LMFDB - Embedded newform 8004.2.a.g.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, 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("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.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:	$63.9122617778$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 31 x^{10} + 80 x^{9} + 347 x^{8} - 697 x^{7} - 1714 x^{6} + 2146 x^{5} + 3304 x^{4} + \cdots + 16 $ x^12 - 3x^11 - 31x^10 + 80x^9 + 347x^8 - 697x^7 - 1714x^6 + 2146x^5 + 3304x^4 - 1156x^3 - 616x^2 + 136*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-3.54084$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.54084 q^{5} +2.31030 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.54084 q^{5} +2.31030 q^{7} +1.00000 q^{9} -0.0711814 q^{11} -0.620759 q^{13} -3.54084 q^{15} -4.97914 q^{17} -6.83275 q^{19} -2.31030 q^{21} +1.00000 q^{23} +7.53753 q^{25} -1.00000 q^{27} -1.00000 q^{29} -2.38311 q^{31} +0.0711814 q^{33} +8.18041 q^{35} -7.78652 q^{37} +0.620759 q^{39} -6.42713 q^{41} +1.04310 q^{43} +3.54084 q^{45} +5.51860 q^{47} -1.66249 q^{49} +4.97914 q^{51} -5.37715 q^{53} -0.252042 q^{55} +6.83275 q^{57} +0.604181 q^{59} -9.02854 q^{61} +2.31030 q^{63} -2.19801 q^{65} -12.1486 q^{67} -1.00000 q^{69} -3.52530 q^{71} -4.78770 q^{73} -7.53753 q^{75} -0.164451 q^{77} +1.72122 q^{79} +1.00000 q^{81} -12.0521 q^{83} -17.6303 q^{85} +1.00000 q^{87} -8.77465 q^{89} -1.43414 q^{91} +2.38311 q^{93} -24.1937 q^{95} +5.44485 q^{97} -0.0711814 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9	$ 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9} - 5 q^{11} - 6 q^{13} + 3 q^{15} - 7 q^{17} - 3 q^{19} - 4 q^{21} + 12 q^{23} + 11 q^{25} - 12 q^{27} - 12 q^{29} + 2 q^{31} + 5 q^{33} - 9 q^{35} - 20 q^{37} + 6 q^{39} - 3 q^{41} + 5 q^{43} - 3 q^{45} - 2 q^{49} + 7 q^{51} - 3 q^{53} + 19 q^{55} + 3 q^{57} - 20 q^{59} - 17 q^{61} + 4 q^{63} - 4 q^{65} - 9 q^{67} - 12 q^{69} + 7 q^{71} - 9 q^{73} - 11 q^{75} - 34 q^{77} + 14 q^{79} + 12 q^{81} + 5 q^{83} - 12 q^{85} + 12 q^{87} - 22 q^{89} - 3 q^{91} - 2 q^{93} - 27 q^{95} + 17 q^{97} - 5 q^{99}+O(q^{100}) $ 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9 - 5 * q^11 - 6 * q^13 + 3 * q^15 - 7 * q^17 - 3 * q^19 - 4 * q^21 + 12 * q^23 + 11 * q^25 - 12 * q^27 - 12 * q^29 + 2 * q^31 + 5 * q^33 - 9 * q^35 - 20 * q^37 + 6 * q^39 - 3 * q^41 + 5 * q^43 - 3 * q^45 - 2 * q^49 + 7 * q^51 - 3 * q^53 + 19 * q^55 + 3 * q^57 - 20 * q^59 - 17 * q^61 + 4 * q^63 - 4 * q^65 - 9 * q^67 - 12 * q^69 + 7 * q^71 - 9 * q^73 - 11 * q^75 - 34 * q^77 + 14 * q^79 + 12 * q^81 + 5 * q^83 - 12 * q^85 + 12 * q^87 - 22 * q^89 - 3 * q^91 - 2 * q^93 - 27 * q^95 + 17 * q^97 - 5 * 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
$5$	3.54084	1.58351	0.791755	−	0.610839i	$-0.209168\pi$
$5$	3.54084	1.58351	0.791755	+	0.610839i	$0.209168\pi$
$6$	0	0
$7$	2.31030	0.873213	0.436606	−	0.899653i	$-0.356180\pi$
$7$	2.31030	0.873213	0.436606	+	0.899653i	$0.356180\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.0711814	−0.0214620	−0.0107310	−	0.999942i	$-0.503416\pi$
$11$	−0.0711814	−0.0214620	−0.0107310	+	0.999942i	$0.503416\pi$
$12$	0	0
$13$	−0.620759	−0.172168	−0.0860838	−	0.996288i	$-0.527435\pi$
$13$	−0.620759	−0.172168	−0.0860838	+	0.996288i	$0.527435\pi$
$14$	0	0
$15$	−3.54084	−0.914240
$16$	0	0
$17$	−4.97914	−1.20762	−0.603810	−	0.797129i	$-0.706351\pi$
$17$	−4.97914	−1.20762	−0.603810	+	0.797129i	$0.706351\pi$
$18$	0	0
$19$	−6.83275	−1.56754	−0.783771	−	0.621051i	$-0.786706\pi$
$19$	−6.83275	−1.56754	−0.783771	+	0.621051i	$0.786706\pi$
$20$	0	0
$21$	−2.31030	−0.504150
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	7.53753	1.50751
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−2.38311	−0.428019	−0.214009	−	0.976832i	$-0.568652\pi$
$31$	−2.38311	−0.428019	−0.214009	+	0.976832i	$0.568652\pi$
$32$	0	0
$33$	0.0711814	0.0123911
$34$	0	0
$35$	8.18041	1.38274
$36$	0	0
$37$	−7.78652	−1.28010	−0.640048	−	0.768335i	$-0.721086\pi$
$37$	−7.78652	−1.28010	−0.640048	+	0.768335i	$0.721086\pi$
$38$	0	0
$39$	0.620759	0.0994010
$40$	0	0
$41$	−6.42713	−1.00375	−0.501874	−	0.864941i	$-0.667356\pi$
$41$	−6.42713	−1.00375	−0.501874	+	0.864941i	$0.667356\pi$
$42$	0	0
$43$	1.04310	0.159071	0.0795355	−	0.996832i	$-0.474656\pi$
$43$	1.04310	0.159071	0.0795355	+	0.996832i	$0.474656\pi$
$44$	0	0
$45$	3.54084	0.527837
$46$	0	0
$47$	5.51860	0.804970	0.402485	−	0.915427i	$-0.368147\pi$
$47$	5.51860	0.804970	0.402485	+	0.915427i	$0.368147\pi$
$48$	0	0
$49$	−1.66249	−0.237499
$50$	0	0
$51$	4.97914	0.697219
$52$	0	0
$53$	−5.37715	−0.738608	−0.369304	−	0.929309i	$-0.620404\pi$
$53$	−5.37715	−0.738608	−0.369304	+	0.929309i	$0.620404\pi$
$54$	0	0
$55$	−0.252042	−0.0339853
$56$	0	0
$57$	6.83275	0.905020
$58$	0	0
$59$	0.604181	0.0786577	0.0393289	−	0.999226i	$-0.487478\pi$
$59$	0.604181	0.0786577	0.0393289	+	0.999226i	$0.487478\pi$
$60$	0	0
$61$	−9.02854	−1.15599	−0.577993	−	0.816042i	$-0.696164\pi$
$61$	−9.02854	−1.15599	−0.577993	+	0.816042i	$0.696164\pi$
$62$	0	0
$63$	2.31030	0.291071
$64$	0	0
$65$	−2.19801	−0.272629
$66$	0	0
$67$	−12.1486	−1.48418	−0.742091	−	0.670299i	$-0.766166\pi$
$67$	−12.1486	−1.48418	−0.742091	+	0.670299i	$0.766166\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−3.52530	−0.418377	−0.209188	−	0.977875i	$-0.567082\pi$
$71$	−3.52530	−0.418377	−0.209188	+	0.977875i	$0.567082\pi$
$72$	0	0
$73$	−4.78770	−0.560358	−0.280179	−	0.959948i	$-0.590394\pi$
$73$	−4.78770	−0.560358	−0.280179	+	0.959948i	$0.590394\pi$
$74$	0	0
$75$	−7.53753	−0.870358
$76$	0	0
$77$	−0.164451	−0.0187409
$78$	0	0
$79$	1.72122	0.193652	0.0968262	−	0.995301i	$-0.469131\pi$
$79$	1.72122	0.193652	0.0968262	+	0.995301i	$0.469131\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.0521	−1.32289	−0.661443	−	0.749996i	$-0.730055\pi$
$83$	−12.0521	−1.32289	−0.661443	+	0.749996i	$0.730055\pi$
$84$	0	0
$85$	−17.6303	−1.91228
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	−8.77465	−0.930111	−0.465056	−	0.885281i	$-0.653966\pi$
$89$	−8.77465	−0.930111	−0.465056	+	0.885281i	$0.653966\pi$
$90$	0	0
$91$	−1.43414	−0.150339
$92$	0	0
$93$	2.38311	0.247117
$94$	0	0
$95$	−24.1937	−2.48222
$96$	0	0
$97$	5.44485	0.552841	0.276420	−	0.961037i	$-0.410852\pi$
$97$	5.44485	0.552841	0.276420	+	0.961037i	$0.410852\pi$
$98$	0	0
$99$	−0.0711814	−0.00715400
$100$	0	0
$101$	−10.2508	−1.01999	−0.509994	−	0.860178i	$-0.670353\pi$
$101$	−10.2508	−1.01999	−0.509994	+	0.860178i	$0.670353\pi$
$102$	0	0
$103$	4.52130	0.445497	0.222748	−	0.974876i	$-0.428497\pi$
$103$	4.52130	0.445497	0.222748	+	0.974876i	$0.428497\pi$
$104$	0	0
$105$	−8.18041	−0.798326
$106$	0	0
$107$	−12.7966	−1.23709	−0.618545	−	0.785750i	$-0.712277\pi$
$107$	−12.7966	−1.23709	−0.618545	+	0.785750i	$0.712277\pi$
$108$	0	0
$109$	1.35997	0.130261	0.0651305	−	0.997877i	$-0.479254\pi$
$109$	1.35997	0.130261	0.0651305	+	0.997877i	$0.479254\pi$
$110$	0	0
$111$	7.78652	0.739064
$112$	0	0
$113$	−5.37462	−0.505601	−0.252801	−	0.967518i	$-0.581352\pi$
$113$	−5.37462	−0.505601	−0.252801	+	0.967518i	$0.581352\pi$
$114$	0	0
$115$	3.54084	0.330185
$116$	0	0
$117$	−0.620759	−0.0573892
$118$	0	0
$119$	−11.5033	−1.05451
$120$	0	0
$121$	−10.9949	−0.999539
$122$	0	0
$123$	6.42713	0.579515
$124$	0	0
$125$	8.98496	0.803639
$126$	0	0
$127$	20.1570	1.78864	0.894320	−	0.447427i	$-0.147660\pi$
$127$	20.1570	1.78864	0.894320	+	0.447427i	$0.147660\pi$
$128$	0	0
$129$	−1.04310	−0.0918397
$130$	0	0
$131$	14.0047	1.22360	0.611799	−	0.791014i	$-0.290446\pi$
$131$	14.0047	1.22360	0.611799	+	0.791014i	$0.290446\pi$
$132$	0	0
$133$	−15.7857	−1.36880
$134$	0	0
$135$	−3.54084	−0.304747
$136$	0	0
$137$	6.14660	0.525140	0.262570	−	0.964913i	$-0.415430\pi$
$137$	6.14660	0.525140	0.262570	+	0.964913i	$0.415430\pi$
$138$	0	0
$139$	14.3036	1.21321	0.606607	−	0.795002i	$-0.292530\pi$
$139$	14.3036	1.21321	0.606607	+	0.795002i	$0.292530\pi$
$140$	0	0
$141$	−5.51860	−0.464750
$142$	0	0
$143$	0.0441865	0.00369506
$144$	0	0
$145$	−3.54084	−0.294050
$146$	0	0
$147$	1.66249	0.137120
$148$	0	0
$149$	20.3437	1.66662	0.833311	−	0.552805i	$-0.186442\pi$
$149$	20.3437	1.66662	0.833311	+	0.552805i	$0.186442\pi$
$150$	0	0
$151$	−2.73246	−0.222365	−0.111182	−	0.993800i	$-0.535464\pi$
$151$	−2.73246	−0.222365	−0.111182	+	0.993800i	$0.535464\pi$
$152$	0	0
$153$	−4.97914	−0.402540
$154$	0	0
$155$	−8.43819	−0.677772
$156$	0	0
$157$	−21.3377	−1.70293	−0.851465	−	0.524411i	$-0.824286\pi$
$157$	−21.3377	−1.70293	−0.851465	+	0.524411i	$0.824286\pi$
$158$	0	0
$159$	5.37715	0.426435
$160$	0	0
$161$	2.31030	0.182077
$162$	0	0
$163$	11.6539	0.912804	0.456402	−	0.889774i	$-0.349138\pi$
$163$	11.6539	0.912804	0.456402	+	0.889774i	$0.349138\pi$
$164$	0	0
$165$	0.252042	0.0196214
$166$	0	0
$167$	−1.01179	−0.0782944	−0.0391472	−	0.999233i	$-0.512464\pi$
$167$	−1.01179	−0.0782944	−0.0391472	+	0.999233i	$0.512464\pi$
$168$	0	0
$169$	−12.6147	−0.970358
$170$	0	0
$171$	−6.83275	−0.522514
$172$	0	0
$173$	−1.34864	−0.102535	−0.0512677	−	0.998685i	$-0.516326\pi$
$173$	−1.34864	−0.102535	−0.0512677	+	0.998685i	$0.516326\pi$
$174$	0	0
$175$	17.4140	1.31637
$176$	0	0
$177$	−0.604181	−0.0454131
$178$	0	0
$179$	−11.9175	−0.890753	−0.445376	−	0.895343i	$-0.646930\pi$
$179$	−11.9175	−0.890753	−0.445376	+	0.895343i	$0.646930\pi$
$180$	0	0
$181$	14.2013	1.05557	0.527786	−	0.849377i	$-0.323022\pi$
$181$	14.2013	1.05557	0.527786	+	0.849377i	$0.323022\pi$
$182$	0	0
$183$	9.02854	0.667409
$184$	0	0
$185$	−27.5708	−2.02705
$186$	0	0
$187$	0.354422	0.0259179
$188$	0	0
$189$	−2.31030	−0.168050
$190$	0	0
$191$	13.8182	0.999851	0.499926	−	0.866068i	$-0.333361\pi$
$191$	13.8182	0.999851	0.499926	+	0.866068i	$0.333361\pi$
$192$	0	0
$193$	−7.49114	−0.539224	−0.269612	−	0.962969i	$-0.586895\pi$
$193$	−7.49114	−0.539224	−0.269612	+	0.962969i	$0.586895\pi$
$194$	0	0
$195$	2.19801	0.157402
$196$	0	0
$197$	12.1766	0.867550	0.433775	−	0.901021i	$-0.357181\pi$
$197$	12.1766	0.867550	0.433775	+	0.901021i	$0.357181\pi$
$198$	0	0
$199$	23.0031	1.63065	0.815323	−	0.579007i	$-0.196559\pi$
$199$	23.0031	1.63065	0.815323	+	0.579007i	$0.196559\pi$
$200$	0	0
$201$	12.1486	0.856893
$202$	0	0
$203$	−2.31030	−0.162152
$204$	0	0
$205$	−22.7574	−1.58945
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	0.486365	0.0336426
$210$	0	0
$211$	−7.53452	−0.518698	−0.259349	−	0.965784i	$-0.583508\pi$
$211$	−7.53452	−0.518698	−0.259349	+	0.965784i	$0.583508\pi$
$212$	0	0
$213$	3.52530	0.241550
$214$	0	0
$215$	3.69344	0.251891
$216$	0	0
$217$	−5.50570	−0.373751
$218$	0	0
$219$	4.78770	0.323523
$220$	0	0
$221$	3.09085	0.207913
$222$	0	0
$223$	−27.4689	−1.83945	−0.919726	−	0.392561i	$-0.871589\pi$
$223$	−27.4689	−1.83945	−0.919726	+	0.392561i	$0.871589\pi$
$224$	0	0
$225$	7.53753	0.502502
$226$	0	0
$227$	15.2865	1.01460	0.507299	−	0.861770i	$-0.330644\pi$
$227$	15.2865	1.01460	0.507299	+	0.861770i	$0.330644\pi$
$228$	0	0
$229$	−10.2081	−0.674570	−0.337285	−	0.941403i	$-0.609509\pi$
$229$	−10.2081	−0.674570	−0.337285	+	0.941403i	$0.609509\pi$
$230$	0	0
$231$	0.164451	0.0108201
$232$	0	0
$233$	12.6697	0.830021	0.415010	−	0.909817i	$-0.363778\pi$
$233$	12.6697	0.830021	0.415010	+	0.909817i	$0.363778\pi$
$234$	0	0
$235$	19.5404	1.27468
$236$	0	0
$237$	−1.72122	−0.111805
$238$	0	0
$239$	17.0568	1.10331	0.551657	−	0.834071i	$-0.313996\pi$
$239$	17.0568	1.10331	0.551657	+	0.834071i	$0.313996\pi$
$240$	0	0
$241$	25.6164	1.65010	0.825050	−	0.565060i	$-0.191147\pi$
$241$	25.6164	1.65010	0.825050	+	0.565060i	$0.191147\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−5.88662	−0.376082
$246$	0	0
$247$	4.24149	0.269880
$248$	0	0
$249$	12.0521	0.763768
$250$	0	0
$251$	12.9643	0.818298	0.409149	−	0.912468i	$-0.365826\pi$
$251$	12.9643	0.818298	0.409149	+	0.912468i	$0.365826\pi$
$252$	0	0
$253$	−0.0711814	−0.00447514
$254$	0	0
$255$	17.6303	1.10405
$256$	0	0
$257$	−7.55820	−0.471467	−0.235734	−	0.971818i	$-0.575749\pi$
$257$	−7.55820	−0.471467	−0.235734	+	0.971818i	$0.575749\pi$
$258$	0	0
$259$	−17.9892	−1.11780
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	4.86890	0.300229	0.150115	−	0.988669i	$-0.452036\pi$
$263$	4.86890	0.300229	0.150115	+	0.988669i	$0.452036\pi$
$264$	0	0
$265$	−19.0396	−1.16959
$266$	0	0
$267$	8.77465	0.537000
$268$	0	0
$269$	1.51039	0.0920898	0.0460449	−	0.998939i	$-0.485338\pi$
$269$	1.51039	0.0920898	0.0460449	+	0.998939i	$0.485338\pi$
$270$	0	0
$271$	10.4038	0.631986	0.315993	−	0.948762i	$-0.397662\pi$
$271$	10.4038	0.631986	0.315993	+	0.948762i	$0.397662\pi$
$272$	0	0
$273$	1.43414	0.0867982
$274$	0	0
$275$	−0.536532	−0.0323541
$276$	0	0
$277$	−6.22112	−0.373791	−0.186896	−	0.982380i	$-0.559843\pi$
$277$	−6.22112	−0.373791	−0.186896	+	0.982380i	$0.559843\pi$
$278$	0	0
$279$	−2.38311	−0.142673
$280$	0	0
$281$	14.1239	0.842562	0.421281	−	0.906930i	$-0.361581\pi$
$281$	14.1239	0.842562	0.421281	+	0.906930i	$0.361581\pi$
$282$	0	0
$283$	−14.4023	−0.856127	−0.428064	−	0.903749i	$-0.640804\pi$
$283$	−14.4023	−0.856127	−0.428064	+	0.903749i	$0.640804\pi$
$284$	0	0
$285$	24.1937	1.43311
$286$	0	0
$287$	−14.8486	−0.876486
$288$	0	0
$289$	7.79184	0.458344
$290$	0	0
$291$	−5.44485	−0.319183
$292$	0	0
$293$	10.4437	0.610127	0.305063	−	0.952332i	$-0.401322\pi$
$293$	10.4437	0.610127	0.305063	+	0.952332i	$0.401322\pi$
$294$	0	0
$295$	2.13931	0.124555
$296$	0	0
$297$	0.0711814	0.00413036
$298$	0	0
$299$	−0.620759	−0.0358994
$300$	0	0
$301$	2.40988	0.138903
$302$	0	0
$303$	10.2508	0.588891
$304$	0	0
$305$	−31.9686	−1.83052
$306$	0	0
$307$	7.71773	0.440474	0.220237	−	0.975446i	$-0.429317\pi$
$307$	7.71773	0.440474	0.220237	+	0.975446i	$0.429317\pi$
$308$	0	0
$309$	−4.52130	−0.257208
$310$	0	0
$311$	21.7132	1.23124	0.615621	−	0.788042i	$-0.288905\pi$
$311$	21.7132	1.23124	0.615621	+	0.788042i	$0.288905\pi$
$312$	0	0
$313$	20.6727	1.16849	0.584244	−	0.811578i	$-0.301391\pi$
$313$	20.6727	1.16849	0.584244	+	0.811578i	$0.301391\pi$
$314$	0	0
$315$	8.18041	0.460914
$316$	0	0
$317$	−29.2949	−1.64537	−0.822683	−	0.568501i	$-0.807524\pi$
$317$	−29.2949	−1.64537	−0.822683	+	0.568501i	$0.807524\pi$
$318$	0	0
$319$	0.0711814	0.00398539
$320$	0	0
$321$	12.7966	0.714234
$322$	0	0
$323$	34.0212	1.89299
$324$	0	0
$325$	−4.67899	−0.259543
$326$	0	0
$327$	−1.35997	−0.0752062
$328$	0	0
$329$	12.7496	0.702910
$330$	0	0
$331$	−15.9585	−0.877156	−0.438578	−	0.898693i	$-0.644518\pi$
$331$	−15.9585	−0.877156	−0.438578	+	0.898693i	$0.644518\pi$
$332$	0	0
$333$	−7.78652	−0.426699
$334$	0	0
$335$	−43.0160	−2.35022
$336$	0	0
$337$	15.3538	0.836373	0.418186	−	0.908361i	$-0.362666\pi$
$337$	15.3538	0.836373	0.418186	+	0.908361i	$0.362666\pi$
$338$	0	0
$339$	5.37462	0.291909
$340$	0	0
$341$	0.169633	0.00918613
$342$	0	0
$343$	−20.0130	−1.08060
$344$	0	0
$345$	−3.54084	−0.190632
$346$	0	0
$347$	−7.09962	−0.381128	−0.190564	−	0.981675i	$-0.561032\pi$
$347$	−7.09962	−0.381128	−0.190564	+	0.981675i	$0.561032\pi$
$348$	0	0
$349$	−6.79974	−0.363982	−0.181991	−	0.983300i	$-0.558254\pi$
$349$	−6.79974	−0.363982	−0.181991	+	0.983300i	$0.558254\pi$
$350$	0	0
$351$	0.620759	0.0331337
$352$	0	0
$353$	14.8482	0.790288	0.395144	−	0.918619i	$-0.370695\pi$
$353$	14.8482	0.790288	0.395144	+	0.918619i	$0.370695\pi$
$354$	0	0
$355$	−12.4825	−0.662504
$356$	0	0
$357$	11.5033	0.608821
$358$	0	0
$359$	−31.6435	−1.67008	−0.835040	−	0.550190i	$-0.814555\pi$
$359$	−31.6435	−1.67008	−0.835040	+	0.550190i	$0.814555\pi$
$360$	0	0
$361$	27.6865	1.45718
$362$	0	0
$363$	10.9949	0.577084
$364$	0	0
$365$	−16.9525	−0.887332
$366$	0	0
$367$	10.9055	0.569261	0.284631	−	0.958637i	$-0.408129\pi$
$367$	10.9055	0.569261	0.284631	+	0.958637i	$0.408129\pi$
$368$	0	0
$369$	−6.42713	−0.334583
$370$	0	0
$371$	−12.4228	−0.644962
$372$	0	0
$373$	22.4072	1.16020	0.580101	−	0.814545i	$-0.303013\pi$
$373$	22.4072	1.16020	0.580101	+	0.814545i	$0.303013\pi$
$374$	0	0
$375$	−8.98496	−0.463981
$376$	0	0
$377$	0.620759	0.0319707
$378$	0	0
$379$	22.0565	1.13297	0.566483	−	0.824074i	$-0.308304\pi$
$379$	22.0565	1.13297	0.566483	+	0.824074i	$0.308304\pi$
$380$	0	0
$381$	−20.1570	−1.03267
$382$	0	0
$383$	−14.8911	−0.760898	−0.380449	−	0.924802i	$-0.624231\pi$
$383$	−14.8911	−0.760898	−0.380449	+	0.924802i	$0.624231\pi$
$384$	0	0
$385$	−0.582293	−0.0296764
$386$	0	0
$387$	1.04310	0.0530237
$388$	0	0
$389$	8.34025	0.422867	0.211434	−	0.977392i	$-0.432187\pi$
$389$	8.34025	0.422867	0.211434	+	0.977392i	$0.432187\pi$
$390$	0	0
$391$	−4.97914	−0.251806
$392$	0	0
$393$	−14.0047	−0.706444
$394$	0	0
$395$	6.09456	0.306651
$396$	0	0
$397$	18.1769	0.912272	0.456136	−	0.889910i	$-0.349233\pi$
$397$	18.1769	0.912272	0.456136	+	0.889910i	$0.349233\pi$
$398$	0	0
$399$	15.7857	0.790275
$400$	0	0
$401$	−31.2426	−1.56018	−0.780090	−	0.625668i	$-0.784827\pi$
$401$	−31.2426	−1.56018	−0.780090	+	0.625668i	$0.784827\pi$
$402$	0	0
$403$	1.47933	0.0736909
$404$	0	0
$405$	3.54084	0.175946
$406$	0	0
$407$	0.554256	0.0274734
$408$	0	0
$409$	−7.88288	−0.389783	−0.194892	−	0.980825i	$-0.562436\pi$
$409$	−7.88288	−0.389783	−0.194892	+	0.980825i	$0.562436\pi$
$410$	0	0
$411$	−6.14660	−0.303190
$412$	0	0
$413$	1.39584	0.0686849
$414$	0	0
$415$	−42.6744	−2.09480
$416$	0	0
$417$	−14.3036	−0.700449
$418$	0	0
$419$	26.0481	1.27253	0.636266	−	0.771470i	$-0.280478\pi$
$419$	26.0481	1.27253	0.636266	+	0.771470i	$0.280478\pi$
$420$	0	0
$421$	−16.0576	−0.782597	−0.391299	−	0.920264i	$-0.627974\pi$
$421$	−16.0576	−0.782597	−0.391299	+	0.920264i	$0.627974\pi$
$422$	0	0
$423$	5.51860	0.268323
$424$	0	0
$425$	−37.5304	−1.82049
$426$	0	0
$427$	−20.8587	−1.00942
$428$	0	0
$429$	−0.0441865	−0.00213334
$430$	0	0
$431$	−1.85883	−0.0895368	−0.0447684	−	0.998997i	$-0.514255\pi$
$431$	−1.85883	−0.0895368	−0.0447684	+	0.998997i	$0.514255\pi$
$432$	0	0
$433$	40.4890	1.94578	0.972890	−	0.231270i	$-0.0742880\pi$
$433$	40.4890	1.94578	0.972890	+	0.231270i	$0.0742880\pi$
$434$	0	0
$435$	3.54084	0.169770
$436$	0	0
$437$	−6.83275	−0.326855
$438$	0	0
$439$	−12.6232	−0.602471	−0.301235	−	0.953550i	$-0.597399\pi$
$439$	−12.6232	−0.602471	−0.301235	+	0.953550i	$0.597399\pi$
$440$	0	0
$441$	−1.66249	−0.0791664
$442$	0	0
$443$	−1.67057	−0.0793712	−0.0396856	−	0.999212i	$-0.512636\pi$
$443$	−1.67057	−0.0793712	−0.0396856	+	0.999212i	$0.512636\pi$
$444$	0	0
$445$	−31.0696	−1.47284
$446$	0	0
$447$	−20.3437	−0.962225
$448$	0	0
$449$	−34.0388	−1.60639	−0.803196	−	0.595715i	$-0.796869\pi$
$449$	−34.0388	−1.60639	−0.803196	+	0.595715i	$0.796869\pi$
$450$	0	0
$451$	0.457492	0.0215425
$452$	0	0
$453$	2.73246	0.128382
$454$	0	0
$455$	−5.07806	−0.238063
$456$	0	0
$457$	−22.0105	−1.02961	−0.514805	−	0.857308i	$-0.672136\pi$
$457$	−22.0105	−1.02961	−0.514805	+	0.857308i	$0.672136\pi$
$458$	0	0
$459$	4.97914	0.232406
$460$	0	0
$461$	−34.2152	−1.59356	−0.796779	−	0.604270i	$-0.793465\pi$
$461$	−34.2152	−1.59356	−0.796779	+	0.604270i	$0.793465\pi$
$462$	0	0
$463$	−19.5768	−0.909810	−0.454905	−	0.890540i	$-0.650327\pi$
$463$	−19.5768	−0.909810	−0.454905	+	0.890540i	$0.650327\pi$
$464$	0	0
$465$	8.43819	0.391312
$466$	0	0
$467$	−4.68553	−0.216820	−0.108410	−	0.994106i	$-0.534576\pi$
$467$	−4.68553	−0.216820	−0.108410	+	0.994106i	$0.534576\pi$
$468$	0	0
$469$	−28.0669	−1.29601
$470$	0	0
$471$	21.3377	0.983187
$472$	0	0
$473$	−0.0742492	−0.00341398
$474$	0	0
$475$	−51.5020	−2.36308
$476$	0	0
$477$	−5.37715	−0.246203
$478$	0	0
$479$	−9.30781	−0.425285	−0.212642	−	0.977130i	$-0.568207\pi$
$479$	−9.30781	−0.425285	−0.212642	+	0.977130i	$0.568207\pi$
$480$	0	0
$481$	4.83355	0.220391
$482$	0	0
$483$	−2.31030	−0.105122
$484$	0	0
$485$	19.2793	0.875429
$486$	0	0
$487$	25.9057	1.17390	0.586950	−	0.809623i	$-0.300329\pi$
$487$	25.9057	1.17390	0.586950	+	0.809623i	$0.300329\pi$
$488$	0	0
$489$	−11.6539	−0.527007
$490$	0	0
$491$	−25.0263	−1.12942	−0.564710	−	0.825290i	$-0.691012\pi$
$491$	−25.0263	−1.12942	−0.564710	+	0.825290i	$0.691012\pi$
$492$	0	0
$493$	4.97914	0.224249
$494$	0	0
$495$	−0.252042	−0.0113284
$496$	0	0
$497$	−8.14453	−0.365332
$498$	0	0
$499$	12.6605	0.566761	0.283381	−	0.959008i	$-0.408544\pi$
$499$	12.6605	0.566761	0.283381	+	0.959008i	$0.408544\pi$
$500$	0	0
$501$	1.01179	0.0452033
$502$	0	0
$503$	19.6887	0.877876	0.438938	−	0.898517i	$-0.355355\pi$
$503$	19.6887	0.877876	0.438938	+	0.898517i	$0.355355\pi$
$504$	0	0
$505$	−36.2963	−1.61516
$506$	0	0
$507$	12.6147	0.560237
$508$	0	0
$509$	13.9225	0.617106	0.308553	−	0.951207i	$-0.400155\pi$
$509$	13.9225	0.617106	0.308553	+	0.951207i	$0.400155\pi$
$510$	0	0
$511$	−11.0610	−0.489311
$512$	0	0
$513$	6.83275	0.301673
$514$	0	0
$515$	16.0092	0.705449
$516$	0	0
$517$	−0.392821	−0.0172763
$518$	0	0
$519$	1.34864	0.0591988
$520$	0	0
$521$	−14.1358	−0.619301	−0.309651	−	0.950850i	$-0.600212\pi$
$521$	−14.1358	−0.619301	−0.309651	+	0.950850i	$0.600212\pi$
$522$	0	0
$523$	1.66533	0.0728197	0.0364099	−	0.999337i	$-0.488408\pi$
$523$	1.66533	0.0728197	0.0364099	+	0.999337i	$0.488408\pi$
$524$	0	0
$525$	−17.4140	−0.760008
$526$	0	0
$527$	11.8658	0.516883
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0.604181	0.0262192
$532$	0	0
$533$	3.98970	0.172813
$534$	0	0
$535$	−45.3105	−1.95894
$536$	0	0
$537$	11.9175	0.514276
$538$	0	0
$539$	0.118339	0.00509721
$540$	0	0
$541$	−3.43339	−0.147613	−0.0738065	−	0.997273i	$-0.523515\pi$
$541$	−3.43339	−0.147613	−0.0738065	+	0.997273i	$0.523515\pi$
$542$	0	0
$543$	−14.2013	−0.609435
$544$	0	0
$545$	4.81541	0.206270
$546$	0	0
$547$	0.855283	0.0365693	0.0182846	−	0.999833i	$-0.494179\pi$
$547$	0.855283	0.0365693	0.0182846	+	0.999833i	$0.494179\pi$
$548$	0	0
$549$	−9.02854	−0.385329
$550$	0	0
$551$	6.83275	0.291085
$552$	0	0
$553$	3.97654	0.169100
$554$	0	0
$555$	27.5708	1.17032
$556$	0	0
$557$	−6.10647	−0.258739	−0.129370	−	0.991596i	$-0.541295\pi$
$557$	−6.10647	−0.258739	−0.129370	+	0.991596i	$0.541295\pi$
$558$	0	0
$559$	−0.647513	−0.0273869
$560$	0	0
$561$	−0.354422	−0.0149637
$562$	0	0
$563$	−12.8657	−0.542224	−0.271112	−	0.962548i	$-0.587391\pi$
$563$	−12.8657	−0.542224	−0.271112	+	0.962548i	$0.587391\pi$
$564$	0	0
$565$	−19.0306	−0.800625
$566$	0	0
$567$	2.31030	0.0970237
$568$	0	0
$569$	−2.30747	−0.0967342	−0.0483671	−	0.998830i	$-0.515402\pi$
$569$	−2.30747	−0.0967342	−0.0483671	+	0.998830i	$0.515402\pi$
$570$	0	0
$571$	36.7084	1.53620	0.768100	−	0.640330i	$-0.221203\pi$
$571$	36.7084	1.53620	0.768100	+	0.640330i	$0.221203\pi$
$572$	0	0
$573$	−13.8182	−0.577264
$574$	0	0
$575$	7.53753	0.314337
$576$	0	0
$577$	−8.21186	−0.341864	−0.170932	−	0.985283i	$-0.554678\pi$
$577$	−8.21186	−0.341864	−0.170932	+	0.985283i	$0.554678\pi$
$578$	0	0
$579$	7.49114	0.311321
$580$	0	0
$581$	−27.8439	−1.15516
$582$	0	0
$583$	0.382753	0.0158520
$584$	0	0
$585$	−2.19801	−0.0908764
$586$	0	0
$587$	27.7611	1.14582	0.572911	−	0.819617i	$-0.305814\pi$
$587$	27.7611	1.14582	0.572911	+	0.819617i	$0.305814\pi$
$588$	0	0
$589$	16.2832	0.670937
$590$	0	0
$591$	−12.1766	−0.500880
$592$	0	0
$593$	−4.40184	−0.180762	−0.0903809	−	0.995907i	$-0.528808\pi$
$593$	−4.40184	−0.180762	−0.0903809	+	0.995907i	$0.528808\pi$
$594$	0	0
$595$	−40.7314	−1.66983
$596$	0	0
$597$	−23.0031	−0.941454
$598$	0	0
$599$	−38.1021	−1.55681	−0.778404	−	0.627763i	$-0.783971\pi$
$599$	−38.1021	−1.55681	−0.778404	+	0.627763i	$0.783971\pi$
$600$	0	0
$601$	8.67479	0.353852	0.176926	−	0.984224i	$-0.443385\pi$
$601$	8.67479	0.353852	0.176926	+	0.984224i	$0.443385\pi$
$602$	0	0
$603$	−12.1486	−0.494727
$604$	0	0
$605$	−38.9313	−1.58278
$606$	0	0
$607$	25.5908	1.03870	0.519349	−	0.854562i	$-0.326174\pi$
$607$	25.5908	1.03870	0.519349	+	0.854562i	$0.326174\pi$
$608$	0	0
$609$	2.31030	0.0936183
$610$	0	0
$611$	−3.42572	−0.138590
$612$	0	0
$613$	15.1226	0.610795	0.305398	−	0.952225i	$-0.401211\pi$
$613$	15.1226	0.610795	0.305398	+	0.952225i	$0.401211\pi$
$614$	0	0
$615$	22.7574	0.917667
$616$	0	0
$617$	−16.7383	−0.673859	−0.336930	−	0.941530i	$-0.609388\pi$
$617$	−16.7383	−0.673859	−0.336930	+	0.941530i	$0.609388\pi$
$618$	0	0
$619$	31.7185	1.27487	0.637436	−	0.770503i	$-0.279995\pi$
$619$	31.7185	1.27487	0.637436	+	0.770503i	$0.279995\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−20.2721	−0.812185
$624$	0	0
$625$	−5.87334	−0.234934
$626$	0	0
$627$	−0.486365	−0.0194235
$628$	0	0
$629$	38.7702	1.54587
$630$	0	0
$631$	14.1918	0.564965	0.282482	−	0.959272i	$-0.408842\pi$
$631$	14.1918	0.564965	0.282482	+	0.959272i	$0.408842\pi$
$632$	0	0
$633$	7.53452	0.299470
$634$	0	0
$635$	71.3725	2.83233
$636$	0	0
$637$	1.03201	0.0408896
$638$	0	0
$639$	−3.52530	−0.139459
$640$	0	0
$641$	−13.3742	−0.528247	−0.264124	−	0.964489i	$-0.585083\pi$
$641$	−13.3742	−0.528247	−0.264124	+	0.964489i	$0.585083\pi$
$642$	0	0
$643$	4.83486	0.190668	0.0953341	−	0.995445i	$-0.469608\pi$
$643$	4.83486	0.190668	0.0953341	+	0.995445i	$0.469608\pi$
$644$	0	0
$645$	−3.69344	−0.145429
$646$	0	0
$647$	−10.6970	−0.420542	−0.210271	−	0.977643i	$-0.567435\pi$
$647$	−10.6970	−0.420542	−0.210271	+	0.977643i	$0.567435\pi$
$648$	0	0
$649$	−0.0430065	−0.00168815
$650$	0	0
$651$	5.50570	0.215785
$652$	0	0
$653$	−23.0895	−0.903563	−0.451781	−	0.892129i	$-0.649211\pi$
$653$	−23.0895	−0.903563	−0.451781	+	0.892129i	$0.649211\pi$
$654$	0	0
$655$	49.5884	1.93758
$656$	0	0
$657$	−4.78770	−0.186786
$658$	0	0
$659$	−18.9876	−0.739653	−0.369826	−	0.929101i	$-0.620583\pi$
$659$	−18.9876	−0.739653	−0.369826	+	0.929101i	$0.620583\pi$
$660$	0	0
$661$	25.3945	0.987730	0.493865	−	0.869539i	$-0.335584\pi$
$661$	25.3945	0.987730	0.493865	+	0.869539i	$0.335584\pi$
$662$	0	0
$663$	−3.09085	−0.120039
$664$	0	0
$665$	−55.8947	−2.16750
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	27.4689	1.06201
$670$	0	0
$671$	0.642664	0.0248098
$672$	0	0
$673$	−12.3188	−0.474857	−0.237428	−	0.971405i	$-0.576304\pi$
$673$	−12.3188	−0.474857	−0.237428	+	0.971405i	$0.576304\pi$
$674$	0	0
$675$	−7.53753	−0.290119
$676$	0	0
$677$	−26.0942	−1.00288	−0.501440	−	0.865193i	$-0.667196\pi$
$677$	−26.0942	−1.00288	−0.501440	+	0.865193i	$0.667196\pi$
$678$	0	0
$679$	12.5793	0.482748
$680$	0	0
$681$	−15.2865	−0.585779
$682$	0	0
$683$	−10.5226	−0.402635	−0.201318	−	0.979526i	$-0.564522\pi$
$683$	−10.5226	−0.402635	−0.201318	+	0.979526i	$0.564522\pi$
$684$	0	0
$685$	21.7641	0.831564
$686$	0	0
$687$	10.2081	0.389463
$688$	0	0
$689$	3.33791	0.127164
$690$	0	0
$691$	20.7754	0.790332	0.395166	−	0.918610i	$-0.370687\pi$
$691$	20.7754	0.790332	0.395166	+	0.918610i	$0.370687\pi$
$692$	0	0
$693$	−0.164451	−0.00624697
$694$	0	0
$695$	50.6466	1.92114
$696$	0	0
$697$	32.0016	1.21215
$698$	0	0
$699$	−12.6697	−0.479213
$700$	0	0
$701$	−4.61954	−0.174478	−0.0872389	−	0.996187i	$-0.527804\pi$
$701$	−4.61954	−0.174478	−0.0872389	+	0.996187i	$0.527804\pi$
$702$	0	0
$703$	53.2034	2.00660
$704$	0	0
$705$	−19.5404	−0.735936
$706$	0	0
$707$	−23.6824	−0.890667
$708$	0	0
$709$	−38.3197	−1.43913	−0.719564	−	0.694426i	$-0.755658\pi$
$709$	−38.3197	−1.43913	−0.719564	+	0.694426i	$0.755658\pi$
$710$	0	0
$711$	1.72122	0.0645508
$712$	0	0
$713$	−2.38311	−0.0892480
$714$	0	0
$715$	0.156457	0.00585117
$716$	0	0
$717$	−17.0568	−0.636999
$718$	0	0
$719$	−39.9640	−1.49040	−0.745202	−	0.666838i	$-0.767647\pi$
$719$	−39.9640	−1.49040	−0.745202	+	0.666838i	$0.767647\pi$
$720$	0	0
$721$	10.4456	0.389013
$722$	0	0
$723$	−25.6164	−0.952686
$724$	0	0
$725$	−7.53753	−0.279937
$726$	0	0
$727$	11.1402	0.413165	0.206583	−	0.978429i	$-0.433766\pi$
$727$	11.1402	0.413165	0.206583	+	0.978429i	$0.433766\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−5.19373	−0.192097
$732$	0	0
$733$	−30.9033	−1.14144	−0.570719	−	0.821145i	$-0.693335\pi$
$733$	−30.9033	−1.14144	−0.570719	+	0.821145i	$0.693335\pi$
$734$	0	0
$735$	5.88662	0.217131
$736$	0	0
$737$	0.864751	0.0318535
$738$	0	0
$739$	−5.38902	−0.198238	−0.0991191	−	0.995076i	$-0.531602\pi$
$739$	−5.38902	−0.198238	−0.0991191	+	0.995076i	$0.531602\pi$
$740$	0	0
$741$	−4.24149	−0.155815
$742$	0	0
$743$	4.30318	0.157868	0.0789342	−	0.996880i	$-0.474848\pi$
$743$	4.30318	0.157868	0.0789342	+	0.996880i	$0.474848\pi$
$744$	0	0
$745$	72.0338	2.63911
$746$	0	0
$747$	−12.0521	−0.440962
$748$	0	0
$749$	−29.5639	−1.08024
$750$	0	0
$751$	−43.2393	−1.57783	−0.788913	−	0.614505i	$-0.789356\pi$
$751$	−43.2393	−1.57783	−0.788913	+	0.614505i	$0.789356\pi$
$752$	0	0
$753$	−12.9643	−0.472444
$754$	0	0
$755$	−9.67520	−0.352117
$756$	0	0
$757$	−38.0765	−1.38391	−0.691957	−	0.721939i	$-0.743251\pi$
$757$	−38.0765	−1.38391	−0.691957	+	0.721939i	$0.743251\pi$
$758$	0	0
$759$	0.0711814	0.00258372
$760$	0	0
$761$	28.1001	1.01863	0.509314	−	0.860581i	$-0.329899\pi$
$761$	28.1001	1.01863	0.509314	+	0.860581i	$0.329899\pi$
$762$	0	0
$763$	3.14193	0.113746
$764$	0	0
$765$	−17.6303	−0.637426
$766$	0	0
$767$	−0.375051	−0.0135423
$768$	0	0
$769$	27.2065	0.981091	0.490546	−	0.871416i	$-0.336798\pi$
$769$	27.2065	0.981091	0.490546	+	0.871416i	$0.336798\pi$
$770$	0	0
$771$	7.55820	0.272202
$772$	0	0
$773$	36.2723	1.30462	0.652312	−	0.757951i	$-0.273799\pi$
$773$	36.2723	1.30462	0.652312	+	0.757951i	$0.273799\pi$
$774$	0	0
$775$	−17.9627	−0.645240
$776$	0	0
$777$	17.9892	0.645360
$778$	0	0
$779$	43.9150	1.57342
$780$	0	0
$781$	0.250936	0.00897920
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	−75.5532	−2.69661
$786$	0	0
$787$	7.29110	0.259900	0.129950	−	0.991521i	$-0.458518\pi$
$787$	7.29110	0.259900	0.129950	+	0.991521i	$0.458518\pi$
$788$	0	0
$789$	−4.86890	−0.173338
$790$	0	0
$791$	−12.4170	−0.441498
$792$	0	0
$793$	5.60455	0.199023
$794$	0	0
$795$	19.0396	0.675265
$796$	0	0
$797$	4.32204	0.153095	0.0765473	−	0.997066i	$-0.475610\pi$
$797$	4.32204	0.153095	0.0765473	+	0.997066i	$0.475610\pi$
$798$	0	0
$799$	−27.4779	−0.972097
$800$	0	0
$801$	−8.77465	−0.310037
$802$	0	0
$803$	0.340795	0.0120264
$804$	0	0
$805$	8.18041	0.288322
$806$	0	0
$807$	−1.51039	−0.0531681
$808$	0	0
$809$	6.45744	0.227032	0.113516	−	0.993536i	$-0.463789\pi$
$809$	6.45744	0.227032	0.113516	+	0.993536i	$0.463789\pi$
$810$	0	0
$811$	−22.0673	−0.774888	−0.387444	−	0.921893i	$-0.626642\pi$
$811$	−22.0673	−0.774888	−0.387444	+	0.921893i	$0.626642\pi$
$812$	0	0
$813$	−10.4038	−0.364877
$814$	0	0
$815$	41.2645	1.44543
$816$	0	0
$817$	−7.12723	−0.249350
$818$	0	0
$819$	−1.43414	−0.0501130
$820$	0	0
$821$	−18.3813	−0.641513	−0.320756	−	0.947162i	$-0.603937\pi$
$821$	−18.3813	−0.641513	−0.320756	+	0.947162i	$0.603937\pi$
$822$	0	0
$823$	23.1189	0.805874	0.402937	−	0.915228i	$-0.367989\pi$
$823$	23.1189	0.805874	0.402937	+	0.915228i	$0.367989\pi$
$824$	0	0
$825$	0.536532	0.0186796
$826$	0	0
$827$	27.4701	0.955229	0.477614	−	0.878570i	$-0.341502\pi$
$827$	27.4701	0.955229	0.477614	+	0.878570i	$0.341502\pi$
$828$	0	0
$829$	−16.9519	−0.588763	−0.294382	−	0.955688i	$-0.595114\pi$
$829$	−16.9519	−0.588763	−0.294382	+	0.955688i	$0.595114\pi$
$830$	0	0
$831$	6.22112	0.215808
$832$	0	0
$833$	8.27779	0.286808
$834$	0	0
$835$	−3.58257	−0.123980
$836$	0	0
$837$	2.38311	0.0823722
$838$	0	0
$839$	11.7048	0.404095	0.202048	−	0.979376i	$-0.435240\pi$
$839$	11.7048	0.404095	0.202048	+	0.979376i	$0.435240\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−14.1239	−0.486454
$844$	0	0
$845$	−44.6664	−1.53657
$846$	0	0
$847$	−25.4016	−0.872811
$848$	0	0
$849$	14.4023	0.494285
$850$	0	0
$851$	−7.78652	−0.266919
$852$	0	0
$853$	−5.09938	−0.174599	−0.0872997	−	0.996182i	$-0.527824\pi$
$853$	−5.09938	−0.174599	−0.0872997	+	0.996182i	$0.527824\pi$
$854$	0	0
$855$	−24.1937	−0.827406
$856$	0	0
$857$	−6.81602	−0.232831	−0.116415	−	0.993201i	$-0.537140\pi$
$857$	−6.81602	−0.232831	−0.116415	+	0.993201i	$0.537140\pi$
$858$	0	0
$859$	−22.6219	−0.771850	−0.385925	−	0.922530i	$-0.626118\pi$
$859$	−22.6219	−0.771850	−0.385925	+	0.922530i	$0.626118\pi$
$860$	0	0
$861$	14.8486	0.506040
$862$	0	0
$863$	−47.7217	−1.62446	−0.812232	−	0.583334i	$-0.801748\pi$
$863$	−47.7217	−1.62446	−0.812232	+	0.583334i	$0.801748\pi$
$864$	0	0
$865$	−4.77532	−0.162366
$866$	0	0
$867$	−7.79184	−0.264625
$868$	0	0
$869$	−0.122519	−0.00415617
$870$	0	0
$871$	7.54132	0.255528
$872$	0	0
$873$	5.44485	0.184280
$874$	0	0
$875$	20.7580	0.701748
$876$	0	0
$877$	−17.9010	−0.604475	−0.302237	−	0.953233i	$-0.597734\pi$
$877$	−17.9010	−0.604475	−0.302237	+	0.953233i	$0.597734\pi$
$878$	0	0
$879$	−10.4437	−0.352257
$880$	0	0
$881$	−21.9553	−0.739693	−0.369846	−	0.929093i	$-0.620590\pi$
$881$	−21.9553	−0.739693	−0.369846	+	0.929093i	$0.620590\pi$
$882$	0	0
$883$	37.8079	1.27234	0.636169	−	0.771550i	$-0.280518\pi$
$883$	37.8079	1.27234	0.636169	+	0.771550i	$0.280518\pi$
$884$	0	0
$885$	−2.13931	−0.0719120
$886$	0	0
$887$	17.2550	0.579366	0.289683	−	0.957123i	$-0.406450\pi$
$887$	17.2550	0.579366	0.289683	+	0.957123i	$0.406450\pi$
$888$	0	0
$889$	46.5687	1.56186
$890$	0	0
$891$	−0.0711814	−0.00238467
$892$	0	0
$893$	−37.7072	−1.26182
$894$	0	0
$895$	−42.1978	−1.41052
$896$	0	0
$897$	0.620759	0.0207265
$898$	0	0
$899$	2.38311	0.0794811
$900$	0	0
$901$	26.7736	0.891957
$902$	0	0
$903$	−2.40988	−0.0801956
$904$	0	0
$905$	50.2844	1.67151
$906$	0	0
$907$	−20.1323	−0.668481	−0.334240	−	0.942488i	$-0.608480\pi$
$907$	−20.1323	−0.668481	−0.334240	+	0.942488i	$0.608480\pi$
$908$	0	0
$909$	−10.2508	−0.339996
$910$	0	0
$911$	16.6963	0.553174	0.276587	−	0.960989i	$-0.410797\pi$
$911$	16.6963	0.553174	0.276587	+	0.960989i	$0.410797\pi$
$912$	0	0
$913$	0.857882	0.0283918
$914$	0	0
$915$	31.9686	1.05685
$916$	0	0
$917$	32.3551	1.06846
$918$	0	0
$919$	18.5764	0.612778	0.306389	−	0.951906i	$-0.400879\pi$
$919$	18.5764	0.612778	0.306389	+	0.951906i	$0.400879\pi$
$920$	0	0
$921$	−7.71773	−0.254308
$922$	0	0
$923$	2.18836	0.0720309
$924$	0	0
$925$	−58.6911	−1.92975
$926$	0	0
$927$	4.52130	0.148499
$928$	0	0
$929$	−1.42951	−0.0469008	−0.0234504	−	0.999725i	$-0.507465\pi$
$929$	−1.42951	−0.0469008	−0.0234504	+	0.999725i	$0.507465\pi$
$930$	0	0
$931$	11.3594	0.372290
$932$	0	0
$933$	−21.7132	−0.710858
$934$	0	0
$935$	1.25495	0.0410413
$936$	0	0
$937$	1.77221	0.0578955	0.0289478	−	0.999581i	$-0.490784\pi$
$937$	1.77221	0.0578955	0.0289478	+	0.999581i	$0.490784\pi$
$938$	0	0
$939$	−20.6727	−0.674627
$940$	0	0
$941$	−52.1196	−1.69905	−0.849526	−	0.527548i	$-0.823112\pi$
$941$	−52.1196	−1.69905	−0.849526	+	0.527548i	$0.823112\pi$
$942$	0	0
$943$	−6.42713	−0.209296
$944$	0	0
$945$	−8.18041	−0.266109
$946$	0	0
$947$	−41.9737	−1.36396	−0.681981	−	0.731369i	$-0.738882\pi$
$947$	−41.9737	−1.36396	−0.681981	+	0.731369i	$0.738882\pi$
$948$	0	0
$949$	2.97201	0.0964754
$950$	0	0
$951$	29.2949	0.949952
$952$	0	0
$953$	16.9920	0.550423	0.275212	−	0.961384i	$-0.411252\pi$
$953$	16.9920	0.550423	0.275212	+	0.961384i	$0.411252\pi$
$954$	0	0
$955$	48.9281	1.58327
$956$	0	0
$957$	−0.0711814	−0.00230097
$958$	0	0
$959$	14.2005	0.458559
$960$	0	0
$961$	−25.3208	−0.816800
$962$	0	0
$963$	−12.7966	−0.412363
$964$	0	0
$965$	−26.5249	−0.853867
$966$	0	0
$967$	14.0030	0.450306	0.225153	−	0.974323i	$-0.427712\pi$
$967$	14.0030	0.450306	0.225153	+	0.974323i	$0.427712\pi$
$968$	0	0
$969$	−34.0212	−1.09292
$970$	0	0
$971$	−10.6948	−0.343212	−0.171606	−	0.985166i	$-0.554896\pi$
$971$	−10.6948	−0.343212	−0.171606	+	0.985166i	$0.554896\pi$
$972$	0	0
$973$	33.0456	1.05939
$974$	0	0
$975$	4.67899	0.149847
$976$	0	0
$977$	−35.3413	−1.13067	−0.565334	−	0.824862i	$-0.691253\pi$
$977$	−35.3413	−1.13067	−0.565334	+	0.824862i	$0.691253\pi$
$978$	0	0
$979$	0.624592	0.0199621
$980$	0	0
$981$	1.35997	0.0434203
$982$	0	0
$983$	−46.6747	−1.48869	−0.744346	−	0.667794i	$-0.767239\pi$
$983$	−46.6747	−1.48869	−0.744346	+	0.667794i	$0.767239\pi$
$984$	0	0
$985$	43.1155	1.37377
$986$	0	0
$987$	−12.7496	−0.405825
$988$	0	0
$989$	1.04310	0.0331686
$990$	0	0
$991$	11.8635	0.376857	0.188428	−	0.982087i	$-0.439661\pi$
$991$	11.8635	0.376857	0.188428	+	0.982087i	$0.439661\pi$
$992$	0	0
$993$	15.9585	0.506426
$994$	0	0
$995$	81.4502	2.58214
$996$	0	0
$997$	−14.9795	−0.474404	−0.237202	−	0.971460i	$-0.576230\pi$
$997$	−14.9795	−0.474404	−0.237202	+	0.971460i	$0.576230\pi$
$998$	0	0
$999$	7.78652	0.246355

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.g.1.12	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.g.1.12	✓	12	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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.54084 q^{5} +2.31030 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.54084 q^{5} +2.31030 q^{7} +1.00000 q^{9} -0.0711814 q^{11} -0.620759 q^{13} -3.54084 q^{15} -4.97914 q^{17} -6.83275 q^{19} -2.31030 q^{21} +1.00000 q^{23} +7.53753 q^{25} -1.00000 q^{27} -1.00000 q^{29} -2.38311 q^{31} +0.0711814 q^{33} +8.18041 q^{35} -7.78652 q^{37} +0.620759 q^{39} -6.42713 q^{41} +1.04310 q^{43} +3.54084 q^{45} +5.51860 q^{47} -1.66249 q^{49} +4.97914 q^{51} -5.37715 q^{53} -0.252042 q^{55} +6.83275 q^{57} +0.604181 q^{59} -9.02854 q^{61} +2.31030 q^{63} -2.19801 q^{65} -12.1486 q^{67} -1.00000 q^{69} -3.52530 q^{71} -4.78770 q^{73} -7.53753 q^{75} -0.164451 q^{77} +1.72122 q^{79} +1.00000 q^{81} -12.0521 q^{83} -17.6303 q^{85} +1.00000 q^{87} -8.77465 q^{89} -1.43414 q^{91} +2.38311 q^{93} -24.1937 q^{95} +5.44485 q^{97} -0.0711814 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9	\( 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9} - 5 q^{11} - 6 q^{13} + 3 q^{15} - 7 q^{17} - 3 q^{19} - 4 q^{21} + 12 q^{23} + 11 q^{25} - 12 q^{27} - 12 q^{29} + 2 q^{31} + 5 q^{33} - 9 q^{35} - 20 q^{37} + 6 q^{39} - 3 q^{41} + 5 q^{43} - 3 q^{45} - 2 q^{49} + 7 q^{51} - 3 q^{53} + 19 q^{55} + 3 q^{57} - 20 q^{59} - 17 q^{61} + 4 q^{63} - 4 q^{65} - 9 q^{67} - 12 q^{69} + 7 q^{71} - 9 q^{73} - 11 q^{75} - 34 q^{77} + 14 q^{79} + 12 q^{81} + 5 q^{83} - 12 q^{85} + 12 q^{87} - 22 q^{89} - 3 q^{91} - 2 q^{93} - 27 q^{95} + 17 q^{97} - 5 q^{99}+O(q^{100}) \) 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9 - 5 * q^11 - 6 * q^13 + 3 * q^15 - 7 * q^17 - 3 * q^19 - 4 * q^21 + 12 * q^23 + 11 * q^25 - 12 * q^27 - 12 * q^29 + 2 * q^31 + 5 * q^33 - 9 * q^35 - 20 * q^37 + 6 * q^39 - 3 * q^41 + 5 * q^43 - 3 * q^45 - 2 * q^49 + 7 * q^51 - 3 * q^53 + 19 * q^55 + 3 * q^57 - 20 * q^59 - 17 * q^61 + 4 * q^63 - 4 * q^65 - 9 * q^67 - 12 * q^69 + 7 * q^71 - 9 * q^73 - 11 * q^75 - 34 * q^77 + 14 * q^79 + 12 * q^81 + 5 * q^83 - 12 * q^85 + 12 * q^87 - 22 * q^89 - 3 * q^91 - 2 * q^93 - 27 * q^95 + 17 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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