 * q^19 + 2 * q^21 - 2 * q^23 + 2 * q^25 + 10 * q^27 + 16 * q^29 - 2 * q^31 - 20 * q^35 + 2 * q^39 + 22 * q^41 - 12 * q^43 - 8 * q^45 + 6 * q^47 - 2 * q^49 - 8 * q^51 + 6 * q^53 + 12 * q^57 - 4 * q^59 + 24 * q^61 + 68 * q^63 + 6 * q^65 - 6 * q^67 - 2 * q^69 - 8 * q^71 - 6 * q^73 - 18 * q^75 - 24 * q^79 + 10 * q^81 + 22 * q^83 + 34 * q^85 - 4 * q^87 - 42 * q^89 - 4 * q^91 - 38 * q^93 + 14 * q^97

Introduction

L-functions

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Database

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