LMFDB - Embedded newform 4056.2.a.bi.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4056,2,Mod(1,4056)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4056, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4056.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4056 = 2^{3} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4056.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:	$32.3873230598$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.27700337.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 19x^{4} + 17x^{3} + 103x^{2} - 71x - 127 $ x^6 - x^5 - 19x^4 + 17x^3 + 103x^2 - 71x - 127
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.71914$ of defining polynomial
Character	$\chi$	$=$	4056.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.16419 q^{5} -3.98252 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.16419 q^{5} -3.98252 q^{7} +1.00000 q^{9} -6.19265 q^{11} +3.16419 q^{15} +0.0827940 q^{17} +2.80729 q^{19} -3.98252 q^{21} +6.47756 q^{23} +5.01207 q^{25} +1.00000 q^{27} +7.19023 q^{29} +4.21310 q^{31} -6.19265 q^{33} -12.6014 q^{35} +0.190284 q^{37} -2.55550 q^{41} -6.72819 q^{43} +3.16419 q^{45} +13.0163 q^{47} +8.86048 q^{49} +0.0827940 q^{51} +1.86294 q^{53} -19.5947 q^{55} +2.80729 q^{57} -4.92990 q^{59} +14.7395 q^{61} -3.98252 q^{63} +6.22172 q^{67} +6.47756 q^{69} -6.19803 q^{71} +13.8297 q^{73} +5.01207 q^{75} +24.6624 q^{77} +14.6130 q^{79} +1.00000 q^{81} -6.44604 q^{83} +0.261976 q^{85} +7.19023 q^{87} +11.3907 q^{89} +4.21310 q^{93} +8.88279 q^{95} +16.3049 q^{97} -6.19265 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} + q^{5} - 5 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 + q^5 - 5 * q^7 + 6 * q^9	$ 6 q + 6 q^{3} + q^{5} - 5 q^{7} + 6 q^{9} - 6 q^{11} + q^{15} + 9 q^{17} + 7 q^{19} - 5 q^{21} + 12 q^{23} + 9 q^{25} + 6 q^{27} + 7 q^{29} - 11 q^{31} - 6 q^{33} + 6 q^{35} + 6 q^{37} + 13 q^{41} + 15 q^{43} + q^{45} - 9 q^{47} + 13 q^{49} + 9 q^{51} + 22 q^{53} + 3 q^{55} + 7 q^{57} - 7 q^{59} + 25 q^{61} - 5 q^{63} + 5 q^{67} + 12 q^{69} + 8 q^{71} + 15 q^{73} + 9 q^{75} + 45 q^{77} + 14 q^{79} + 6 q^{81} - 13 q^{83} - 35 q^{85} + 7 q^{87} + 33 q^{89} - 11 q^{93} + 47 q^{95} + 50 q^{97} - 6 q^{99}+O(q^{100}) $ 6 * q + 6 * q^3 + q^5 - 5 * q^7 + 6 * q^9 - 6 * q^11 + q^15 + 9 * q^17 + 7 * q^19 - 5 * q^21 + 12 * q^23 + 9 * q^25 + 6 * q^27 + 7 * q^29 - 11 * q^31 - 6 * q^33 + 6 * q^35 + 6 * q^37 + 13 * q^41 + 15 * q^43 + q^45 - 9 * q^47 + 13 * q^49 + 9 * q^51 + 22 * q^53 + 3 * q^55 + 7 * q^57 - 7 * q^59 + 25 * q^61 - 5 * q^63 + 5 * q^67 + 12 * q^69 + 8 * q^71 + 15 * q^73 + 9 * q^75 + 45 * q^77 + 14 * q^79 + 6 * q^81 - 13 * q^83 - 35 * q^85 + 7 * q^87 + 33 * q^89 - 11 * q^93 + 47 * q^95 + 50 * q^97 - 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	3.16419	1.41507	0.707533	−	0.706680i	$-0.249808\pi$
$5$	3.16419	1.41507	0.707533	+	0.706680i	$0.249808\pi$
$6$	0	0
$7$	−3.98252	−1.50525	−0.752626	−	0.658448i	$-0.771213\pi$
$7$	−3.98252	−1.50525	−0.752626	+	0.658448i	$0.771213\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.19265	−1.86716	−0.933578	−	0.358375i	$-0.883331\pi$
$11$	−6.19265	−1.86716	−0.933578	+	0.358375i	$0.883331\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	3.16419	0.816989
$16$	0	0
$17$	0.0827940	0.0200805	0.0100402	−	0.999950i	$-0.496804\pi$
$17$	0.0827940	0.0200805	0.0100402	+	0.999950i	$0.496804\pi$
$18$	0	0
$19$	2.80729	0.644036	0.322018	−	0.946733i	$-0.395639\pi$
$19$	2.80729	0.644036	0.322018	+	0.946733i	$0.395639\pi$
$20$	0	0
$21$	−3.98252	−0.869057
$22$	0	0
$23$	6.47756	1.35066	0.675332	−	0.737513i	$-0.264000\pi$
$23$	6.47756	1.35066	0.675332	+	0.737513i	$0.264000\pi$
$24$	0	0
$25$	5.01207	1.00241
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	7.19023	1.33519	0.667596	−	0.744524i	$-0.267323\pi$
$29$	7.19023	1.33519	0.667596	+	0.744524i	$0.267323\pi$
$30$	0	0
$31$	4.21310	0.756696	0.378348	−	0.925664i	$-0.376492\pi$
$31$	4.21310	0.756696	0.378348	+	0.925664i	$0.376492\pi$
$32$	0	0
$33$	−6.19265	−1.07800
$34$	0	0
$35$	−12.6014	−2.13003
$36$	0	0
$37$	0.190284	0.0312824	0.0156412	−	0.999878i	$-0.495021\pi$
$37$	0.190284	0.0312824	0.0156412	+	0.999878i	$0.495021\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.55550	−0.399102	−0.199551	−	0.979887i	$-0.563948\pi$
$41$	−2.55550	−0.399102	−0.199551	+	0.979887i	$0.563948\pi$
$42$	0	0
$43$	−6.72819	−1.02604	−0.513020	−	0.858377i	$-0.671473\pi$
$43$	−6.72819	−1.02604	−0.513020	+	0.858377i	$0.671473\pi$
$44$	0	0
$45$	3.16419	0.471689
$46$	0	0
$47$	13.0163	1.89862	0.949312	−	0.314336i	$-0.101782\pi$
$47$	13.0163	1.89862	0.949312	+	0.314336i	$0.101782\pi$
$48$	0	0
$49$	8.86048	1.26578
$50$	0	0
$51$	0.0827940	0.0115935
$52$	0	0
$53$	1.86294	0.255894	0.127947	−	0.991781i	$-0.459161\pi$
$53$	1.86294	0.255894	0.127947	+	0.991781i	$0.459161\pi$
$54$	0	0
$55$	−19.5947	−2.64215
$56$	0	0
$57$	2.80729	0.371835
$58$	0	0
$59$	−4.92990	−0.641818	−0.320909	−	0.947110i	$-0.603988\pi$
$59$	−4.92990	−0.641818	−0.320909	+	0.947110i	$0.603988\pi$
$60$	0	0
$61$	14.7395	1.88720	0.943602	−	0.331082i	$-0.107414\pi$
$61$	14.7395	1.88720	0.943602	+	0.331082i	$0.107414\pi$
$62$	0	0
$63$	−3.98252	−0.501751
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.22172	0.760104	0.380052	−	0.924965i	$-0.375906\pi$
$67$	6.22172	0.760104	0.380052	+	0.924965i	$0.375906\pi$
$68$	0	0
$69$	6.47756	0.779807
$70$	0	0
$71$	−6.19803	−0.735570	−0.367785	−	0.929911i	$-0.619884\pi$
$71$	−6.19803	−0.735570	−0.367785	+	0.929911i	$0.619884\pi$
$72$	0	0
$73$	13.8297	1.61864	0.809320	−	0.587367i	$-0.199836\pi$
$73$	13.8297	1.61864	0.809320	+	0.587367i	$0.199836\pi$
$74$	0	0
$75$	5.01207	0.578744
$76$	0	0
$77$	24.6624	2.81054
$78$	0	0
$79$	14.6130	1.64409	0.822043	−	0.569425i	$-0.192834\pi$
$79$	14.6130	1.64409	0.822043	+	0.569425i	$0.192834\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.44604	−0.707545	−0.353772	−	0.935332i	$-0.615101\pi$
$83$	−6.44604	−0.707545	−0.353772	+	0.935332i	$0.615101\pi$
$84$	0	0
$85$	0.261976	0.0284152
$86$	0	0
$87$	7.19023	0.770873
$88$	0	0
$89$	11.3907	1.20742	0.603708	−	0.797206i	$-0.293689\pi$
$89$	11.3907	1.20742	0.603708	+	0.797206i	$0.293689\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.21310	0.436878
$94$	0	0
$95$	8.88279	0.911355
$96$	0	0
$97$	16.3049	1.65551	0.827756	−	0.561089i	$-0.189617\pi$
$97$	16.3049	1.65551	0.827756	+	0.561089i	$0.189617\pi$
$98$	0	0
$99$	−6.19265	−0.622385
$100$	0	0
$101$	−2.84245	−0.282835	−0.141417	−	0.989950i	$-0.545166\pi$
$101$	−2.84245	−0.282835	−0.141417	+	0.989950i	$0.545166\pi$
$102$	0	0
$103$	−0.789223	−0.0777645	−0.0388822	−	0.999244i	$-0.512380\pi$
$103$	−0.789223	−0.0777645	−0.0388822	+	0.999244i	$0.512380\pi$
$104$	0	0
$105$	−12.6014	−1.22977
$106$	0	0
$107$	6.15452	0.594980	0.297490	−	0.954725i	$-0.403851\pi$
$107$	6.15452	0.594980	0.297490	+	0.954725i	$0.403851\pi$
$108$	0	0
$109$	−17.4999	−1.67619	−0.838094	−	0.545526i	$-0.816330\pi$
$109$	−17.4999	−1.67619	−0.838094	+	0.545526i	$0.816330\pi$
$110$	0	0
$111$	0.190284	0.0180609
$112$	0	0
$113$	−4.32296	−0.406670	−0.203335	−	0.979109i	$-0.565178\pi$
$113$	−4.32296	−0.406670	−0.203335	+	0.979109i	$0.565178\pi$
$114$	0	0
$115$	20.4962	1.91128
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.329729	−0.0302262
$120$	0	0
$121$	27.3490	2.48627
$122$	0	0
$123$	−2.55550	−0.230422
$124$	0	0
$125$	0.0381934	0.00341613
$126$	0	0
$127$	−13.1138	−1.16366	−0.581832	−	0.813309i	$-0.697664\pi$
$127$	−13.1138	−1.16366	−0.581832	+	0.813309i	$0.697664\pi$
$128$	0	0
$129$	−6.72819	−0.592384
$130$	0	0
$131$	7.47175	0.652810	0.326405	−	0.945230i	$-0.394163\pi$
$131$	7.47175	0.652810	0.326405	+	0.945230i	$0.394163\pi$
$132$	0	0
$133$	−11.1801	−0.969437
$134$	0	0
$135$	3.16419	0.272330
$136$	0	0
$137$	−4.35367	−0.371959	−0.185980	−	0.982554i	$-0.559546\pi$
$137$	−4.35367	−0.371959	−0.185980	+	0.982554i	$0.559546\pi$
$138$	0	0
$139$	−13.0060	−1.10315	−0.551576	−	0.834125i	$-0.685973\pi$
$139$	−13.0060	−1.10315	−0.551576	+	0.834125i	$0.685973\pi$
$140$	0	0
$141$	13.0163	1.09617
$142$	0	0
$143$	0	0
$144$	0	0
$145$	22.7512	1.88939
$146$	0	0
$147$	8.86048	0.730800
$148$	0	0
$149$	−16.6182	−1.36142	−0.680709	−	0.732554i	$-0.738328\pi$
$149$	−16.6182	−1.36142	−0.680709	+	0.732554i	$0.738328\pi$
$150$	0	0
$151$	14.3250	1.16575	0.582875	−	0.812562i	$-0.301928\pi$
$151$	14.3250	1.16575	0.582875	+	0.812562i	$0.301928\pi$
$152$	0	0
$153$	0.0827940	0.00669350
$154$	0	0
$155$	13.3310	1.07077
$156$	0	0
$157$	1.35614	0.108232	0.0541158	−	0.998535i	$-0.482766\pi$
$157$	1.35614	0.108232	0.0541158	+	0.998535i	$0.482766\pi$
$158$	0	0
$159$	1.86294	0.147740
$160$	0	0
$161$	−25.7970	−2.03309
$162$	0	0
$163$	−9.26015	−0.725311	−0.362656	−	0.931923i	$-0.618130\pi$
$163$	−9.26015	−0.725311	−0.362656	+	0.931923i	$0.618130\pi$
$164$	0	0
$165$	−19.5947	−1.52545
$166$	0	0
$167$	−23.6785	−1.83229	−0.916147	−	0.400842i	$-0.868717\pi$
$167$	−23.6785	−1.83229	−0.916147	+	0.400842i	$0.868717\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	2.80729	0.214679
$172$	0	0
$173$	16.3593	1.24377	0.621887	−	0.783107i	$-0.286366\pi$
$173$	16.3593	1.24377	0.621887	+	0.783107i	$0.286366\pi$
$174$	0	0
$175$	−19.9607	−1.50889
$176$	0	0
$177$	−4.92990	−0.370554
$178$	0	0
$179$	6.04357	0.451717	0.225859	−	0.974160i	$-0.427481\pi$
$179$	6.04357	0.451717	0.225859	+	0.974160i	$0.427481\pi$
$180$	0	0
$181$	−11.8062	−0.877548	−0.438774	−	0.898597i	$-0.644587\pi$
$181$	−11.8062	−0.877548	−0.438774	+	0.898597i	$0.644587\pi$
$182$	0	0
$183$	14.7395	1.08958
$184$	0	0
$185$	0.602093	0.0442668
$186$	0	0
$187$	−0.512715	−0.0374934
$188$	0	0
$189$	−3.98252	−0.289686
$190$	0	0
$191$	−2.79423	−0.202183	−0.101092	−	0.994877i	$-0.532234\pi$
$191$	−2.79423	−0.202183	−0.101092	+	0.994877i	$0.532234\pi$
$192$	0	0
$193$	−11.7655	−0.846903	−0.423451	−	0.905919i	$-0.639182\pi$
$193$	−11.7655	−0.846903	−0.423451	+	0.905919i	$0.639182\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.21802	0.514263	0.257131	−	0.966376i	$-0.417223\pi$
$197$	7.21802	0.514263	0.257131	+	0.966376i	$0.417223\pi$
$198$	0	0
$199$	−1.28987	−0.0914363	−0.0457181	−	0.998954i	$-0.514558\pi$
$199$	−1.28987	−0.0914363	−0.0457181	+	0.998954i	$0.514558\pi$
$200$	0	0
$201$	6.22172	0.438846
$202$	0	0
$203$	−28.6352	−2.00980
$204$	0	0
$205$	−8.08608	−0.564756
$206$	0	0
$207$	6.47756	0.450222
$208$	0	0
$209$	−17.3846	−1.20252
$210$	0	0
$211$	10.4565	0.719856	0.359928	−	0.932980i	$-0.382801\pi$
$211$	10.4565	0.719856	0.359928	+	0.932980i	$0.382801\pi$
$212$	0	0
$213$	−6.19803	−0.424682
$214$	0	0
$215$	−21.2892	−1.45191
$216$	0	0
$217$	−16.7788	−1.13902
$218$	0	0
$219$	13.8297	0.934523
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−15.4999	−1.03795	−0.518974	−	0.854790i	$-0.673686\pi$
$223$	−15.4999	−1.03795	−0.518974	+	0.854790i	$0.673686\pi$
$224$	0	0
$225$	5.01207	0.334138
$226$	0	0
$227$	21.8170	1.44804	0.724021	−	0.689777i	$-0.242292\pi$
$227$	21.8170	1.44804	0.724021	+	0.689777i	$0.242292\pi$
$228$	0	0
$229$	4.15934	0.274857	0.137428	−	0.990512i	$-0.456116\pi$
$229$	4.15934	0.274857	0.137428	+	0.990512i	$0.456116\pi$
$230$	0	0
$231$	24.6624	1.62267
$232$	0	0
$233$	14.1371	0.926154	0.463077	−	0.886318i	$-0.346745\pi$
$233$	14.1371	0.926154	0.463077	+	0.886318i	$0.346745\pi$
$234$	0	0
$235$	41.1860	2.68668
$236$	0	0
$237$	14.6130	0.949214
$238$	0	0
$239$	−5.16741	−0.334252	−0.167126	−	0.985936i	$-0.553449\pi$
$239$	−5.16741	−0.334252	−0.167126	+	0.985936i	$0.553449\pi$
$240$	0	0
$241$	−23.8587	−1.53687	−0.768437	−	0.639925i	$-0.778965\pi$
$241$	−23.8587	−1.53687	−0.768437	+	0.639925i	$0.778965\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	28.0362	1.79117
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−6.44604	−0.408501
$250$	0	0
$251$	19.6285	1.23894	0.619471	−	0.785019i	$-0.287347\pi$
$251$	19.6285	1.23894	0.619471	+	0.785019i	$0.287347\pi$
$252$	0	0
$253$	−40.1133	−2.52190
$254$	0	0
$255$	0.261976	0.0164055
$256$	0	0
$257$	−6.75173	−0.421161	−0.210581	−	0.977577i	$-0.567535\pi$
$257$	−6.75173	−0.421161	−0.210581	+	0.977577i	$0.567535\pi$
$258$	0	0
$259$	−0.757809	−0.0470880
$260$	0	0
$261$	7.19023	0.445064
$262$	0	0
$263$	−1.92820	−0.118898	−0.0594488	−	0.998231i	$-0.518934\pi$
$263$	−1.92820	−0.118898	−0.0594488	+	0.998231i	$0.518934\pi$
$264$	0	0
$265$	5.89468	0.362107
$266$	0	0
$267$	11.3907	0.697102
$268$	0	0
$269$	14.6231	0.891587	0.445794	−	0.895136i	$-0.352922\pi$
$269$	14.6231	0.891587	0.445794	+	0.895136i	$0.352922\pi$
$270$	0	0
$271$	4.94455	0.300360	0.150180	−	0.988659i	$-0.452015\pi$
$271$	4.94455	0.300360	0.150180	+	0.988659i	$0.452015\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−31.0380	−1.87166
$276$	0	0
$277$	−10.6112	−0.637562	−0.318781	−	0.947828i	$-0.603274\pi$
$277$	−10.6112	−0.637562	−0.318781	+	0.947828i	$0.603274\pi$
$278$	0	0
$279$	4.21310	0.252232
$280$	0	0
$281$	−2.55001	−0.152121	−0.0760603	−	0.997103i	$-0.524234\pi$
$281$	−2.55001	−0.152121	−0.0760603	+	0.997103i	$0.524234\pi$
$282$	0	0
$283$	0.568008	0.0337646	0.0168823	−	0.999857i	$-0.494626\pi$
$283$	0.568008	0.0337646	0.0168823	+	0.999857i	$0.494626\pi$
$284$	0	0
$285$	8.88279	0.526171
$286$	0	0
$287$	10.1773	0.600749
$288$	0	0
$289$	−16.9931	−0.999597
$290$	0	0
$291$	16.3049	0.955810
$292$	0	0
$293$	0.898419	0.0524862	0.0262431	−	0.999656i	$-0.491646\pi$
$293$	0.898419	0.0524862	0.0262431	+	0.999656i	$0.491646\pi$
$294$	0	0
$295$	−15.5991	−0.908216
$296$	0	0
$297$	−6.19265	−0.359334
$298$	0	0
$299$	0	0
$300$	0	0
$301$	26.7952	1.54445
$302$	0	0
$303$	−2.84245	−0.163295
$304$	0	0
$305$	46.6386	2.67052
$306$	0	0
$307$	−8.53020	−0.486844	−0.243422	−	0.969920i	$-0.578270\pi$
$307$	−8.53020	−0.486844	−0.243422	+	0.969920i	$0.578270\pi$
$308$	0	0
$309$	−0.789223	−0.0448973
$310$	0	0
$311$	−12.2062	−0.692151	−0.346075	−	0.938207i	$-0.612486\pi$
$311$	−12.2062	−0.692151	−0.346075	+	0.938207i	$0.612486\pi$
$312$	0	0
$313$	−4.23781	−0.239535	−0.119768	−	0.992802i	$-0.538215\pi$
$313$	−4.23781	−0.239535	−0.119768	+	0.992802i	$0.538215\pi$
$314$	0	0
$315$	−12.6014	−0.710011
$316$	0	0
$317$	−18.7797	−1.05477	−0.527387	−	0.849625i	$-0.676828\pi$
$317$	−18.7797	−1.05477	−0.527387	+	0.849625i	$0.676828\pi$
$318$	0	0
$319$	−44.5266	−2.49301
$320$	0	0
$321$	6.15452	0.343512
$322$	0	0
$323$	0.232427	0.0129326
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−17.4999	−0.967748
$328$	0	0
$329$	−51.8377	−2.85791
$330$	0	0
$331$	34.9816	1.92276	0.961381	−	0.275220i	$-0.0887508\pi$
$331$	34.9816	1.92276	0.961381	+	0.275220i	$0.0887508\pi$
$332$	0	0
$333$	0.190284	0.0104275
$334$	0	0
$335$	19.6867	1.07560
$336$	0	0
$337$	−12.2540	−0.667518	−0.333759	−	0.942658i	$-0.608317\pi$
$337$	−12.2540	−0.667518	−0.333759	+	0.942658i	$0.608317\pi$
$338$	0	0
$339$	−4.32296	−0.234791
$340$	0	0
$341$	−26.0903	−1.41287
$342$	0	0
$343$	−7.40940	−0.400070
$344$	0	0
$345$	20.4962	1.10348
$346$	0	0
$347$	−19.9199	−1.06936	−0.534678	−	0.845056i	$-0.679567\pi$
$347$	−19.9199	−1.06936	−0.534678	+	0.845056i	$0.679567\pi$
$348$	0	0
$349$	−3.95305	−0.211602	−0.105801	−	0.994387i	$-0.533741\pi$
$349$	−3.95305	−0.211602	−0.105801	+	0.994387i	$0.533741\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	16.8290	0.895716	0.447858	−	0.894105i	$-0.352187\pi$
$353$	16.8290	0.895716	0.447858	+	0.894105i	$0.352187\pi$
$354$	0	0
$355$	−19.6117	−1.04088
$356$	0	0
$357$	−0.329729	−0.0174511
$358$	0	0
$359$	−20.2835	−1.07052	−0.535261	−	0.844686i	$-0.679787\pi$
$359$	−20.2835	−1.07052	−0.535261	+	0.844686i	$0.679787\pi$
$360$	0	0
$361$	−11.1191	−0.585217
$362$	0	0
$363$	27.3490	1.43545
$364$	0	0
$365$	43.7597	2.29048
$366$	0	0
$367$	−2.33366	−0.121816	−0.0609081	−	0.998143i	$-0.519400\pi$
$367$	−2.33366	−0.121816	−0.0609081	+	0.998143i	$0.519400\pi$
$368$	0	0
$369$	−2.55550	−0.133034
$370$	0	0
$371$	−7.41919	−0.385185
$372$	0	0
$373$	4.77414	0.247195	0.123598	−	0.992332i	$-0.460557\pi$
$373$	4.77414	0.247195	0.123598	+	0.992332i	$0.460557\pi$
$374$	0	0
$375$	0.0381934	0.00197230
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−2.83628	−0.145690	−0.0728450	−	0.997343i	$-0.523208\pi$
$379$	−2.83628	−0.145690	−0.0728450	+	0.997343i	$0.523208\pi$
$380$	0	0
$381$	−13.1138	−0.671842
$382$	0	0
$383$	21.3723	1.09207	0.546037	−	0.837761i	$-0.316136\pi$
$383$	21.3723	1.09207	0.546037	+	0.837761i	$0.316136\pi$
$384$	0	0
$385$	78.0364	3.97710
$386$	0	0
$387$	−6.72819	−0.342013
$388$	0	0
$389$	15.7652	0.799326	0.399663	−	0.916662i	$-0.369127\pi$
$389$	15.7652	0.799326	0.399663	+	0.916662i	$0.369127\pi$
$390$	0	0
$391$	0.536303	0.0271220
$392$	0	0
$393$	7.47175	0.376900
$394$	0	0
$395$	46.2381	2.32649
$396$	0	0
$397$	7.48617	0.375720	0.187860	−	0.982196i	$-0.439845\pi$
$397$	7.48617	0.375720	0.187860	+	0.982196i	$0.439845\pi$
$398$	0	0
$399$	−11.1801	−0.559705
$400$	0	0
$401$	30.7534	1.53575	0.767876	−	0.640598i	$-0.221314\pi$
$401$	30.7534	1.53575	0.767876	+	0.640598i	$0.221314\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	3.16419	0.157230
$406$	0	0
$407$	−1.17836	−0.0584092
$408$	0	0
$409$	−33.6493	−1.66385	−0.831924	−	0.554889i	$-0.812761\pi$
$409$	−33.6493	−1.66385	−0.831924	+	0.554889i	$0.812761\pi$
$410$	0	0
$411$	−4.35367	−0.214751
$412$	0	0
$413$	19.6334	0.966098
$414$	0	0
$415$	−20.3965	−1.00122
$416$	0	0
$417$	−13.0060	−0.636905
$418$	0	0
$419$	7.89083	0.385492	0.192746	−	0.981249i	$-0.438261\pi$
$419$	7.89083	0.385492	0.192746	+	0.981249i	$0.438261\pi$
$420$	0	0
$421$	19.8316	0.966534	0.483267	−	0.875473i	$-0.339450\pi$
$421$	19.8316	0.966534	0.483267	+	0.875473i	$0.339450\pi$
$422$	0	0
$423$	13.0163	0.632874
$424$	0	0
$425$	0.414969	0.0201290
$426$	0	0
$427$	−58.7005	−2.84072
$428$	0	0
$429$	0	0
$430$	0	0
$431$	26.7639	1.28917	0.644585	−	0.764533i	$-0.277030\pi$
$431$	26.7639	1.28917	0.644585	+	0.764533i	$0.277030\pi$
$432$	0	0
$433$	4.94694	0.237735	0.118867	−	0.992910i	$-0.462074\pi$
$433$	4.94694	0.237735	0.118867	+	0.992910i	$0.462074\pi$
$434$	0	0
$435$	22.7512	1.09084
$436$	0	0
$437$	18.1844	0.869877
$438$	0	0
$439$	−3.74064	−0.178531	−0.0892654	−	0.996008i	$-0.528452\pi$
$439$	−3.74064	−0.178531	−0.0892654	+	0.996008i	$0.528452\pi$
$440$	0	0
$441$	8.86048	0.421928
$442$	0	0
$443$	−31.5291	−1.49799	−0.748996	−	0.662575i	$-0.769464\pi$
$443$	−31.5291	−1.49799	−0.748996	+	0.662575i	$0.769464\pi$
$444$	0	0
$445$	36.0424	1.70857
$446$	0	0
$447$	−16.6182	−0.786015
$448$	0	0
$449$	17.3073	0.816782	0.408391	−	0.912807i	$-0.366090\pi$
$449$	17.3073	0.816782	0.408391	+	0.912807i	$0.366090\pi$
$450$	0	0
$451$	15.8253	0.745186
$452$	0	0
$453$	14.3250	0.673046
$454$	0	0
$455$	0	0
$456$	0	0
$457$	17.4384	0.815734	0.407867	−	0.913041i	$-0.366273\pi$
$457$	17.4384	0.815734	0.407867	+	0.913041i	$0.366273\pi$
$458$	0	0
$459$	0.0827940	0.00386449
$460$	0	0
$461$	−8.78961	−0.409373	−0.204687	−	0.978828i	$-0.565618\pi$
$461$	−8.78961	−0.409373	−0.204687	+	0.978828i	$0.565618\pi$
$462$	0	0
$463$	−4.90074	−0.227757	−0.113878	−	0.993495i	$-0.536327\pi$
$463$	−4.90074	−0.227757	−0.113878	+	0.993495i	$0.536327\pi$
$464$	0	0
$465$	13.3310	0.618212
$466$	0	0
$467$	22.3449	1.03400	0.517000	−	0.855985i	$-0.327049\pi$
$467$	22.3449	1.03400	0.517000	+	0.855985i	$0.327049\pi$
$468$	0	0
$469$	−24.7781	−1.14415
$470$	0	0
$471$	1.35614	0.0624875
$472$	0	0
$473$	41.6653	1.91577
$474$	0	0
$475$	14.0703	0.645591
$476$	0	0
$477$	1.86294	0.0852980
$478$	0	0
$479$	24.5728	1.12276	0.561379	−	0.827559i	$-0.310271\pi$
$479$	24.5728	1.12276	0.561379	+	0.827559i	$0.310271\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−25.7970	−1.17381
$484$	0	0
$485$	51.5917	2.34266
$486$	0	0
$487$	−3.28538	−0.148875	−0.0744374	−	0.997226i	$-0.523716\pi$
$487$	−3.28538	−0.148875	−0.0744374	+	0.997226i	$0.523716\pi$
$488$	0	0
$489$	−9.26015	−0.418759
$490$	0	0
$491$	17.3868	0.784657	0.392329	−	0.919825i	$-0.371670\pi$
$491$	17.3868	0.784657	0.392329	+	0.919825i	$0.371670\pi$
$492$	0	0
$493$	0.595308	0.0268113
$494$	0	0
$495$	−19.5947	−0.880717
$496$	0	0
$497$	24.6838	1.10722
$498$	0	0
$499$	−39.6459	−1.77479	−0.887397	−	0.461006i	$-0.847489\pi$
$499$	−39.6459	−1.77479	−0.887397	+	0.461006i	$0.847489\pi$
$500$	0	0
$501$	−23.6785	−1.05788
$502$	0	0
$503$	18.5478	0.827008	0.413504	−	0.910502i	$-0.364305\pi$
$503$	18.5478	0.827008	0.413504	+	0.910502i	$0.364305\pi$
$504$	0	0
$505$	−8.99405	−0.400230
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−0.333314	−0.0147739	−0.00738694	−	0.999973i	$-0.502351\pi$
$509$	−0.333314	−0.0147739	−0.00738694	+	0.999973i	$0.502351\pi$
$510$	0	0
$511$	−55.0770	−2.43646
$512$	0	0
$513$	2.80729	0.123945
$514$	0	0
$515$	−2.49725	−0.110042
$516$	0	0
$517$	−80.6055	−3.54503
$518$	0	0
$519$	16.3593	0.718093
$520$	0	0
$521$	−31.4940	−1.37978	−0.689890	−	0.723915i	$-0.742341\pi$
$521$	−31.4940	−1.37978	−0.689890	+	0.723915i	$0.742341\pi$
$522$	0	0
$523$	0.401134	0.0175404	0.00877019	−	0.999962i	$-0.497208\pi$
$523$	0.401134	0.0175404	0.00877019	+	0.999962i	$0.497208\pi$
$524$	0	0
$525$	−19.9607	−0.871155
$526$	0	0
$527$	0.348820	0.0151948
$528$	0	0
$529$	18.9588	0.824295
$530$	0	0
$531$	−4.92990	−0.213939
$532$	0	0
$533$	0	0
$534$	0	0
$535$	19.4740	0.841936
$536$	0	0
$537$	6.04357	0.260799
$538$	0	0
$539$	−54.8699	−2.36341
$540$	0	0
$541$	−16.0200	−0.688753	−0.344376	−	0.938832i	$-0.611910\pi$
$541$	−16.0200	−0.688753	−0.344376	+	0.938832i	$0.611910\pi$
$542$	0	0
$543$	−11.8062	−0.506652
$544$	0	0
$545$	−55.3730	−2.37192
$546$	0	0
$547$	5.28441	0.225945	0.112972	−	0.993598i	$-0.463963\pi$
$547$	5.28441	0.225945	0.112972	+	0.993598i	$0.463963\pi$
$548$	0	0
$549$	14.7395	0.629068
$550$	0	0
$551$	20.1851	0.859912
$552$	0	0
$553$	−58.1964	−2.47476
$554$	0	0
$555$	0.602093	0.0255574
$556$	0	0
$557$	19.2730	0.816625	0.408312	−	0.912842i	$-0.366117\pi$
$557$	19.2730	0.816625	0.408312	+	0.912842i	$0.366117\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−0.512715	−0.0216468
$562$	0	0
$563$	−5.15319	−0.217181	−0.108591	−	0.994087i	$-0.534634\pi$
$563$	−5.15319	−0.217181	−0.108591	+	0.994087i	$0.534634\pi$
$564$	0	0
$565$	−13.6787	−0.575465
$566$	0	0
$567$	−3.98252	−0.167250
$568$	0	0
$569$	5.80951	0.243547	0.121774	−	0.992558i	$-0.461142\pi$
$569$	5.80951	0.243547	0.121774	+	0.992558i	$0.461142\pi$
$570$	0	0
$571$	−36.1094	−1.51113	−0.755565	−	0.655074i	$-0.772637\pi$
$571$	−36.1094	−1.51113	−0.755565	+	0.655074i	$0.772637\pi$
$572$	0	0
$573$	−2.79423	−0.116731
$574$	0	0
$575$	32.4660	1.35393
$576$	0	0
$577$	−25.0463	−1.04269	−0.521345	−	0.853346i	$-0.674569\pi$
$577$	−25.0463	−1.04269	−0.521345	+	0.853346i	$0.674569\pi$
$578$	0	0
$579$	−11.7655	−0.488959
$580$	0	0
$581$	25.6715	1.06503
$582$	0	0
$583$	−11.5365	−0.477794
$584$	0	0
$585$	0	0
$586$	0	0
$587$	30.3713	1.25356	0.626778	−	0.779198i	$-0.284373\pi$
$587$	30.3713	1.25356	0.626778	+	0.779198i	$0.284373\pi$
$588$	0	0
$589$	11.8274	0.487340
$590$	0	0
$591$	7.21802	0.296910
$592$	0	0
$593$	33.7414	1.38559	0.692797	−	0.721133i	$-0.256378\pi$
$593$	33.7414	1.38559	0.692797	+	0.721133i	$0.256378\pi$
$594$	0	0
$595$	−1.04332	−0.0427721
$596$	0	0
$597$	−1.28987	−0.0527907
$598$	0	0
$599$	−0.438398	−0.0179124	−0.00895622	−	0.999960i	$-0.502851\pi$
$599$	−0.438398	−0.0179124	−0.00895622	+	0.999960i	$0.502851\pi$
$600$	0	0
$601$	−2.75023	−0.112184	−0.0560921	−	0.998426i	$-0.517864\pi$
$601$	−2.75023	−0.112184	−0.0560921	+	0.998426i	$0.517864\pi$
$602$	0	0
$603$	6.22172	0.253368
$604$	0	0
$605$	86.5372	3.51824
$606$	0	0
$607$	−7.80766	−0.316903	−0.158451	−	0.987367i	$-0.550650\pi$
$607$	−7.80766	−0.316903	−0.158451	+	0.987367i	$0.550650\pi$
$608$	0	0
$609$	−28.6352	−1.16036
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−33.3698	−1.34779	−0.673897	−	0.738825i	$-0.735381\pi$
$613$	−33.3698	−1.34779	−0.673897	+	0.738825i	$0.735381\pi$
$614$	0	0
$615$	−8.08608	−0.326062
$616$	0	0
$617$	37.1931	1.49734	0.748670	−	0.662943i	$-0.230693\pi$
$617$	37.1931	1.49734	0.748670	+	0.662943i	$0.230693\pi$
$618$	0	0
$619$	39.1414	1.57323	0.786613	−	0.617447i	$-0.211833\pi$
$619$	39.1414	1.57323	0.786613	+	0.617447i	$0.211833\pi$
$620$	0	0
$621$	6.47756	0.259936
$622$	0	0
$623$	−45.3639	−1.81746
$624$	0	0
$625$	−24.9395	−0.997580
$626$	0	0
$627$	−17.3846	−0.694273
$628$	0	0
$629$	0.0157543	0.000628167 0
$630$	0	0
$631$	30.7642	1.22470	0.612352	−	0.790585i	$-0.290223\pi$
$631$	30.7642	1.22470	0.612352	+	0.790585i	$0.290223\pi$
$632$	0	0
$633$	10.4565	0.415609
$634$	0	0
$635$	−41.4946	−1.64666
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−6.19803	−0.245190
$640$	0	0
$641$	23.6028	0.932253	0.466127	−	0.884718i	$-0.345649\pi$
$641$	23.6028	0.932253	0.466127	+	0.884718i	$0.345649\pi$
$642$	0	0
$643$	−24.5921	−0.969820	−0.484910	−	0.874564i	$-0.661148\pi$
$643$	−24.5921	−0.969820	−0.484910	+	0.874564i	$0.661148\pi$
$644$	0	0
$645$	−21.2892	−0.838263
$646$	0	0
$647$	16.6054	0.652825	0.326413	−	0.945227i	$-0.394160\pi$
$647$	16.6054	0.652825	0.326413	+	0.945227i	$0.394160\pi$
$648$	0	0
$649$	30.5292	1.19837
$650$	0	0
$651$	−16.7788	−0.657612
$652$	0	0
$653$	−23.4251	−0.916697	−0.458348	−	0.888773i	$-0.651559\pi$
$653$	−23.4251	−0.916697	−0.458348	+	0.888773i	$0.651559\pi$
$654$	0	0
$655$	23.6420	0.923770
$656$	0	0
$657$	13.8297	0.539547
$658$	0	0
$659$	9.93522	0.387021	0.193511	−	0.981098i	$-0.438013\pi$
$659$	9.93522	0.387021	0.193511	+	0.981098i	$0.438013\pi$
$660$	0	0
$661$	−1.03778	−0.0403650	−0.0201825	−	0.999796i	$-0.506425\pi$
$661$	−1.03778	−0.0403650	−0.0201825	+	0.999796i	$0.506425\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−35.3759	−1.37182
$666$	0	0
$667$	46.5751	1.80340
$668$	0	0
$669$	−15.4999	−0.599260
$670$	0	0
$671$	−91.2769	−3.52370
$672$	0	0
$673$	15.1608	0.584404	0.292202	−	0.956357i	$-0.405612\pi$
$673$	15.1608	0.584404	0.292202	+	0.956357i	$0.405612\pi$
$674$	0	0
$675$	5.01207	0.192915
$676$	0	0
$677$	−45.1709	−1.73606	−0.868030	−	0.496512i	$-0.834614\pi$
$677$	−45.1709	−1.73606	−0.868030	+	0.496512i	$0.834614\pi$
$678$	0	0
$679$	−64.9346	−2.49196
$680$	0	0
$681$	21.8170	0.836028
$682$	0	0
$683$	−1.60007	−0.0612250	−0.0306125	−	0.999531i	$-0.509746\pi$
$683$	−1.60007	−0.0612250	−0.0306125	+	0.999531i	$0.509746\pi$
$684$	0	0
$685$	−13.7758	−0.526347
$686$	0	0
$687$	4.15934	0.158689
$688$	0	0
$689$	0	0
$690$	0	0
$691$	32.6934	1.24372	0.621858	−	0.783130i	$-0.286378\pi$
$691$	32.6934	1.24372	0.621858	+	0.783130i	$0.286378\pi$
$692$	0	0
$693$	24.6624	0.936846
$694$	0	0
$695$	−41.1533	−1.56103
$696$	0	0
$697$	−0.211580	−0.00801417
$698$	0	0
$699$	14.1371	0.534715
$700$	0	0
$701$	−28.2377	−1.06652	−0.533262	−	0.845950i	$-0.679034\pi$
$701$	−28.2377	−1.06652	−0.533262	+	0.845950i	$0.679034\pi$
$702$	0	0
$703$	0.534181	0.0201470
$704$	0	0
$705$	41.1860	1.55115
$706$	0	0
$707$	11.3201	0.425737
$708$	0	0
$709$	48.6844	1.82838	0.914191	−	0.405283i	$-0.132827\pi$
$709$	48.6844	1.82838	0.914191	+	0.405283i	$0.132827\pi$
$710$	0	0
$711$	14.6130	0.548029
$712$	0	0
$713$	27.2906	1.02204
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−5.16741	−0.192980
$718$	0	0
$719$	−20.0864	−0.749097	−0.374549	−	0.927207i	$-0.622202\pi$
$719$	−20.0864	−0.749097	−0.374549	+	0.927207i	$0.622202\pi$
$720$	0	0
$721$	3.14310	0.117055
$722$	0	0
$723$	−23.8587	−0.887315
$724$	0	0
$725$	36.0379	1.33842
$726$	0	0
$727$	39.8799	1.47906	0.739531	−	0.673122i	$-0.235047\pi$
$727$	39.8799	1.47906	0.739531	+	0.673122i	$0.235047\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.557054	−0.0206034
$732$	0	0
$733$	−7.27839	−0.268834	−0.134417	−	0.990925i	$-0.542916\pi$
$733$	−7.27839	−0.268834	−0.134417	+	0.990925i	$0.542916\pi$
$734$	0	0
$735$	28.0362	1.03413
$736$	0	0
$737$	−38.5290	−1.41923
$738$	0	0
$739$	−17.2706	−0.635310	−0.317655	−	0.948206i	$-0.602895\pi$
$739$	−17.2706	−0.635310	−0.317655	+	0.948206i	$0.602895\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−35.3555	−1.29707	−0.648533	−	0.761186i	$-0.724617\pi$
$743$	−35.3555	−1.29707	−0.648533	+	0.761186i	$0.724617\pi$
$744$	0	0
$745$	−52.5831	−1.92650
$746$	0	0
$747$	−6.44604	−0.235848
$748$	0	0
$749$	−24.5105	−0.895594
$750$	0	0
$751$	−35.5229	−1.29625	−0.648125	−	0.761534i	$-0.724446\pi$
$751$	−35.5229	−1.29625	−0.648125	+	0.761534i	$0.724446\pi$
$752$	0	0
$753$	19.6285	0.715304
$754$	0	0
$755$	45.3269	1.64961
$756$	0	0
$757$	−40.2066	−1.46133	−0.730667	−	0.682734i	$-0.760791\pi$
$757$	−40.2066	−1.46133	−0.730667	+	0.682734i	$0.760791\pi$
$758$	0	0
$759$	−40.1133	−1.45602
$760$	0	0
$761$	27.2434	0.987572	0.493786	−	0.869583i	$-0.335613\pi$
$761$	27.2434	0.987572	0.493786	+	0.869583i	$0.335613\pi$
$762$	0	0
$763$	69.6938	2.52309
$764$	0	0
$765$	0.261976	0.00947175
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−20.3872	−0.735181	−0.367591	−	0.929988i	$-0.619817\pi$
$769$	−20.3872	−0.735181	−0.367591	+	0.929988i	$0.619817\pi$
$770$	0	0
$771$	−6.75173	−0.243157
$772$	0	0
$773$	36.7361	1.32131	0.660653	−	0.750691i	$-0.270279\pi$
$773$	36.7361	1.32131	0.660653	+	0.750691i	$0.270279\pi$
$774$	0	0
$775$	21.1164	0.758522
$776$	0	0
$777$	−0.757809	−0.0271862
$778$	0	0
$779$	−7.17403	−0.257036
$780$	0	0
$781$	38.3822	1.37342
$782$	0	0
$783$	7.19023	0.256958
$784$	0	0
$785$	4.29107	0.153155
$786$	0	0
$787$	55.6292	1.98297	0.991483	−	0.130237i	$-0.0415739\pi$
$787$	55.6292	1.98297	0.991483	+	0.130237i	$0.0415739\pi$
$788$	0	0
$789$	−1.92820	−0.0686456
$790$	0	0
$791$	17.2163	0.612141
$792$	0	0
$793$	0	0
$794$	0	0
$795$	5.89468	0.209063
$796$	0	0
$797$	10.0129	0.354675	0.177337	−	0.984150i	$-0.443252\pi$
$797$	10.0129	0.354675	0.177337	+	0.984150i	$0.443252\pi$
$798$	0	0
$799$	1.07767	0.0381253
$800$	0	0
$801$	11.3907	0.402472
$802$	0	0
$803$	−85.6424	−3.02225
$804$	0	0
$805$	−81.6266	−2.87696
$806$	0	0
$807$	14.6231	0.514758
$808$	0	0
$809$	−3.96404	−0.139368	−0.0696842	−	0.997569i	$-0.522199\pi$
$809$	−3.96404	−0.139368	−0.0696842	+	0.997569i	$0.522199\pi$
$810$	0	0
$811$	0.636536	0.0223518	0.0111759	−	0.999938i	$-0.496443\pi$
$811$	0.636536	0.0223518	0.0111759	+	0.999938i	$0.496443\pi$
$812$	0	0
$813$	4.94455	0.173413
$814$	0	0
$815$	−29.3008	−1.02636
$816$	0	0
$817$	−18.8880	−0.660807
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−41.9136	−1.46280	−0.731398	−	0.681951i	$-0.761132\pi$
$821$	−41.9136	−1.46280	−0.731398	+	0.681951i	$0.761132\pi$
$822$	0	0
$823$	−16.3072	−0.568433	−0.284217	−	0.958760i	$-0.591734\pi$
$823$	−16.3072	−0.568433	−0.284217	+	0.958760i	$0.591734\pi$
$824$	0	0
$825$	−31.0380	−1.08061
$826$	0	0
$827$	7.01398	0.243900	0.121950	−	0.992536i	$-0.461085\pi$
$827$	7.01398	0.243900	0.121950	+	0.992536i	$0.461085\pi$
$828$	0	0
$829$	−23.7360	−0.824384	−0.412192	−	0.911097i	$-0.635237\pi$
$829$	−23.7360	−0.824384	−0.412192	+	0.911097i	$0.635237\pi$
$830$	0	0
$831$	−10.6112	−0.368097
$832$	0	0
$833$	0.733594	0.0254175
$834$	0	0
$835$	−74.9230	−2.59282
$836$	0	0
$837$	4.21310	0.145626
$838$	0	0
$839$	−3.72151	−0.128481	−0.0642405	−	0.997934i	$-0.520462\pi$
$839$	−3.72151	−0.128481	−0.0642405	+	0.997934i	$0.520462\pi$
$840$	0	0
$841$	22.6994	0.782737
$842$	0	0
$843$	−2.55001	−0.0878269
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−108.918	−3.74246
$848$	0	0
$849$	0.568008	0.0194940
$850$	0	0
$851$	1.23257	0.0422521
$852$	0	0
$853$	−25.2289	−0.863821	−0.431910	−	0.901917i	$-0.642160\pi$
$853$	−25.2289	−0.863821	−0.431910	+	0.901917i	$0.642160\pi$
$854$	0	0
$855$	8.88279	0.303785
$856$	0	0
$857$	36.7216	1.25438	0.627192	−	0.778864i	$-0.284204\pi$
$857$	36.7216	1.25438	0.627192	+	0.778864i	$0.284204\pi$
$858$	0	0
$859$	42.1582	1.43842	0.719210	−	0.694793i	$-0.244504\pi$
$859$	42.1582	1.43842	0.719210	+	0.694793i	$0.244504\pi$
$860$	0	0
$861$	10.1773	0.346843
$862$	0	0
$863$	−29.1708	−0.992987	−0.496493	−	0.868041i	$-0.665379\pi$
$863$	−29.1708	−0.992987	−0.496493	+	0.868041i	$0.665379\pi$
$864$	0	0
$865$	51.7638	1.76002
$866$	0	0
$867$	−16.9931	−0.577117
$868$	0	0
$869$	−90.4930	−3.06977
$870$	0	0
$871$	0	0
$872$	0	0
$873$	16.3049	0.551837
$874$	0	0
$875$	−0.152106	−0.00514213
$876$	0	0
$877$	−10.1028	−0.341147	−0.170574	−	0.985345i	$-0.554562\pi$
$877$	−10.1028	−0.341147	−0.170574	+	0.985345i	$0.554562\pi$
$878$	0	0
$879$	0.898419	0.0303029
$880$	0	0
$881$	7.73774	0.260691	0.130345	−	0.991469i	$-0.458391\pi$
$881$	7.73774	0.260691	0.130345	+	0.991469i	$0.458391\pi$
$882$	0	0
$883$	−50.8006	−1.70958	−0.854789	−	0.518976i	$-0.826313\pi$
$883$	−50.8006	−1.70958	−0.854789	+	0.518976i	$0.826313\pi$
$884$	0	0
$885$	−15.5991	−0.524359
$886$	0	0
$887$	17.1414	0.575551	0.287776	−	0.957698i	$-0.407084\pi$
$887$	17.1414	0.575551	0.287776	+	0.957698i	$0.407084\pi$
$888$	0	0
$889$	52.2261	1.75161
$890$	0	0
$891$	−6.19265	−0.207462
$892$	0	0
$893$	36.5405	1.22278
$894$	0	0
$895$	19.1230	0.639210
$896$	0	0
$897$	0	0
$898$	0	0
$899$	30.2932	1.01033
$900$	0	0
$901$	0.154240	0.00513848
$902$	0	0
$903$	26.7952	0.891687
$904$	0	0
$905$	−37.3570	−1.24179
$906$	0	0
$907$	7.97808	0.264908	0.132454	−	0.991189i	$-0.457714\pi$
$907$	7.97808	0.264908	0.132454	+	0.991189i	$0.457714\pi$
$908$	0	0
$909$	−2.84245	−0.0942782
$910$	0	0
$911$	−0.545514	−0.0180737	−0.00903685	−	0.999959i	$-0.502877\pi$
$911$	−0.545514	−0.0180737	−0.00903685	+	0.999959i	$0.502877\pi$
$912$	0	0
$913$	39.9181	1.32110
$914$	0	0
$915$	46.6386	1.54183
$916$	0	0
$917$	−29.7564	−0.982644
$918$	0	0
$919$	28.0429	0.925050	0.462525	−	0.886606i	$-0.346943\pi$
$919$	28.0429	0.925050	0.462525	+	0.886606i	$0.346943\pi$
$920$	0	0
$921$	−8.53020	−0.281080
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0.953715	0.0313580
$926$	0	0
$927$	−0.789223	−0.0259215
$928$	0	0
$929$	6.51220	0.213658	0.106829	−	0.994277i	$-0.465930\pi$
$929$	6.51220	0.213658	0.106829	+	0.994277i	$0.465930\pi$
$930$	0	0
$931$	24.8739	0.815210
$932$	0	0
$933$	−12.2062	−0.399614
$934$	0	0
$935$	−1.62232	−0.0530557
$936$	0	0
$937$	26.3675	0.861389	0.430695	−	0.902498i	$-0.358269\pi$
$937$	26.3675	0.861389	0.430695	+	0.902498i	$0.358269\pi$
$938$	0	0
$939$	−4.23781	−0.138296
$940$	0	0
$941$	−8.14965	−0.265671	−0.132835	−	0.991138i	$-0.542408\pi$
$941$	−8.14965	−0.265671	−0.132835	+	0.991138i	$0.542408\pi$
$942$	0	0
$943$	−16.5534	−0.539053
$944$	0	0
$945$	−12.6014	−0.409925
$946$	0	0
$947$	−43.8774	−1.42583	−0.712913	−	0.701253i	$-0.752624\pi$
$947$	−43.8774	−1.42583	−0.712913	+	0.701253i	$0.752624\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−18.7797	−0.608974
$952$	0	0
$953$	−39.7316	−1.28703	−0.643517	−	0.765432i	$-0.722525\pi$
$953$	−39.7316	−1.28703	−0.643517	+	0.765432i	$0.722525\pi$
$954$	0	0
$955$	−8.84146	−0.286103
$956$	0	0
$957$	−44.5266	−1.43934
$958$	0	0
$959$	17.3386	0.559892
$960$	0	0
$961$	−13.2498	−0.427412
$962$	0	0
$963$	6.15452	0.198327
$964$	0	0
$965$	−37.2284	−1.19842
$966$	0	0
$967$	19.6413	0.631621	0.315810	−	0.948822i	$-0.397724\pi$
$967$	19.6413	0.631621	0.315810	+	0.948822i	$0.397724\pi$
$968$	0	0
$969$	0.232427	0.00746662
$970$	0	0
$971$	18.0293	0.578589	0.289294	−	0.957240i	$-0.406579\pi$
$971$	18.0293	0.578589	0.289294	+	0.957240i	$0.406579\pi$
$972$	0	0
$973$	51.7965	1.66052
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−26.8610	−0.859361	−0.429680	−	0.902981i	$-0.641374\pi$
$977$	−26.8610	−0.859361	−0.429680	+	0.902981i	$0.641374\pi$
$978$	0	0
$979$	−70.5389	−2.25443
$980$	0	0
$981$	−17.4999	−0.558729
$982$	0	0
$983$	12.0402	0.384022	0.192011	−	0.981393i	$-0.438499\pi$
$983$	12.0402	0.384022	0.192011	+	0.981393i	$0.438499\pi$
$984$	0	0
$985$	22.8392	0.727716
$986$	0	0
$987$	−51.8377	−1.65001
$988$	0	0
$989$	−43.5822	−1.38583
$990$	0	0
$991$	−8.79841	−0.279491	−0.139745	−	0.990187i	$-0.544628\pi$
$991$	−8.79841	−0.279491	−0.139745	+	0.990187i	$0.544628\pi$
$992$	0	0
$993$	34.9816	1.11011
$994$	0	0
$995$	−4.08138	−0.129388
$996$	0	0
$997$	−42.9102	−1.35898	−0.679489	−	0.733685i	$-0.737799\pi$
$997$	−42.9102	−1.35898	−0.679489	+	0.733685i	$0.737799\pi$
$998$	0	0
$999$	0.190284	0.00602031

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4056.2.a.bi.1.6	yes	6
4.3	odd	2		8112.2.a.cu.1.6		6
13.5	odd	4		4056.2.c.r.337.1		12
13.8	odd	4		4056.2.c.r.337.12		12
13.12	even	2		4056.2.a.bh.1.1	✓	6
52.51	odd	2		8112.2.a.ct.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4056.2.a.bh.1.1	✓	6	13.12	even	2
4056.2.a.bi.1.6	yes	6	1.1	even	1	trivial
4056.2.c.r.337.1		12	13.5	odd	4
4056.2.c.r.337.12		12	13.8	odd	4
8112.2.a.ct.1.1		6	52.51	odd	2
8112.2.a.cu.1.6		6	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4056.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.16419 q^{5} -3.98252 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.16419 q^{5} -3.98252 q^{7} +1.00000 q^{9} -6.19265 q^{11} +3.16419 q^{15} +0.0827940 q^{17} +2.80729 q^{19} -3.98252 q^{21} +6.47756 q^{23} +5.01207 q^{25} +1.00000 q^{27} +7.19023 q^{29} +4.21310 q^{31} -6.19265 q^{33} -12.6014 q^{35} +0.190284 q^{37} -2.55550 q^{41} -6.72819 q^{43} +3.16419 q^{45} +13.0163 q^{47} +8.86048 q^{49} +0.0827940 q^{51} +1.86294 q^{53} -19.5947 q^{55} +2.80729 q^{57} -4.92990 q^{59} +14.7395 q^{61} -3.98252 q^{63} +6.22172 q^{67} +6.47756 q^{69} -6.19803 q^{71} +13.8297 q^{73} +5.01207 q^{75} +24.6624 q^{77} +14.6130 q^{79} +1.00000 q^{81} -6.44604 q^{83} +0.261976 q^{85} +7.19023 q^{87} +11.3907 q^{89} +4.21310 q^{93} +8.88279 q^{95} +16.3049 q^{97} -6.19265 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} + q^{5} - 5 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 + q^5 - 5 * q^7 + 6 * q^9	\( 6 q + 6 q^{3} + q^{5} - 5 q^{7} + 6 q^{9} - 6 q^{11} + q^{15} + 9 q^{17} + 7 q^{19} - 5 q^{21} + 12 q^{23} + 9 q^{25} + 6 q^{27} + 7 q^{29} - 11 q^{31} - 6 q^{33} + 6 q^{35} + 6 q^{37} + 13 q^{41} + 15 q^{43} + q^{45} - 9 q^{47} + 13 q^{49} + 9 q^{51} + 22 q^{53} + 3 q^{55} + 7 q^{57} - 7 q^{59} + 25 q^{61} - 5 q^{63} + 5 q^{67} + 12 q^{69} + 8 q^{71} + 15 q^{73} + 9 q^{75} + 45 q^{77} + 14 q^{79} + 6 q^{81} - 13 q^{83} - 35 q^{85} + 7 q^{87} + 33 q^{89} - 11 q^{93} + 47 q^{95} + 50 q^{97} - 6 q^{99}+O(q^{100}) \) 6 * q + 6 * q^3 + q^5 - 5 * q^7 + 6 * q^9 - 6 * q^11 + q^15 + 9 * q^17 + 7 * q^19 - 5 * q^21 + 12 * q^23 + 9 * q^25 + 6 * q^27 + 7 * q^29 - 11 * q^31 - 6 * q^33 + 6 * q^35 + 6 * q^37 + 13 * q^41 + 15 * q^43 + q^45 - 9 * q^47 + 13 * q^49 + 9 * q^51 + 22 * q^53 + 3 * q^55 + 7 * q^57 - 7 * q^59 + 25 * q^61 - 5 * q^63 + 5 * q^67 + 12 * q^69 + 8 * q^71 + 15 * q^73 + 9 * q^75 + 45 * q^77 + 14 * q^79 + 6 * q^81 - 13 * q^83 - 35 * q^85 + 7 * q^87 + 33 * q^89 - 11 * q^93 + 47 * q^95 + 50 * q^97 - 6 * q^99

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