LMFDB - Embedded newform 6534.2.a.cs.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6534,2,Mod(1,6534)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6534, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6534.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6534 = 2 \cdot 3^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6534.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:	$52.1742526807$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.4752.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 3x^{2} + 4x + 1 $ x^4 - 2x^3 - 3x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.49551$ of defining polynomial
Character	$\chi$	$=$	6534.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -2.59030 q^{5} +0.164177 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -2.59030 q^{5} +0.164177 q^{7} +1.00000 q^{8} -2.59030 q^{10} -0.573882 q^{13} +0.164177 q^{14} +1.00000 q^{16} -2.30593 q^{19} -2.59030 q^{20} -0.409705 q^{23} +1.70963 q^{25} -0.573882 q^{26} +0.164177 q^{28} +8.48052 q^{29} +5.48652 q^{31} +1.00000 q^{32} -0.425267 q^{35} -7.19615 q^{37} -2.30593 q^{38} -2.59030 q^{40} +0.284363 q^{41} -6.28436 q^{43} -0.409705 q^{46} +9.96704 q^{47} -6.97305 q^{49} +1.70963 q^{50} -0.573882 q^{52} -8.68868 q^{53} +0.164177 q^{56} +8.48052 q^{58} +3.90224 q^{59} -8.18059 q^{61} +5.48652 q^{62} +1.00000 q^{64} +1.48652 q^{65} +6.48052 q^{67} -0.425267 q^{70} -11.1806 q^{71} +6.51850 q^{73} -7.19615 q^{74} -2.30593 q^{76} -10.1582 q^{79} -2.59030 q^{80} +0.284363 q^{82} -1.79784 q^{83} -6.28436 q^{86} -3.39414 q^{89} -0.0942181 q^{91} -0.409705 q^{92} +9.96704 q^{94} +5.97305 q^{95} -12.9730 q^{97} -6.97305 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} - 6 q^{7} + 4 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^4 - 6 * q^7 + 4 * q^8	$ 4 q + 4 q^{2} + 4 q^{4} - 6 q^{7} + 4 q^{8} - 6 q^{13} - 6 q^{14} + 4 q^{16} - 6 q^{19} - 12 q^{23} + 10 q^{25} - 6 q^{26} - 6 q^{28} + 6 q^{29} - 2 q^{31} + 4 q^{32} - 12 q^{35} - 8 q^{37} - 6 q^{38} - 6 q^{41} - 18 q^{43} - 12 q^{46} - 12 q^{47} + 20 q^{49} + 10 q^{50} - 6 q^{52} + 6 q^{53} - 6 q^{56} + 6 q^{58} + 6 q^{59} - 12 q^{61} - 2 q^{62} + 4 q^{64} - 18 q^{65} - 2 q^{67} - 12 q^{70} - 24 q^{71} - 12 q^{73} - 8 q^{74} - 6 q^{76} - 30 q^{79} - 6 q^{82} - 24 q^{83} - 18 q^{86} - 24 q^{89} - 18 q^{91} - 12 q^{92} - 12 q^{94} - 24 q^{95} - 4 q^{97} + 20 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 - 6 * q^7 + 4 * q^8 - 6 * q^13 - 6 * q^14 + 4 * q^16 - 6 * q^19 - 12 * q^23 + 10 * q^25 - 6 * q^26 - 6 * q^28 + 6 * q^29 - 2 * q^31 + 4 * q^32 - 12 * q^35 - 8 * q^37 - 6 * q^38 - 6 * q^41 - 18 * q^43 - 12 * q^46 - 12 * q^47 + 20 * q^49 + 10 * q^50 - 6 * q^52 + 6 * q^53 - 6 * q^56 + 6 * q^58 + 6 * q^59 - 12 * q^61 - 2 * q^62 + 4 * q^64 - 18 * q^65 - 2 * q^67 - 12 * q^70 - 24 * q^71 - 12 * q^73 - 8 * q^74 - 6 * q^76 - 30 * q^79 - 6 * q^82 - 24 * q^83 - 18 * q^86 - 24 * q^89 - 18 * q^91 - 12 * q^92 - 12 * q^94 - 24 * q^95 - 4 * q^97 + 20 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−2.59030	−1.15842	−0.579208	−	0.815180i	$-0.696638\pi$
$5$	−2.59030	−1.15842	−0.579208	+	0.815180i	$0.696638\pi$
$6$	0	0
$7$	0.164177	0.0620530	0.0310265	−	0.999519i	$-0.490122\pi$
$7$	0.164177	0.0620530	0.0310265	+	0.999519i	$0.490122\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−2.59030	−0.819123
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−0.573882	−0.159166	−0.0795831	−	0.996828i	$-0.525359\pi$
$13$	−0.573882	−0.159166	−0.0795831	+	0.996828i	$0.525359\pi$
$14$	0.164177	0.0438781
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−2.30593	−0.529017	−0.264509	−	0.964383i	$-0.585210\pi$
$19$	−2.30593	−0.529017	−0.264509	+	0.964383i	$0.585210\pi$
$20$	−2.59030	−0.579208
$21$	0	0
$22$	0	0
$23$	−0.409705	−0.0854293	−0.0427147	−	0.999087i	$-0.513601\pi$
$23$	−0.409705	−0.0854293	−0.0427147	+	0.999087i	$0.513601\pi$
$24$	0	0
$25$	1.70963	0.341926
$26$	−0.573882	−0.112547
$27$	0	0
$28$	0.164177	0.0310265
$29$	8.48052	1.57479	0.787396	−	0.616447i	$-0.211429\pi$
$29$	8.48052	1.57479	0.787396	+	0.616447i	$0.211429\pi$
$30$	0	0
$31$	5.48652	0.985409	0.492704	−	0.870197i	$-0.336008\pi$
$31$	5.48652	0.985409	0.492704	+	0.870197i	$0.336008\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0	0
$35$	−0.425267	−0.0718832
$36$	0	0
$37$	−7.19615	−1.18304	−0.591520	−	0.806290i	$-0.701472\pi$
$37$	−7.19615	−1.18304	−0.591520	+	0.806290i	$0.701472\pi$
$38$	−2.30593	−0.374072
$39$	0	0
$40$	−2.59030	−0.409562
$41$	0.284363	0.0444100	0.0222050	−	0.999753i	$-0.492931\pi$
$41$	0.284363	0.0444100	0.0222050	+	0.999753i	$0.492931\pi$
$42$	0	0
$43$	−6.28436	−0.958356	−0.479178	−	0.877718i	$-0.659065\pi$
$43$	−6.28436	−0.958356	−0.479178	+	0.877718i	$0.659065\pi$
$44$	0	0
$45$	0	0
$46$	−0.409705	−0.0604077
$47$	9.96704	1.45384	0.726921	−	0.686721i	$-0.240951\pi$
$47$	9.96704	1.45384	0.726921	+	0.686721i	$0.240951\pi$
$48$	0	0
$49$	−6.97305	−0.996149
$50$	1.70963	0.241778
$51$	0	0
$52$	−0.573882	−0.0795831
$53$	−8.68868	−1.19348	−0.596741	−	0.802434i	$-0.703538\pi$
$53$	−8.68868	−1.19348	−0.596741	+	0.802434i	$0.703538\pi$
$54$	0	0
$55$	0	0
$56$	0.164177	0.0219391
$57$	0	0
$58$	8.48052	1.11355
$59$	3.90224	0.508028	0.254014	−	0.967201i	$-0.418249\pi$
$59$	3.90224	0.508028	0.254014	+	0.967201i	$0.418249\pi$
$60$	0	0
$61$	−8.18059	−1.04742	−0.523709	−	0.851897i	$-0.675452\pi$
$61$	−8.18059	−1.04742	−0.523709	+	0.851897i	$0.675452\pi$
$62$	5.48652	0.696789
$63$	0	0
$64$	1.00000	0.125000
$65$	1.48652	0.184380
$66$	0	0
$67$	6.48052	0.791721	0.395860	−	0.918311i	$-0.370446\pi$
$67$	6.48052	0.791721	0.395860	+	0.918311i	$0.370446\pi$
$68$	0	0
$69$	0	0
$70$	−0.425267	−0.0508291
$71$	−11.1806	−1.32689	−0.663446	−	0.748224i	$-0.730907\pi$
$71$	−11.1806	−1.32689	−0.663446	+	0.748224i	$0.730907\pi$
$72$	0	0
$73$	6.51850	0.762933	0.381466	−	0.924383i	$-0.375419\pi$
$73$	6.51850	0.762933	0.381466	+	0.924383i	$0.375419\pi$
$74$	−7.19615	−0.836536
$75$	0	0
$76$	−2.30593	−0.264509
$77$	0	0
$78$	0	0
$79$	−10.1582	−1.14288	−0.571442	−	0.820643i	$-0.693616\pi$
$79$	−10.1582	−1.14288	−0.571442	+	0.820643i	$0.693616\pi$
$80$	−2.59030	−0.289604
$81$	0	0
$82$	0.284363	0.0314026
$83$	−1.79784	−0.197339	−0.0986693	−	0.995120i	$-0.531459\pi$
$83$	−1.79784	−0.197339	−0.0986693	+	0.995120i	$0.531459\pi$
$84$	0	0
$85$	0	0
$86$	−6.28436	−0.677660
$87$	0	0
$88$	0	0
$89$	−3.39414	−0.359778	−0.179889	−	0.983687i	$-0.557574\pi$
$89$	−3.39414	−0.359778	−0.179889	+	0.983687i	$0.557574\pi$
$90$	0	0
$91$	−0.0942181	−0.00987674
$92$	−0.409705	−0.0427147
$93$	0	0
$94$	9.96704	1.02802
$95$	5.97305	0.612822
$96$	0	0
$97$	−12.9730	−1.31721	−0.658607	−	0.752487i	$-0.728854\pi$
$97$	−12.9730	−1.31721	−0.658607	+	0.752487i	$0.728854\pi$
$98$	−6.97305	−0.704384
$99$	0	0
$100$	1.70963	0.170963
$101$	−4.39230	−0.437051	−0.218525	−	0.975831i	$-0.570125\pi$
$101$	−4.39230	−0.437051	−0.218525	+	0.975831i	$0.570125\pi$
$102$	0	0
$103$	−17.6497	−1.73908	−0.869539	−	0.493864i	$-0.835584\pi$
$103$	−17.6497	−1.73908	−0.869539	+	0.493864i	$0.835584\pi$
$104$	−0.573882	−0.0562737
$105$	0	0
$106$	−8.68868	−0.843920
$107$	14.4043	1.39252	0.696259	−	0.717791i	$-0.254846\pi$
$107$	14.4043	1.39252	0.696259	+	0.717791i	$0.254846\pi$
$108$	0	0
$109$	−5.78143	−0.553760	−0.276880	−	0.960904i	$-0.589300\pi$
$109$	−5.78143	−0.553760	−0.276880	+	0.960904i	$0.589300\pi$
$110$	0	0
$111$	0	0
$112$	0.164177	0.0155133
$113$	−1.29392	−0.121721	−0.0608607	−	0.998146i	$-0.519385\pi$
$113$	−1.29392	−0.121721	−0.0608607	+	0.998146i	$0.519385\pi$
$114$	0	0
$115$	1.06126	0.0989627
$116$	8.48052	0.787396
$117$	0	0
$118$	3.90224	0.359230
$119$	0	0
$120$	0	0
$121$	0	0
$122$	−8.18059	−0.740636
$123$	0	0
$124$	5.48652	0.492704
$125$	8.52303	0.762323
$126$	0	0
$127$	−17.9073	−1.58901	−0.794506	−	0.607256i	$-0.792270\pi$
$127$	−17.9073	−1.58901	−0.794506	+	0.607256i	$0.792270\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	1.48652	0.130377
$131$	−5.19615	−0.453990	−0.226995	−	0.973896i	$-0.572890\pi$
$131$	−5.19615	−0.453990	−0.226995	+	0.973896i	$0.572890\pi$
$132$	0	0
$133$	−0.378581	−0.0328271
$134$	6.48052	0.559831
$135$	0	0
$136$	0	0
$137$	−8.60586	−0.735248	−0.367624	−	0.929975i	$-0.619829\pi$
$137$	−8.60586	−0.735248	−0.367624	+	0.929975i	$0.619829\pi$
$138$	0	0
$139$	−17.8244	−1.51185	−0.755924	−	0.654659i	$-0.772812\pi$
$139$	−17.8244	−1.51185	−0.755924	+	0.654659i	$0.772812\pi$
$140$	−0.425267	−0.0359416
$141$	0	0
$142$	−11.1806	−0.938254
$143$	0	0
$144$	0	0
$145$	−21.9670	−1.82426
$146$	6.51850	0.539475
$147$	0	0
$148$	−7.19615	−0.591520
$149$	12.3983	1.01571	0.507855	−	0.861443i	$-0.330439\pi$
$149$	12.3983	1.01571	0.507855	+	0.861443i	$0.330439\pi$
$150$	0	0
$151$	4.77604	0.388669	0.194334	−	0.980935i	$-0.437745\pi$
$151$	4.77604	0.388669	0.194334	+	0.980935i	$0.437745\pi$
$152$	−2.30593	−0.187036
$153$	0	0
$154$	0	0
$155$	−14.2117	−1.14151
$156$	0	0
$157$	17.6005	1.40467	0.702335	−	0.711846i	$-0.252141\pi$
$157$	17.6005	1.40467	0.702335	+	0.711846i	$0.252141\pi$
$158$	−10.1582	−0.808141
$159$	0	0
$160$	−2.59030	−0.204781
$161$	−0.0672641	−0.00530115
$162$	0	0
$163$	−2.48652	−0.194760	−0.0973798	−	0.995247i	$-0.531046\pi$
$163$	−2.48652	−0.194760	−0.0973798	+	0.995247i	$0.531046\pi$
$164$	0.284363	0.0222050
$165$	0	0
$166$	−1.79784	−0.139539
$167$	−20.6707	−1.59954	−0.799772	−	0.600304i	$-0.795046\pi$
$167$	−20.6707	−1.59954	−0.799772	+	0.600304i	$0.795046\pi$
$168$	0	0
$169$	−12.6707	−0.974666
$170$	0	0
$171$	0	0
$172$	−6.28436	−0.479178
$173$	−12.4925	−0.949790	−0.474895	−	0.880043i	$-0.657514\pi$
$173$	−12.4925	−0.949790	−0.474895	+	0.880043i	$0.657514\pi$
$174$	0	0
$175$	0.280682	0.0212175
$176$	0	0
$177$	0	0
$178$	−3.39414	−0.254402
$179$	−8.29454	−0.619963	−0.309982	−	0.950743i	$-0.600323\pi$
$179$	−8.29454	−0.619963	−0.309982	+	0.950743i	$0.600323\pi$
$180$	0	0
$181$	−16.7589	−1.24568	−0.622839	−	0.782350i	$-0.714021\pi$
$181$	−16.7589	−1.24568	−0.622839	+	0.782350i	$0.714021\pi$
$182$	−0.0942181	−0.00698391
$183$	0	0
$184$	−0.409705	−0.0302038
$185$	18.6402	1.37045
$186$	0	0
$187$	0	0
$188$	9.96704	0.726921
$189$	0	0
$190$	5.97305	0.433330
$191$	6.39414	0.462664	0.231332	−	0.972875i	$-0.425692\pi$
$191$	6.39414	0.462664	0.231332	+	0.972875i	$0.425692\pi$
$192$	0	0
$193$	21.1476	1.52224	0.761120	−	0.648611i	$-0.224650\pi$
$193$	21.1476	1.52224	0.761120	+	0.648611i	$0.224650\pi$
$194$	−12.9730	−0.931410
$195$	0	0
$196$	−6.97305	−0.498075
$197$	14.4043	1.02627	0.513133	−	0.858309i	$-0.328485\pi$
$197$	14.4043	1.02627	0.513133	+	0.858309i	$0.328485\pi$
$198$	0	0
$199$	23.3654	1.65633	0.828163	−	0.560487i	$-0.189386\pi$
$199$	23.3654	1.65633	0.828163	+	0.560487i	$0.189386\pi$
$200$	1.70963	0.120889
$201$	0	0
$202$	−4.39230	−0.309041
$203$	1.39230	0.0977206
$204$	0	0
$205$	−0.736584	−0.0514452
$206$	−17.6497	−1.22971
$207$	0	0
$208$	−0.573882	−0.0397915
$209$	0	0
$210$	0	0
$211$	2.46912	0.169981	0.0849907	−	0.996382i	$-0.472914\pi$
$211$	2.46912	0.169981	0.0849907	+	0.996382i	$0.472914\pi$
$212$	−8.68868	−0.596741
$213$	0	0
$214$	14.4043	0.984659
$215$	16.2784	1.11017
$216$	0	0
$217$	0.900760	0.0611476
$218$	−5.78143	−0.391568
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	3.20216	0.214433	0.107216	−	0.994236i	$-0.465806\pi$
$223$	3.20216	0.214433	0.107216	+	0.994236i	$0.465806\pi$
$224$	0.164177	0.0109695
$225$	0	0
$226$	−1.29392	−0.0860701
$227$	−1.56272	−0.103721	−0.0518606	−	0.998654i	$-0.516515\pi$
$227$	−1.56272	−0.103721	−0.0518606	+	0.998654i	$0.516515\pi$
$228$	0	0
$229$	11.9791	0.791598	0.395799	−	0.918337i	$-0.370468\pi$
$229$	11.9791	0.791598	0.395799	+	0.918337i	$0.370468\pi$
$230$	1.06126	0.0699772
$231$	0	0
$232$	8.48052	0.556773
$233$	−26.8519	−1.75912	−0.879562	−	0.475784i	$-0.842165\pi$
$233$	−26.8519	−1.75912	−0.879562	+	0.475784i	$0.842165\pi$
$234$	0	0
$235$	−25.8176	−1.68415
$236$	3.90224	0.254014
$237$	0	0
$238$	0	0
$239$	−1.58074	−0.102250	−0.0511248	−	0.998692i	$-0.516281\pi$
$239$	−1.58074	−0.102250	−0.0511248	+	0.998692i	$0.516281\pi$
$240$	0	0
$241$	−22.1208	−1.42493	−0.712463	−	0.701709i	$-0.752421\pi$
$241$	−22.1208	−1.42493	−0.712463	+	0.701709i	$0.752421\pi$
$242$	0	0
$243$	0	0
$244$	−8.18059	−0.523709
$245$	18.0622	1.15395
$246$	0	0
$247$	1.32333	0.0842016
$248$	5.48652	0.348395
$249$	0	0
$250$	8.52303	0.539044
$251$	2.54105	0.160390	0.0801950	−	0.996779i	$-0.474446\pi$
$251$	2.54105	0.160390	0.0801950	+	0.996779i	$0.474446\pi$
$252$	0	0
$253$	0	0
$254$	−17.9073	−1.12360
$255$	0	0
$256$	1.00000	0.0625000
$257$	15.0492	0.938746	0.469373	−	0.883000i	$-0.344480\pi$
$257$	15.0492	0.938746	0.469373	+	0.883000i	$0.344480\pi$
$258$	0	0
$259$	−1.18144	−0.0734112
$260$	1.48652	0.0921902
$261$	0	0
$262$	−5.19615	−0.321019
$263$	−14.9730	−0.923278	−0.461639	−	0.887068i	$-0.652738\pi$
$263$	−14.9730	−0.923278	−0.461639	+	0.887068i	$0.652738\pi$
$264$	0	0
$265$	22.5063	1.38255
$266$	−0.378581	−0.0232123
$267$	0	0
$268$	6.48052	0.395860
$269$	16.4596	1.00356	0.501779	−	0.864996i	$-0.332679\pi$
$269$	16.4596	1.00356	0.501779	+	0.864996i	$0.332679\pi$
$270$	0	0
$271$	3.04153	0.184760	0.0923799	−	0.995724i	$-0.470553\pi$
$271$	3.04153	0.184760	0.0923799	+	0.995724i	$0.470553\pi$
$272$	0	0
$273$	0	0
$274$	−8.60586	−0.519899
$275$	0	0
$276$	0	0
$277$	−29.4820	−1.77140	−0.885701	−	0.464257i	$-0.846321\pi$
$277$	−29.4820	−1.77140	−0.885701	+	0.464257i	$0.846321\pi$
$278$	−17.8244	−1.06904
$279$	0	0
$280$	−0.425267	−0.0254145
$281$	−24.9178	−1.48647	−0.743236	−	0.669030i	$-0.766710\pi$
$281$	−24.9178	−1.48647	−0.743236	+	0.669030i	$0.766710\pi$
$282$	0	0
$283$	18.6438	1.10826	0.554131	−	0.832430i	$-0.313051\pi$
$283$	18.6438	1.10826	0.554131	+	0.832430i	$0.313051\pi$
$284$	−11.1806	−0.663446
$285$	0	0
$286$	0	0
$287$	0.0466858	0.00275578
$288$	0	0
$289$	−17.0000	−1.00000
$290$	−21.9670	−1.28995
$291$	0	0
$292$	6.51850	0.381466
$293$	−6.58675	−0.384802	−0.192401	−	0.981316i	$-0.561627\pi$
$293$	−6.58675	−0.384802	−0.192401	+	0.981316i	$0.561627\pi$
$294$	0	0
$295$	−10.1079	−0.588507
$296$	−7.19615	−0.418268
$297$	0	0
$298$	12.3983	0.718215
$299$	0.235122	0.0135975
$300$	0	0
$301$	−1.03175	−0.0594689
$302$	4.77604	0.274830
$303$	0	0
$304$	−2.30593	−0.132254
$305$	21.1901	1.21334
$306$	0	0
$307$	−8.60316	−0.491008	−0.245504	−	0.969396i	$-0.578954\pi$
$307$	−8.60316	−0.491008	−0.245504	+	0.969396i	$0.578954\pi$
$308$	0	0
$309$	0	0
$310$	−14.2117	−0.807171
$311$	−28.5735	−1.62026	−0.810128	−	0.586253i	$-0.800602\pi$
$311$	−28.5735	−1.62026	−0.810128	+	0.586253i	$0.800602\pi$
$312$	0	0
$313$	−5.70963	−0.322727	−0.161364	−	0.986895i	$-0.551589\pi$
$313$	−5.70963	−0.322727	−0.161364	+	0.986895i	$0.551589\pi$
$314$	17.6005	0.993252
$315$	0	0
$316$	−10.1582	−0.571442
$317$	26.7535	1.50263	0.751313	−	0.659946i	$-0.229421\pi$
$317$	26.7535	1.50263	0.751313	+	0.659946i	$0.229421\pi$
$318$	0	0
$319$	0	0
$320$	−2.59030	−0.144802
$321$	0	0
$322$	−0.0672641	−0.00374848
$323$	0	0
$324$	0	0
$325$	−0.981125	−0.0544230
$326$	−2.48652	−0.137716
$327$	0	0
$328$	0.284363	0.0157013
$329$	1.63636	0.0902153
$330$	0	0
$331$	23.8459	1.31069	0.655344	−	0.755331i	$-0.272524\pi$
$331$	23.8459	1.31069	0.655344	+	0.755331i	$0.272524\pi$
$332$	−1.79784	−0.0986693
$333$	0	0
$334$	−20.6707	−1.13105
$335$	−16.7864	−0.917142
$336$	0	0
$337$	−13.1997	−0.719033	−0.359517	−	0.933139i	$-0.617058\pi$
$337$	−13.1997	−0.719033	−0.359517	+	0.933139i	$0.617058\pi$
$338$	−12.6707	−0.689193
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−2.29405	−0.123867
$344$	−6.28436	−0.338830
$345$	0	0
$346$	−12.4925	−0.671603
$347$	−14.3593	−0.770850	−0.385425	−	0.922739i	$-0.625945\pi$
$347$	−14.3593	−0.770850	−0.385425	+	0.922739i	$0.625945\pi$
$348$	0	0
$349$	−5.92134	−0.316962	−0.158481	−	0.987362i	$-0.550660\pi$
$349$	−5.92134	−0.316962	−0.158481	+	0.987362i	$0.550660\pi$
$350$	0.280682	0.0150031
$351$	0	0
$352$	0	0
$353$	−19.9652	−1.06264	−0.531320	−	0.847171i	$-0.678304\pi$
$353$	−19.9652	−1.06264	−0.531320	+	0.847171i	$0.678304\pi$
$354$	0	0
$355$	28.9610	1.53709
$356$	−3.39414	−0.179889
$357$	0	0
$358$	−8.29454	−0.438380
$359$	−13.9610	−0.736835	−0.368418	−	0.929660i	$-0.620100\pi$
$359$	−13.9610	−0.736835	−0.368418	+	0.929660i	$0.620100\pi$
$360$	0	0
$361$	−13.6827	−0.720141
$362$	−16.7589	−0.880827
$363$	0	0
$364$	−0.0942181	−0.00493837
$365$	−16.8848	−0.883793
$366$	0	0
$367$	4.80986	0.251072	0.125536	−	0.992089i	$-0.459935\pi$
$367$	4.80986	0.251072	0.125536	+	0.992089i	$0.459935\pi$
$368$	−0.409705	−0.0213573
$369$	0	0
$370$	18.6402	0.969056
$371$	−1.42648	−0.0740592
$372$	0	0
$373$	−21.8961	−1.13374	−0.566868	−	0.823808i	$-0.691845\pi$
$373$	−21.8961	−1.13374	−0.566868	+	0.823808i	$0.691845\pi$
$374$	0	0
$375$	0	0
$376$	9.96704	0.514011
$377$	−4.86681	−0.250654
$378$	0	0
$379$	0.562718	0.0289049	0.0144524	−	0.999896i	$-0.495399\pi$
$379$	0.562718	0.0289049	0.0144524	+	0.999896i	$0.495399\pi$
$380$	5.97305	0.306411
$381$	0	0
$382$	6.39414	0.327153
$383$	18.0156	0.920552	0.460276	−	0.887776i	$-0.347750\pi$
$383$	18.0156	0.920552	0.460276	+	0.887776i	$0.347750\pi$
$384$	0	0
$385$	0	0
$386$	21.1476	1.07639
$387$	0	0
$388$	−12.9730	−0.658607
$389$	−0.0648026	−0.00328562	−0.00164281	−	0.999999i	$-0.500523\pi$
$389$	−0.0648026	−0.00328562	−0.00164281	+	0.999999i	$0.500523\pi$
$390$	0	0
$391$	0	0
$392$	−6.97305	−0.352192
$393$	0	0
$394$	14.4043	0.725679
$395$	26.3127	1.32393
$396$	0	0
$397$	−0.592757	−0.0297496	−0.0148748	−	0.999889i	$-0.504735\pi$
$397$	−0.592757	−0.0297496	−0.0148748	+	0.999889i	$0.504735\pi$
$398$	23.3654	1.17120
$399$	0	0
$400$	1.70963	0.0854815
$401$	11.6240	0.580474	0.290237	−	0.956955i	$-0.406266\pi$
$401$	11.6240	0.580474	0.290237	+	0.956955i	$0.406266\pi$
$402$	0	0
$403$	−3.14861	−0.156844
$404$	−4.39230	−0.218525
$405$	0	0
$406$	1.39230	0.0690989
$407$	0	0
$408$	0	0
$409$	−7.78228	−0.384809	−0.192404	−	0.981316i	$-0.561629\pi$
$409$	−7.78228	−0.384809	−0.192404	+	0.981316i	$0.561629\pi$
$410$	−0.736584	−0.0363773
$411$	0	0
$412$	−17.6497	−0.869539
$413$	0.640657	0.0315247
$414$	0	0
$415$	4.65694	0.228600
$416$	−0.573882	−0.0281369
$417$	0	0
$418$	0	0
$419$	12.5418	0.612706	0.306353	−	0.951918i	$-0.400891\pi$
$419$	12.5418	0.612706	0.306353	+	0.951918i	$0.400891\pi$
$420$	0	0
$421$	11.7529	0.572799	0.286400	−	0.958110i	$-0.407541\pi$
$421$	11.7529	0.572799	0.286400	+	0.958110i	$0.407541\pi$
$422$	2.46912	0.120195
$423$	0	0
$424$	−8.68868	−0.421960
$425$	0	0
$426$	0	0
$427$	−1.34306	−0.0649954
$428$	14.4043	0.696259
$429$	0	0
$430$	16.2784	0.785012
$431$	−6.56873	−0.316404	−0.158202	−	0.987407i	$-0.550570\pi$
$431$	−6.56873	−0.316404	−0.158202	+	0.987407i	$0.550570\pi$
$432$	0	0
$433$	−13.6827	−0.657547	−0.328774	−	0.944409i	$-0.606635\pi$
$433$	−13.6827	−0.657547	−0.328774	+	0.944409i	$0.606635\pi$
$434$	0.900760	0.0432379
$435$	0	0
$436$	−5.78143	−0.276880
$437$	0.944752	0.0451936
$438$	0	0
$439$	−21.9774	−1.04893	−0.524463	−	0.851433i	$-0.675734\pi$
$439$	−21.9774	−1.04893	−0.524463	+	0.851433i	$0.675734\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	13.9652	0.663507	0.331753	−	0.943366i	$-0.392360\pi$
$443$	13.9652	0.663507	0.331753	+	0.943366i	$0.392360\pi$
$444$	0	0
$445$	8.79183	0.416773
$446$	3.20216	0.151627
$447$	0	0
$448$	0.164177	0.00775663
$449$	−24.6533	−1.16346	−0.581730	−	0.813382i	$-0.697624\pi$
$449$	−24.6533	−1.16346	−0.581730	+	0.813382i	$0.697624\pi$
$450$	0	0
$451$	0	0
$452$	−1.29392	−0.0608607
$453$	0	0
$454$	−1.56272	−0.0733420
$455$	0.244053	0.0114414
$456$	0	0
$457$	39.2994	1.83835	0.919175	−	0.393850i	$-0.128857\pi$
$457$	39.2994	1.83835	0.919175	+	0.393850i	$0.128857\pi$
$458$	11.9791	0.559744
$459$	0	0
$460$	1.06126	0.0494813
$461$	−2.10023	−0.0978173	−0.0489086	−	0.998803i	$-0.515574\pi$
$461$	−2.10023	−0.0978173	−0.0489086	+	0.998803i	$0.515574\pi$
$462$	0	0
$463$	28.1512	1.30830	0.654148	−	0.756367i	$-0.273027\pi$
$463$	28.1512	1.30830	0.654148	+	0.756367i	$0.273027\pi$
$464$	8.48052	0.393698
$465$	0	0
$466$	−26.8519	−1.24389
$467$	3.45895	0.160061	0.0800305	−	0.996792i	$-0.474498\pi$
$467$	3.45895	0.160061	0.0800305	+	0.996792i	$0.474498\pi$
$468$	0	0
$469$	1.06395	0.0491287
$470$	−25.8176	−1.19088
$471$	0	0
$472$	3.90224	0.179615
$473$	0	0
$474$	0	0
$475$	−3.94229	−0.180885
$476$	0	0
$477$	0	0
$478$	−1.58074	−0.0723014
$479$	−39.0870	−1.78593	−0.892965	−	0.450126i	$-0.851379\pi$
$479$	−39.0870	−1.78593	−0.892965	+	0.450126i	$0.851379\pi$
$480$	0	0
$481$	4.12974	0.188300
$482$	−22.1208	−1.00758
$483$	0	0
$484$	0	0
$485$	33.6040	1.52588
$486$	0	0
$487$	8.10794	0.367406	0.183703	−	0.982982i	$-0.441192\pi$
$487$	8.10794	0.367406	0.183703	+	0.982982i	$0.441192\pi$
$488$	−8.18059	−0.370318
$489$	0	0
$490$	18.0622	0.815969
$491$	21.1543	0.954678	0.477339	−	0.878719i	$-0.341601\pi$
$491$	21.1543	0.954678	0.477339	+	0.878719i	$0.341601\pi$
$492$	0	0
$493$	0	0
$494$	1.32333	0.0595395
$495$	0	0
$496$	5.48652	0.246352
$497$	−1.83559	−0.0823377
$498$	0	0
$499$	17.4776	0.782404	0.391202	−	0.920305i	$-0.372059\pi$
$499$	17.4776	0.782404	0.391202	+	0.920305i	$0.372059\pi$
$500$	8.52303	0.381162
$501$	0	0
$502$	2.54105	0.113413
$503$	40.6317	1.81168	0.905839	−	0.423623i	$-0.139242\pi$
$503$	40.6317	1.81168	0.905839	+	0.423623i	$0.139242\pi$
$504$	0	0
$505$	11.3774	0.506286
$506$	0	0
$507$	0	0
$508$	−17.9073	−0.794506
$509$	30.0648	1.33260	0.666299	−	0.745684i	$-0.267877\pi$
$509$	30.0648	1.33260	0.666299	+	0.745684i	$0.267877\pi$
$510$	0	0
$511$	1.07019	0.0473423
$512$	1.00000	0.0441942
$513$	0	0
$514$	15.0492	0.663794
$515$	45.7180	2.01457
$516$	0	0
$517$	0	0
$518$	−1.18144	−0.0519096
$519$	0	0
$520$	1.48652	0.0651883
$521$	10.0834	0.441764	0.220882	−	0.975301i	$-0.429106\pi$
$521$	10.0834	0.441764	0.220882	+	0.975301i	$0.429106\pi$
$522$	0	0
$523$	−2.78707	−0.121870	−0.0609351	−	0.998142i	$-0.519408\pi$
$523$	−2.78707	−0.121870	−0.0609351	+	0.998142i	$0.519408\pi$
$524$	−5.19615	−0.226995
$525$	0	0
$526$	−14.9730	−0.652856
$527$	0	0
$528$	0	0
$529$	−22.8321	−0.992702
$530$	22.5063	0.977609
$531$	0	0
$532$	−0.378581	−0.0164136
$533$	−0.163191	−0.00706857
$534$	0	0
$535$	−37.3114	−1.61311
$536$	6.48052	0.279916
$537$	0	0
$538$	16.4596	0.709622
$539$	0	0
$540$	0	0
$541$	−33.9054	−1.45771	−0.728854	−	0.684669i	$-0.759947\pi$
$541$	−33.9054	−1.45771	−0.728854	+	0.684669i	$0.759947\pi$
$542$	3.04153	0.130645
$543$	0	0
$544$	0	0
$545$	14.9756	0.641484
$546$	0	0
$547$	13.8797	0.593452	0.296726	−	0.954963i	$-0.404105\pi$
$547$	13.8797	0.593452	0.296726	+	0.954963i	$0.404105\pi$
$548$	−8.60586	−0.367624
$549$	0	0
$550$	0	0
$551$	−19.5555	−0.833092
$552$	0	0
$553$	−1.66774	−0.0709194
$554$	−29.4820	−1.25257
$555$	0	0
$556$	−17.8244	−0.755924
$557$	5.62397	0.238295	0.119148	−	0.992877i	$-0.461984\pi$
$557$	5.62397	0.238295	0.119148	+	0.992877i	$0.461984\pi$
$558$	0	0
$559$	3.60648	0.152538
$560$	−0.425267	−0.0179708
$561$	0	0
$562$	−24.9178	−1.05109
$563$	26.5028	1.11696	0.558480	−	0.829518i	$-0.311385\pi$
$563$	26.5028	1.11696	0.558480	+	0.829518i	$0.311385\pi$
$564$	0	0
$565$	3.35163	0.141004
$566$	18.6438	0.783659
$567$	0	0
$568$	−11.1806	−0.469127
$569$	−7.89206	−0.330852	−0.165426	−	0.986222i	$-0.552900\pi$
$569$	−7.89206	−0.330852	−0.165426	+	0.986222i	$0.552900\pi$
$570$	0	0
$571$	−19.5098	−0.816460	−0.408230	−	0.912879i	$-0.633854\pi$
$571$	−19.5098	−0.816460	−0.408230	+	0.912879i	$0.633854\pi$
$572$	0	0
$573$	0	0
$574$	0.0466858	0.00194863
$575$	−0.700443	−0.0292105
$576$	0	0
$577$	−17.5418	−0.730274	−0.365137	−	0.930954i	$-0.618978\pi$
$577$	−17.5418	−0.730274	−0.365137	+	0.930954i	$0.618978\pi$
$578$	−17.0000	−0.707107
$579$	0	0
$580$	−21.9670	−0.912132
$581$	−0.295164	−0.0122455
$582$	0	0
$583$	0	0
$584$	6.51850	0.269737
$585$	0	0
$586$	−6.58675	−0.272096
$587$	9.42782	0.389128	0.194564	−	0.980890i	$-0.437671\pi$
$587$	9.42782	0.389128	0.194564	+	0.980890i	$0.437671\pi$
$588$	0	0
$589$	−12.6516	−0.521298
$590$	−10.1079	−0.416137
$591$	0	0
$592$	−7.19615	−0.295760
$593$	30.5435	1.25427	0.627135	−	0.778910i	$-0.284227\pi$
$593$	30.5435	1.25427	0.627135	+	0.778910i	$0.284227\pi$
$594$	0	0
$595$	0	0
$596$	12.3983	0.507855
$597$	0	0
$598$	0.235122	0.00961486
$599$	−18.5237	−0.756860	−0.378430	−	0.925630i	$-0.623536\pi$
$599$	−18.5237	−0.756860	−0.378430	+	0.925630i	$0.623536\pi$
$600$	0	0
$601$	−11.8217	−0.482219	−0.241109	−	0.970498i	$-0.577511\pi$
$601$	−11.8217	−0.482219	−0.241109	+	0.970498i	$0.577511\pi$
$602$	−1.03175	−0.0420509
$603$	0	0
$604$	4.77604	0.194334
$605$	0	0
$606$	0	0
$607$	39.9547	1.62171	0.810855	−	0.585247i	$-0.199003\pi$
$607$	39.9547	1.62171	0.810855	+	0.585247i	$0.199003\pi$
$608$	−2.30593	−0.0935179
$609$	0	0
$610$	21.1901	0.857964
$611$	−5.71990	−0.231402
$612$	0	0
$613$	16.4803	0.665633	0.332816	−	0.942992i	$-0.392001\pi$
$613$	16.4803	0.665633	0.332816	+	0.942992i	$0.392001\pi$
$614$	−8.60316	−0.347195
$615$	0	0
$616$	0	0
$617$	35.3593	1.42351	0.711757	−	0.702426i	$-0.247900\pi$
$617$	35.3593	1.42351	0.711757	+	0.702426i	$0.247900\pi$
$618$	0	0
$619$	10.4985	0.421972	0.210986	−	0.977489i	$-0.432333\pi$
$619$	10.4985	0.421972	0.210986	+	0.977489i	$0.432333\pi$
$620$	−14.2117	−0.570756
$621$	0	0
$622$	−28.5735	−1.14569
$623$	−0.557240	−0.0223253
$624$	0	0
$625$	−30.6253	−1.22501
$626$	−5.70963	−0.228203
$627$	0	0
$628$	17.6005	0.702335
$629$	0	0
$630$	0	0
$631$	−20.4163	−0.812762	−0.406381	−	0.913704i	$-0.633209\pi$
$631$	−20.4163	−0.812762	−0.406381	+	0.913704i	$0.633209\pi$
$632$	−10.1582	−0.404070
$633$	0	0
$634$	26.7535	1.06252
$635$	46.3851	1.84074
$636$	0	0
$637$	4.00170	0.158553
$638$	0	0
$639$	0	0
$640$	−2.59030	−0.102390
$641$	35.8830	1.41729	0.708647	−	0.705564i	$-0.249306\pi$
$641$	35.8830	1.41729	0.708647	+	0.705564i	$0.249306\pi$
$642$	0	0
$643$	35.9581	1.41805	0.709025	−	0.705184i	$-0.249135\pi$
$643$	35.9581	1.41805	0.709025	+	0.705184i	$0.249135\pi$
$644$	−0.0672641	−0.00265058
$645$	0	0
$646$	0	0
$647$	−24.3419	−0.956981	−0.478490	−	0.878093i	$-0.658816\pi$
$647$	−24.3419	−0.956981	−0.478490	+	0.878093i	$0.658816\pi$
$648$	0	0
$649$	0	0
$650$	−0.981125	−0.0384829
$651$	0	0
$652$	−2.48652	−0.0973798
$653$	23.3462	0.913609	0.456805	−	0.889567i	$-0.348994\pi$
$653$	23.3462	0.913609	0.456805	+	0.889567i	$0.348994\pi$
$654$	0	0
$655$	13.4596	0.525909
$656$	0.284363	0.0111025
$657$	0	0
$658$	1.63636	0.0637919
$659$	−44.9281	−1.75015	−0.875075	−	0.483988i	$-0.839188\pi$
$659$	−44.9281	−1.75015	−0.875075	+	0.483988i	$0.839188\pi$
$660$	0	0
$661$	−32.0913	−1.24821	−0.624103	−	0.781342i	$-0.714535\pi$
$661$	−32.0913	−1.24821	−0.624103	+	0.781342i	$0.714535\pi$
$662$	23.8459	0.926796
$663$	0	0
$664$	−1.79784	−0.0697697
$665$	0.980636	0.0380274
$666$	0	0
$667$	−3.47451	−0.134533
$668$	−20.6707	−0.799772
$669$	0	0
$670$	−16.7864	−0.648517
$671$	0	0
$672$	0	0
$673$	9.78214	0.377074	0.188537	−	0.982066i	$-0.439625\pi$
$673$	9.78214	0.377074	0.188537	+	0.982066i	$0.439625\pi$
$674$	−13.1997	−0.508433
$675$	0	0
$676$	−12.6707	−0.487333
$677$	−44.9773	−1.72862	−0.864309	−	0.502961i	$-0.832244\pi$
$677$	−44.9773	−1.72862	−0.864309	+	0.502961i	$0.832244\pi$
$678$	0	0
$679$	−2.12987	−0.0817371
$680$	0	0
$681$	0	0
$682$	0	0
$683$	31.6047	1.20932	0.604661	−	0.796483i	$-0.293309\pi$
$683$	31.6047	1.20932	0.604661	+	0.796483i	$0.293309\pi$
$684$	0	0
$685$	22.2917	0.851722
$686$	−2.29405	−0.0875873
$687$	0	0
$688$	−6.28436	−0.239589
$689$	4.98628	0.189962
$690$	0	0
$691$	34.3324	1.30607	0.653033	−	0.757330i	$-0.273496\pi$
$691$	34.3324	1.30607	0.653033	+	0.757330i	$0.273496\pi$
$692$	−12.4925	−0.474895
$693$	0	0
$694$	−14.3593	−0.545073
$695$	46.1705	1.75135
$696$	0	0
$697$	0	0
$698$	−5.92134	−0.224126
$699$	0	0
$700$	0.280682	0.0106088
$701$	25.1555	0.950109	0.475055	−	0.879956i	$-0.342428\pi$
$701$	25.1555	0.950109	0.475055	+	0.879956i	$0.342428\pi$
$702$	0	0
$703$	16.5938	0.625849
$704$	0	0
$705$	0	0
$706$	−19.9652	−0.751400
$707$	−0.721115	−0.0271203
$708$	0	0
$709$	−43.7110	−1.64160	−0.820800	−	0.571216i	$-0.806472\pi$
$709$	−43.7110	−1.64160	−0.820800	+	0.571216i	$0.806472\pi$
$710$	28.9610	1.08689
$711$	0	0
$712$	−3.39414	−0.127201
$713$	−2.24785	−0.0841828
$714$	0	0
$715$	0	0
$716$	−8.29454	−0.309982
$717$	0	0
$718$	−13.9610	−0.521021
$719$	49.1614	1.83341	0.916705	−	0.399566i	$-0.130839\pi$
$719$	49.1614	1.83341	0.916705	+	0.399566i	$0.130839\pi$
$720$	0	0
$721$	−2.89768	−0.107915
$722$	−13.6827	−0.509216
$723$	0	0
$724$	−16.7589	−0.622839
$725$	14.4985	0.538462
$726$	0	0
$727$	−14.3803	−0.533335	−0.266668	−	0.963789i	$-0.585923\pi$
$727$	−14.3803	−0.533335	−0.266668	+	0.963789i	$0.585923\pi$
$728$	−0.0942181	−0.00349196
$729$	0	0
$730$	−16.8848	−0.624936
$731$	0	0
$732$	0	0
$733$	−5.10193	−0.188444	−0.0942221	−	0.995551i	$-0.530036\pi$
$733$	−5.10193	−0.188444	−0.0942221	+	0.995551i	$0.530036\pi$
$734$	4.80986	0.177535
$735$	0	0
$736$	−0.409705	−0.0151019
$737$	0	0
$738$	0	0
$739$	4.58859	0.168794	0.0843970	−	0.996432i	$-0.473104\pi$
$739$	4.58859	0.168794	0.0843970	+	0.996432i	$0.473104\pi$
$740$	18.6402	0.685226
$741$	0	0
$742$	−1.42648	−0.0523678
$743$	−36.0331	−1.32193	−0.660963	−	0.750419i	$-0.729852\pi$
$743$	−36.0331	−1.32193	−0.660963	+	0.750419i	$0.729852\pi$
$744$	0	0
$745$	−32.1153	−1.17661
$746$	−21.8961	−0.801673
$747$	0	0
$748$	0	0
$749$	2.36486	0.0864100
$750$	0	0
$751$	3.52976	0.128803	0.0644013	−	0.997924i	$-0.479486\pi$
$751$	3.52976	0.128803	0.0644013	+	0.997924i	$0.479486\pi$
$752$	9.96704	0.363460
$753$	0	0
$754$	−4.86681	−0.177239
$755$	−12.3714	−0.450240
$756$	0	0
$757$	14.9143	0.542071	0.271036	−	0.962569i	$-0.412634\pi$
$757$	14.9143	0.542071	0.271036	+	0.962569i	$0.412634\pi$
$758$	0.562718	0.0204388
$759$	0	0
$760$	5.97305	0.216665
$761$	5.43383	0.196976	0.0984881	−	0.995138i	$-0.468599\pi$
$761$	5.43383	0.196976	0.0984881	+	0.995138i	$0.468599\pi$
$762$	0	0
$763$	−0.949177	−0.0343625
$764$	6.39414	0.231332
$765$	0	0
$766$	18.0156	0.650929
$767$	−2.23942	−0.0808608
$768$	0	0
$769$	−10.7117	−0.386275	−0.193137	−	0.981172i	$-0.561866\pi$
$769$	−10.7117	−0.386275	−0.193137	+	0.981172i	$0.561866\pi$
$770$	0	0
$771$	0	0
$772$	21.1476	0.761120
$773$	−17.4733	−0.628471	−0.314235	−	0.949345i	$-0.601748\pi$
$773$	−17.4733	−0.628471	−0.314235	+	0.949345i	$0.601748\pi$
$774$	0	0
$775$	9.37992	0.336937
$776$	−12.9730	−0.465705
$777$	0	0
$778$	−0.0648026	−0.00232329
$779$	−0.655721	−0.0234937
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−6.97305	−0.249037
$785$	−45.5904	−1.62719
$786$	0	0
$787$	−37.8124	−1.34787	−0.673934	−	0.738792i	$-0.735397\pi$
$787$	−37.8124	−1.34787	−0.673934	+	0.738792i	$0.735397\pi$
$788$	14.4043	0.513133
$789$	0	0
$790$	26.3127	0.936162
$791$	−0.212431	−0.00755319
$792$	0	0
$793$	4.69469	0.166713
$794$	−0.592757	−0.0210361
$795$	0	0
$796$	23.3654	0.828163
$797$	35.2117	1.24726	0.623631	−	0.781718i	$-0.285657\pi$
$797$	35.2117	1.24726	0.623631	+	0.781718i	$0.285657\pi$
$798$	0	0
$799$	0	0
$800$	1.70963	0.0604445
$801$	0	0
$802$	11.6240	0.410457
$803$	0	0
$804$	0	0
$805$	0.174234	0.00614093
$806$	−3.14861	−0.110905
$807$	0	0
$808$	−4.39230	−0.154521
$809$	8.74259	0.307373	0.153687	−	0.988120i	$-0.450885\pi$
$809$	8.74259	0.307373	0.153687	+	0.988120i	$0.450885\pi$
$810$	0	0
$811$	−33.2110	−1.16619	−0.583097	−	0.812402i	$-0.698159\pi$
$811$	−33.2110	−1.16619	−0.583097	+	0.812402i	$0.698159\pi$
$812$	1.39230	0.0488603
$813$	0	0
$814$	0	0
$815$	6.44083	0.225612
$816$	0	0
$817$	14.4913	0.506987
$818$	−7.78228	−0.272101
$819$	0	0
$820$	−0.736584	−0.0257226
$821$	26.2562	0.916347	0.458174	−	0.888863i	$-0.348504\pi$
$821$	26.2562	0.916347	0.458174	+	0.888863i	$0.348504\pi$
$822$	0	0
$823$	6.09714	0.212533	0.106266	−	0.994338i	$-0.466110\pi$
$823$	6.09714	0.212533	0.106266	+	0.994338i	$0.466110\pi$
$824$	−17.6497	−0.614857
$825$	0	0
$826$	0.640657	0.0222913
$827$	−2.74515	−0.0954581	−0.0477290	−	0.998860i	$-0.515198\pi$
$827$	−2.74515	−0.0954581	−0.0477290	+	0.998860i	$0.515198\pi$
$828$	0	0
$829$	36.3295	1.26177	0.630887	−	0.775875i	$-0.282691\pi$
$829$	36.3295	1.26177	0.630887	+	0.775875i	$0.282691\pi$
$830$	4.65694	0.161645
$831$	0	0
$832$	−0.573882	−0.0198958
$833$	0	0
$834$	0	0
$835$	53.5431	1.85294
$836$	0	0
$837$	0	0
$838$	12.5418	0.433248
$839$	50.4734	1.74253	0.871267	−	0.490809i	$-0.163299\pi$
$839$	50.4734	1.74253	0.871267	+	0.490809i	$0.163299\pi$
$840$	0	0
$841$	42.9191	1.47997
$842$	11.7529	0.405030
$843$	0	0
$844$	2.46912	0.0849907
$845$	32.8207	1.12907
$846$	0	0
$847$	0	0
$848$	−8.68868	−0.298371
$849$	0	0
$850$	0	0
$851$	2.94830	0.101066
$852$	0	0
$853$	36.5937	1.25294	0.626472	−	0.779444i	$-0.284498\pi$
$853$	36.5937	1.25294	0.626472	+	0.779444i	$0.284498\pi$
$854$	−1.34306	−0.0459587
$855$	0	0
$856$	14.4043	0.492330
$857$	44.8939	1.53355	0.766773	−	0.641918i	$-0.221861\pi$
$857$	44.8939	1.53355	0.766773	+	0.641918i	$0.221861\pi$
$858$	0	0
$859$	9.24969	0.315595	0.157798	−	0.987471i	$-0.449561\pi$
$859$	9.24969	0.315595	0.157798	+	0.987471i	$0.449561\pi$
$860$	16.2784	0.555087
$861$	0	0
$862$	−6.56873	−0.223732
$863$	6.22790	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$863$	6.22790	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$864$	0	0
$865$	32.3593	1.10025
$866$	−13.6827	−0.464956
$867$	0	0
$868$	0.900760	0.0305738
$869$	0	0
$870$	0	0
$871$	−3.71905	−0.126015
$872$	−5.78143	−0.195784
$873$	0	0
$874$	0.944752	0.0319567
$875$	1.39928	0.0473045
$876$	0	0
$877$	15.5771	0.526000	0.263000	−	0.964796i	$-0.415288\pi$
$877$	15.5771	0.526000	0.263000	+	0.964796i	$0.415288\pi$
$878$	−21.9774	−0.741703
$879$	0	0
$880$	0	0
$881$	32.0143	1.07859	0.539295	−	0.842117i	$-0.318691\pi$
$881$	32.0143	1.07859	0.539295	+	0.842117i	$0.318691\pi$
$882$	0	0
$883$	−49.2651	−1.65790	−0.828952	−	0.559320i	$-0.811062\pi$
$883$	−49.2651	−1.65790	−0.828952	+	0.559320i	$0.811062\pi$
$884$	0	0
$885$	0	0
$886$	13.9652	0.469170
$887$	−14.8116	−0.497324	−0.248662	−	0.968590i	$-0.579991\pi$
$887$	−14.8116	−0.497324	−0.248662	+	0.968590i	$0.579991\pi$
$888$	0	0
$889$	−2.93996	−0.0986030
$890$	8.79183	0.294703
$891$	0	0
$892$	3.20216	0.107216
$893$	−22.9833	−0.769107
$894$	0	0
$895$	21.4853	0.718175
$896$	0.164177	0.00548477
$897$	0	0
$898$	−24.6533	−0.822690
$899$	46.5285	1.55181
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−1.29392	−0.0430350
$905$	43.4104	1.44301
$906$	0	0
$907$	−23.6120	−0.784022	−0.392011	−	0.919960i	$-0.628221\pi$
$907$	−23.6120	−0.784022	−0.392011	+	0.919960i	$0.628221\pi$
$908$	−1.56272	−0.0518606
$909$	0	0
$910$	0.244053	0.00809027
$911$	39.6869	1.31489	0.657443	−	0.753504i	$-0.271638\pi$
$911$	39.6869	1.31489	0.657443	+	0.753504i	$0.271638\pi$
$912$	0	0
$913$	0	0
$914$	39.2994	1.29991
$915$	0	0
$916$	11.9791	0.395799
$917$	−0.853088	−0.0281715
$918$	0	0
$919$	3.86094	0.127361	0.0636803	−	0.997970i	$-0.479716\pi$
$919$	3.86094	0.127361	0.0636803	+	0.997970i	$0.479716\pi$
$920$	1.06126	0.0349886
$921$	0	0
$922$	−2.10023	−0.0691673
$923$	6.41634	0.211196
$924$	0	0
$925$	−12.3028	−0.404512
$926$	28.1512	0.925105
$927$	0	0
$928$	8.48052	0.278387
$929$	34.8668	1.14394	0.571972	−	0.820273i	$-0.306179\pi$
$929$	34.8668	1.14394	0.571972	+	0.820273i	$0.306179\pi$
$930$	0	0
$931$	16.0794	0.526980
$932$	−26.8519	−0.879562
$933$	0	0
$934$	3.45895	0.113180
$935$	0	0
$936$	0	0
$937$	58.0513	1.89645	0.948227	−	0.317593i	$-0.102874\pi$
$937$	58.0513	1.89645	0.948227	+	0.317593i	$0.102874\pi$
$938$	1.06395	0.0347392
$939$	0	0
$940$	−25.8176	−0.842076
$941$	6.41634	0.209167	0.104583	−	0.994516i	$-0.466649\pi$
$941$	6.41634	0.209167	0.104583	+	0.994516i	$0.466649\pi$
$942$	0	0
$943$	−0.116505	−0.00379392
$944$	3.90224	0.127007
$945$	0	0
$946$	0	0
$947$	−55.7498	−1.81163	−0.905813	−	0.423678i	$-0.860739\pi$
$947$	−55.7498	−1.81163	−0.905813	+	0.423678i	$0.860739\pi$
$948$	0	0
$949$	−3.74085	−0.121433
$950$	−3.94229	−0.127905
$951$	0	0
$952$	0	0
$953$	−1.42181	−0.0460571	−0.0230285	−	0.999735i	$-0.507331\pi$
$953$	−1.42181	−0.0460571	−0.0230285	+	0.999735i	$0.507331\pi$
$954$	0	0
$955$	−16.5627	−0.535957
$956$	−1.58074	−0.0511248
$957$	0	0
$958$	−39.0870	−1.26284
$959$	−1.41288	−0.0456244
$960$	0	0
$961$	−0.898066	−0.0289699
$962$	4.12974	0.133148
$963$	0	0
$964$	−22.1208	−0.712463
$965$	−54.7786	−1.76339
$966$	0	0
$967$	−37.5892	−1.20879	−0.604393	−	0.796686i	$-0.706584\pi$
$967$	−37.5892	−1.20879	−0.604393	+	0.796686i	$0.706584\pi$
$968$	0	0
$969$	0	0
$970$	33.6040	1.07896
$971$	−53.5070	−1.71712	−0.858560	−	0.512713i	$-0.828641\pi$
$971$	−53.5070	−1.71712	−0.858560	+	0.512713i	$0.828641\pi$
$972$	0	0
$973$	−2.92636	−0.0938148
$974$	8.10794	0.259795
$975$	0	0
$976$	−8.18059	−0.261854
$977$	−27.5729	−0.882135	−0.441068	−	0.897474i	$-0.645400\pi$
$977$	−27.5729	−0.882135	−0.441068	+	0.897474i	$0.645400\pi$
$978$	0	0
$979$	0	0
$980$	18.0622	0.576977
$981$	0	0
$982$	21.1543	0.675060
$983$	10.5424	0.336250	0.168125	−	0.985766i	$-0.446229\pi$
$983$	10.5424	0.336250	0.168125	+	0.985766i	$0.446229\pi$
$984$	0	0
$985$	−37.3114	−1.18884
$986$	0	0
$987$	0	0
$988$	1.32333	0.0421008
$989$	2.57473	0.0818718
$990$	0	0
$991$	−47.8656	−1.52050	−0.760250	−	0.649630i	$-0.774924\pi$
$991$	−47.8656	−1.52050	−0.760250	+	0.649630i	$0.774924\pi$
$992$	5.48652	0.174197
$993$	0	0
$994$	−1.83559	−0.0582215
$995$	−60.5232	−1.91871
$996$	0	0
$997$	59.5121	1.88477	0.942384	−	0.334533i	$-0.108579\pi$
$997$	59.5121	1.88477	0.942384	+	0.334533i	$0.108579\pi$
$998$	17.4776	0.553244
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6534.2.a.cs.1.1	yes	4
3.2	odd	2		6534.2.a.cq.1.4	✓	4
11.10	odd	2		6534.2.a.cr.1.1	yes	4
33.32	even	2		6534.2.a.ct.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6534.2.a.cq.1.4	✓	4	3.2	odd	2
6534.2.a.cr.1.1	yes	4	11.10	odd	2
6534.2.a.cs.1.1	yes	4	1.1	even	1	trivial
6534.2.a.ct.1.4	yes	4	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6534 = 2 \cdot 3^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6534.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.59030 q^{5} +0.164177 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.59030 q^{5} +0.164177 q^{7} +1.00000 q^{8} -2.59030 q^{10} -0.573882 q^{13} +0.164177 q^{14} +1.00000 q^{16} -2.30593 q^{19} -2.59030 q^{20} -0.409705 q^{23} +1.70963 q^{25} -0.573882 q^{26} +0.164177 q^{28} +8.48052 q^{29} +5.48652 q^{31} +1.00000 q^{32} -0.425267 q^{35} -7.19615 q^{37} -2.30593 q^{38} -2.59030 q^{40} +0.284363 q^{41} -6.28436 q^{43} -0.409705 q^{46} +9.96704 q^{47} -6.97305 q^{49} +1.70963 q^{50} -0.573882 q^{52} -8.68868 q^{53} +0.164177 q^{56} +8.48052 q^{58} +3.90224 q^{59} -8.18059 q^{61} +5.48652 q^{62} +1.00000 q^{64} +1.48652 q^{65} +6.48052 q^{67} -0.425267 q^{70} -11.1806 q^{71} +6.51850 q^{73} -7.19615 q^{74} -2.30593 q^{76} -10.1582 q^{79} -2.59030 q^{80} +0.284363 q^{82} -1.79784 q^{83} -6.28436 q^{86} -3.39414 q^{89} -0.0942181 q^{91} -0.409705 q^{92} +9.96704 q^{94} +5.97305 q^{95} -12.9730 q^{97} -6.97305 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} - 6 q^{7} + 4 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^4 - 6 * q^7 + 4 * q^8	\( 4 q + 4 q^{2} + 4 q^{4} - 6 q^{7} + 4 q^{8} - 6 q^{13} - 6 q^{14} + 4 q^{16} - 6 q^{19} - 12 q^{23} + 10 q^{25} - 6 q^{26} - 6 q^{28} + 6 q^{29} - 2 q^{31} + 4 q^{32} - 12 q^{35} - 8 q^{37} - 6 q^{38} - 6 q^{41} - 18 q^{43} - 12 q^{46} - 12 q^{47} + 20 q^{49} + 10 q^{50} - 6 q^{52} + 6 q^{53} - 6 q^{56} + 6 q^{58} + 6 q^{59} - 12 q^{61} - 2 q^{62} + 4 q^{64} - 18 q^{65} - 2 q^{67} - 12 q^{70} - 24 q^{71} - 12 q^{73} - 8 q^{74} - 6 q^{76} - 30 q^{79} - 6 q^{82} - 24 q^{83} - 18 q^{86} - 24 q^{89} - 18 q^{91} - 12 q^{92} - 12 q^{94} - 24 q^{95} - 4 q^{97} + 20 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 - 6 * q^7 + 4 * q^8 - 6 * q^13 - 6 * q^14 + 4 * q^16 - 6 * q^19 - 12 * q^23 + 10 * q^25 - 6 * q^26 - 6 * q^28 + 6 * q^29 - 2 * q^31 + 4 * q^32 - 12 * q^35 - 8 * q^37 - 6 * q^38 - 6 * q^41 - 18 * q^43 - 12 * q^46 - 12 * q^47 + 20 * q^49 + 10 * q^50 - 6 * q^52 + 6 * q^53 - 6 * q^56 + 6 * q^58 + 6 * q^59 - 12 * q^61 - 2 * q^62 + 4 * q^64 - 18 * q^65 - 2 * q^67 - 12 * q^70 - 24 * q^71 - 12 * q^73 - 8 * q^74 - 6 * q^76 - 30 * q^79 - 6 * q^82 - 24 * q^83 - 18 * q^86 - 24 * q^89 - 18 * q^91 - 12 * q^92 - 12 * q^94 - 24 * q^95 - 4 * q^97 + 20 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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