LMFDB - Embedded newform 2001.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2001,2,Mod(1,2001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2001, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2001 = 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2001.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:	$15.9780654445$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	2001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.00000 q^{4} -2.44949 q^{5} +0.449490 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.00000 q^{4} -2.44949 q^{5} +0.449490 q^{7} +1.00000 q^{9} -2.00000 q^{12} +1.00000 q^{13} -2.44949 q^{15} +4.00000 q^{16} +3.44949 q^{17} -1.00000 q^{19} +4.89898 q^{20} +0.449490 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -0.898979 q^{28} +1.00000 q^{29} -5.55051 q^{31} -1.10102 q^{35} -2.00000 q^{36} +0.101021 q^{37} +1.00000 q^{39} +5.34847 q^{41} -9.89898 q^{43} -2.44949 q^{45} -7.55051 q^{47} +4.00000 q^{48} -6.79796 q^{49} +3.44949 q^{51} -2.00000 q^{52} -2.00000 q^{53} -1.00000 q^{57} -5.44949 q^{59} +4.89898 q^{60} -12.8990 q^{61} +0.449490 q^{63} -8.00000 q^{64} -2.44949 q^{65} +0.898979 q^{67} -6.89898 q^{68} -1.00000 q^{69} +2.34847 q^{71} +10.2474 q^{73} +1.00000 q^{75} +2.00000 q^{76} -3.89898 q^{79} -9.79796 q^{80} +1.00000 q^{81} -12.4495 q^{83} -0.898979 q^{84} -8.44949 q^{85} +1.00000 q^{87} +0.550510 q^{89} +0.449490 q^{91} +2.00000 q^{92} -5.55051 q^{93} +2.44949 q^{95} +7.10102 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 4 q^{4} - 4 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 4 * q^4 - 4 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 4 q^{4} - 4 q^{7} + 2 q^{9} - 4 q^{12} + 2 q^{13} + 8 q^{16} + 2 q^{17} - 2 q^{19} - 4 q^{21} - 2 q^{23} + 2 q^{25} + 2 q^{27} + 8 q^{28} + 2 q^{29} - 16 q^{31} - 12 q^{35} - 4 q^{36} + 10 q^{37} + 2 q^{39} - 4 q^{41} - 10 q^{43} - 20 q^{47} + 8 q^{48} + 6 q^{49} + 2 q^{51} - 4 q^{52} - 4 q^{53} - 2 q^{57} - 6 q^{59} - 16 q^{61} - 4 q^{63} - 16 q^{64} - 8 q^{67} - 4 q^{68} - 2 q^{69} - 10 q^{71} - 4 q^{73} + 2 q^{75} + 4 q^{76} + 2 q^{79} + 2 q^{81} - 20 q^{83} + 8 q^{84} - 12 q^{85} + 2 q^{87} + 6 q^{89} - 4 q^{91} + 4 q^{92} - 16 q^{93} + 24 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 - 4 * q^4 - 4 * q^7 + 2 * q^9 - 4 * q^12 + 2 * q^13 + 8 * q^16 + 2 * q^17 - 2 * q^19 - 4 * q^21 - 2 * q^23 + 2 * q^25 + 2 * q^27 + 8 * q^28 + 2 * q^29 - 16 * q^31 - 12 * q^35 - 4 * q^36 + 10 * q^37 + 2 * q^39 - 4 * q^41 - 10 * q^43 - 20 * q^47 + 8 * q^48 + 6 * q^49 + 2 * q^51 - 4 * q^52 - 4 * q^53 - 2 * q^57 - 6 * q^59 - 16 * q^61 - 4 * q^63 - 16 * q^64 - 8 * q^67 - 4 * q^68 - 2 * q^69 - 10 * q^71 - 4 * q^73 + 2 * q^75 + 4 * q^76 + 2 * q^79 + 2 * q^81 - 20 * q^83 + 8 * q^84 - 12 * q^85 + 2 * q^87 + 6 * q^89 - 4 * q^91 + 4 * q^92 - 16 * q^93 + 24 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−2.00000	−1.00000
$5$	−2.44949	−1.09545	−0.547723	−	0.836660i	$-0.684505\pi$
$5$	−2.44949	−1.09545	−0.547723	+	0.836660i	$0.684505\pi$
$6$	0	0
$7$	0.449490	0.169891	0.0849456	−	0.996386i	$-0.472928\pi$
$7$	0.449490	0.169891	0.0849456	+	0.996386i	$0.472928\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−2.00000	−0.577350
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	−2.44949	−0.632456
$16$	4.00000	1.00000
$17$	3.44949	0.836624	0.418312	−	0.908303i	$-0.362622\pi$
$17$	3.44949	0.836624	0.418312	+	0.908303i	$0.362622\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	4.89898	1.09545
$21$	0.449490	0.0980867
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	−0.898979	−0.169891
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−5.55051	−0.996901	−0.498451	−	0.866918i	$-0.666097\pi$
$31$	−5.55051	−0.996901	−0.498451	+	0.866918i	$0.666097\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.10102	−0.186106
$36$	−2.00000	−0.333333
$37$	0.101021	0.0166077	0.00830384	−	0.999966i	$-0.497357\pi$
$37$	0.101021	0.0166077	0.00830384	+	0.999966i	$0.497357\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	5.34847	0.835291	0.417645	−	0.908610i	$-0.362855\pi$
$41$	5.34847	0.835291	0.417645	+	0.908610i	$0.362855\pi$
$42$	0	0
$43$	−9.89898	−1.50958	−0.754790	−	0.655966i	$-0.772261\pi$
$43$	−9.89898	−1.50958	−0.754790	+	0.655966i	$0.772261\pi$
$44$	0	0
$45$	−2.44949	−0.365148
$46$	0	0
$47$	−7.55051	−1.10136	−0.550678	−	0.834718i	$-0.685631\pi$
$47$	−7.55051	−1.10136	−0.550678	+	0.834718i	$0.685631\pi$
$48$	4.00000	0.577350
$49$	−6.79796	−0.971137
$50$	0	0
$51$	3.44949	0.483025
$52$	−2.00000	−0.277350
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−5.44949	−0.709463	−0.354732	−	0.934968i	$-0.615428\pi$
$59$	−5.44949	−0.709463	−0.354732	+	0.934968i	$0.615428\pi$
$60$	4.89898	0.632456
$61$	−12.8990	−1.65155	−0.825773	−	0.564003i	$-0.809261\pi$
$61$	−12.8990	−1.65155	−0.825773	+	0.564003i	$0.809261\pi$
$62$	0	0
$63$	0.449490	0.0566304
$64$	−8.00000	−1.00000
$65$	−2.44949	−0.303822
$66$	0	0
$67$	0.898979	0.109828	0.0549139	−	0.998491i	$-0.482512\pi$
$67$	0.898979	0.109828	0.0549139	+	0.998491i	$0.482512\pi$
$68$	−6.89898	−0.836624
$69$	−1.00000	−0.120386
$70$	0	0
$71$	2.34847	0.278712	0.139356	−	0.990242i	$-0.455497\pi$
$71$	2.34847	0.278712	0.139356	+	0.990242i	$0.455497\pi$
$72$	0	0
$73$	10.2474	1.19937	0.599687	−	0.800235i	$-0.295292\pi$
$73$	10.2474	1.19937	0.599687	+	0.800235i	$0.295292\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	2.00000	0.229416
$77$	0	0
$78$	0	0
$79$	−3.89898	−0.438669	−0.219335	−	0.975650i	$-0.570389\pi$
$79$	−3.89898	−0.438669	−0.219335	+	0.975650i	$0.570389\pi$
$80$	−9.79796	−1.09545
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.4495	−1.36651	−0.683255	−	0.730180i	$-0.739436\pi$
$83$	−12.4495	−1.36651	−0.683255	+	0.730180i	$0.739436\pi$
$84$	−0.898979	−0.0980867
$85$	−8.44949	−0.916476
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	0.550510	0.0583540	0.0291770	−	0.999574i	$-0.490711\pi$
$89$	0.550510	0.0583540	0.0291770	+	0.999574i	$0.490711\pi$
$90$	0	0
$91$	0.449490	0.0471193
$92$	2.00000	0.208514
$93$	−5.55051	−0.575561
$94$	0	0
$95$	2.44949	0.251312
$96$	0	0
$97$	7.10102	0.720999	0.360500	−	0.932759i	$-0.382606\pi$
$97$	7.10102	0.720999	0.360500	+	0.932759i	$0.382606\pi$
$98$	0	0
$99$	0	0
$100$	−2.00000	−0.200000
$101$	5.55051	0.552296	0.276148	−	0.961115i	$-0.410942\pi$
$101$	5.55051	0.552296	0.276148	+	0.961115i	$0.410942\pi$
$102$	0	0
$103$	−17.7980	−1.75369	−0.876843	−	0.480778i	$-0.840354\pi$
$103$	−17.7980	−1.75369	−0.876843	+	0.480778i	$0.840354\pi$
$104$	0	0
$105$	−1.10102	−0.107449
$106$	0	0
$107$	10.2474	0.990658	0.495329	−	0.868705i	$-0.335047\pi$
$107$	10.2474	0.990658	0.495329	+	0.868705i	$0.335047\pi$
$108$	−2.00000	−0.192450
$109$	−1.55051	−0.148512	−0.0742560	−	0.997239i	$-0.523658\pi$
$109$	−1.55051	−0.148512	−0.0742560	+	0.997239i	$0.523658\pi$
$110$	0	0
$111$	0.101021	0.00958844
$112$	1.79796	0.169891
$113$	−7.10102	−0.668008	−0.334004	−	0.942572i	$-0.608400\pi$
$113$	−7.10102	−0.668008	−0.334004	+	0.942572i	$0.608400\pi$
$114$	0	0
$115$	2.44949	0.228416
$116$	−2.00000	−0.185695
$117$	1.00000	0.0924500
$118$	0	0
$119$	1.55051	0.142135
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	5.34847	0.482255
$124$	11.1010	0.996901
$125$	9.79796	0.876356
$126$	0	0
$127$	−14.2474	−1.26426	−0.632128	−	0.774864i	$-0.717818\pi$
$127$	−14.2474	−1.26426	−0.632128	+	0.774864i	$0.717818\pi$
$128$	0	0
$129$	−9.89898	−0.871557
$130$	0	0
$131$	−10.6969	−0.934596	−0.467298	−	0.884100i	$-0.654772\pi$
$131$	−10.6969	−0.934596	−0.467298	+	0.884100i	$0.654772\pi$
$132$	0	0
$133$	−0.449490	−0.0389757
$134$	0	0
$135$	−2.44949	−0.210819
$136$	0	0
$137$	−14.1464	−1.20861	−0.604305	−	0.796753i	$-0.706549\pi$
$137$	−14.1464	−1.20861	−0.604305	+	0.796753i	$0.706549\pi$
$138$	0	0
$139$	6.00000	0.508913	0.254457	−	0.967084i	$-0.418103\pi$
$139$	6.00000	0.508913	0.254457	+	0.967084i	$0.418103\pi$
$140$	2.20204	0.186106
$141$	−7.55051	−0.635868
$142$	0	0
$143$	0	0
$144$	4.00000	0.333333
$145$	−2.44949	−0.203419
$146$	0	0
$147$	−6.79796	−0.560686
$148$	−0.202041	−0.0166077
$149$	8.89898	0.729033	0.364516	−	0.931197i	$-0.381234\pi$
$149$	8.89898	0.729033	0.364516	+	0.931197i	$0.381234\pi$
$150$	0	0
$151$	−20.5959	−1.67607	−0.838036	−	0.545615i	$-0.816296\pi$
$151$	−20.5959	−1.67607	−0.838036	+	0.545615i	$0.816296\pi$
$152$	0	0
$153$	3.44949	0.278875
$154$	0	0
$155$	13.5959	1.09205
$156$	−2.00000	−0.160128
$157$	18.5959	1.48412	0.742058	−	0.670336i	$-0.233850\pi$
$157$	18.5959	1.48412	0.742058	+	0.670336i	$0.233850\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	0	0
$161$	−0.449490	−0.0354248
$162$	0	0
$163$	3.10102	0.242891	0.121445	−	0.992598i	$-0.461247\pi$
$163$	3.10102	0.242891	0.121445	+	0.992598i	$0.461247\pi$
$164$	−10.6969	−0.835291
$165$	0	0
$166$	0	0
$167$	−8.34847	−0.646024	−0.323012	−	0.946395i	$-0.604695\pi$
$167$	−8.34847	−0.646024	−0.323012	+	0.946395i	$0.604695\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	19.7980	1.50958
$173$	−1.24745	−0.0948418	−0.0474209	−	0.998875i	$-0.515100\pi$
$173$	−1.24745	−0.0948418	−0.0474209	+	0.998875i	$0.515100\pi$
$174$	0	0
$175$	0.449490	0.0339782
$176$	0	0
$177$	−5.44949	−0.409609
$178$	0	0
$179$	16.3485	1.22194	0.610971	−	0.791653i	$-0.290779\pi$
$179$	16.3485	1.22194	0.610971	+	0.791653i	$0.290779\pi$
$180$	4.89898	0.365148
$181$	3.79796	0.282300	0.141150	−	0.989988i	$-0.454920\pi$
$181$	3.79796	0.282300	0.141150	+	0.989988i	$0.454920\pi$
$182$	0	0
$183$	−12.8990	−0.953520
$184$	0	0
$185$	−0.247449	−0.0181928
$186$	0	0
$187$	0	0
$188$	15.1010	1.10136
$189$	0.449490	0.0326956
$190$	0	0
$191$	12.1464	0.878885	0.439442	−	0.898271i	$-0.355176\pi$
$191$	12.1464	0.878885	0.439442	+	0.898271i	$0.355176\pi$
$192$	−8.00000	−0.577350
$193$	3.79796	0.273383	0.136692	−	0.990614i	$-0.456353\pi$
$193$	3.79796	0.273383	0.136692	+	0.990614i	$0.456353\pi$
$194$	0	0
$195$	−2.44949	−0.175412
$196$	13.5959	0.971137
$197$	−10.1464	−0.722903	−0.361452	−	0.932391i	$-0.617719\pi$
$197$	−10.1464	−0.722903	−0.361452	+	0.932391i	$0.617719\pi$
$198$	0	0
$199$	4.24745	0.301094	0.150547	−	0.988603i	$-0.451897\pi$
$199$	4.24745	0.301094	0.150547	+	0.988603i	$0.451897\pi$
$200$	0	0
$201$	0.898979	0.0634091
$202$	0	0
$203$	0.449490	0.0315480
$204$	−6.89898	−0.483025
$205$	−13.1010	−0.915015
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	4.00000	0.277350
$209$	0	0
$210$	0	0
$211$	26.0454	1.79304	0.896520	−	0.443003i	$-0.146087\pi$
$211$	26.0454	1.79304	0.896520	+	0.443003i	$0.146087\pi$
$212$	4.00000	0.274721
$213$	2.34847	0.160914
$214$	0	0
$215$	24.2474	1.65366
$216$	0	0
$217$	−2.49490	−0.169365
$218$	0	0
$219$	10.2474	0.692458
$220$	0	0
$221$	3.44949	0.232038
$222$	0	0
$223$	−12.1010	−0.810344	−0.405172	−	0.914240i	$-0.632788\pi$
$223$	−12.1010	−0.810344	−0.405172	+	0.914240i	$0.632788\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	14.0000	0.929213	0.464606	−	0.885517i	$-0.346196\pi$
$227$	14.0000	0.929213	0.464606	+	0.885517i	$0.346196\pi$
$228$	2.00000	0.132453
$229$	−26.7980	−1.77086	−0.885429	−	0.464774i	$-0.846136\pi$
$229$	−26.7980	−1.77086	−0.885429	+	0.464774i	$0.846136\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	22.1464	1.45086	0.725430	−	0.688296i	$-0.241641\pi$
$233$	22.1464	1.45086	0.725430	+	0.688296i	$0.241641\pi$
$234$	0	0
$235$	18.4949	1.20647
$236$	10.8990	0.709463
$237$	−3.89898	−0.253266
$238$	0	0
$239$	13.4495	0.869975	0.434988	−	0.900436i	$-0.356753\pi$
$239$	13.4495	0.869975	0.434988	+	0.900436i	$0.356753\pi$
$240$	−9.79796	−0.632456
$241$	−8.69694	−0.560219	−0.280110	−	0.959968i	$-0.590371\pi$
$241$	−8.69694	−0.560219	−0.280110	+	0.959968i	$0.590371\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	25.7980	1.65155
$245$	16.6515	1.06383
$246$	0	0
$247$	−1.00000	−0.0636285
$248$	0	0
$249$	−12.4495	−0.788954
$250$	0	0
$251$	−13.4495	−0.848924	−0.424462	−	0.905446i	$-0.639537\pi$
$251$	−13.4495	−0.848924	−0.424462	+	0.905446i	$0.639537\pi$
$252$	−0.898979	−0.0566304
$253$	0	0
$254$	0	0
$255$	−8.44949	−0.529128
$256$	16.0000	1.00000
$257$	10.5505	0.658123	0.329061	−	0.944309i	$-0.393268\pi$
$257$	10.5505	0.658123	0.329061	+	0.944309i	$0.393268\pi$
$258$	0	0
$259$	0.0454077	0.00282150
$260$	4.89898	0.303822
$261$	1.00000	0.0618984
$262$	0	0
$263$	−7.10102	−0.437868	−0.218934	−	0.975740i	$-0.570258\pi$
$263$	−7.10102	−0.437868	−0.218934	+	0.975740i	$0.570258\pi$
$264$	0	0
$265$	4.89898	0.300942
$266$	0	0
$267$	0.550510	0.0336907
$268$	−1.79796	−0.109828
$269$	15.5959	0.950900	0.475450	−	0.879743i	$-0.342285\pi$
$269$	15.5959	0.950900	0.475450	+	0.879743i	$0.342285\pi$
$270$	0	0
$271$	16.8990	1.02654	0.513270	−	0.858227i	$-0.328434\pi$
$271$	16.8990	1.02654	0.513270	+	0.858227i	$0.328434\pi$
$272$	13.7980	0.836624
$273$	0.449490	0.0272044
$274$	0	0
$275$	0	0
$276$	2.00000	0.120386
$277$	−7.00000	−0.420589	−0.210295	−	0.977638i	$-0.567442\pi$
$277$	−7.00000	−0.420589	−0.210295	+	0.977638i	$0.567442\pi$
$278$	0	0
$279$	−5.55051	−0.332300
$280$	0	0
$281$	−24.4949	−1.46124	−0.730622	−	0.682783i	$-0.760770\pi$
$281$	−24.4949	−1.46124	−0.730622	+	0.682783i	$0.760770\pi$
$282$	0	0
$283$	−22.0454	−1.31046	−0.655232	−	0.755428i	$-0.727429\pi$
$283$	−22.0454	−1.31046	−0.655232	+	0.755428i	$0.727429\pi$
$284$	−4.69694	−0.278712
$285$	2.44949	0.145095
$286$	0	0
$287$	2.40408	0.141908
$288$	0	0
$289$	−5.10102	−0.300060
$290$	0	0
$291$	7.10102	0.416269
$292$	−20.4949	−1.19937
$293$	−27.2474	−1.59181	−0.795906	−	0.605420i	$-0.793005\pi$
$293$	−27.2474	−1.59181	−0.795906	+	0.605420i	$0.793005\pi$
$294$	0	0
$295$	13.3485	0.777178
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.00000	−0.0578315
$300$	−2.00000	−0.115470
$301$	−4.44949	−0.256464
$302$	0	0
$303$	5.55051	0.318868
$304$	−4.00000	−0.229416
$305$	31.5959	1.80918
$306$	0	0
$307$	12.6515	0.722061	0.361030	−	0.932554i	$-0.382425\pi$
$307$	12.6515	0.722061	0.361030	+	0.932554i	$0.382425\pi$
$308$	0	0
$309$	−17.7980	−1.01249
$310$	0	0
$311$	−9.79796	−0.555591	−0.277796	−	0.960640i	$-0.589604\pi$
$311$	−9.79796	−0.555591	−0.277796	+	0.960640i	$0.589604\pi$
$312$	0	0
$313$	0.651531	0.0368267	0.0184133	−	0.999830i	$-0.494139\pi$
$313$	0.651531	0.0368267	0.0184133	+	0.999830i	$0.494139\pi$
$314$	0	0
$315$	−1.10102	−0.0620355
$316$	7.79796	0.438669
$317$	18.2474	1.02488	0.512439	−	0.858723i	$-0.328742\pi$
$317$	18.2474	1.02488	0.512439	+	0.858723i	$0.328742\pi$
$318$	0	0
$319$	0	0
$320$	19.5959	1.09545
$321$	10.2474	0.571957
$322$	0	0
$323$	−3.44949	−0.191935
$324$	−2.00000	−0.111111
$325$	1.00000	0.0554700
$326$	0	0
$327$	−1.55051	−0.0857434
$328$	0	0
$329$	−3.39388	−0.187110
$330$	0	0
$331$	−5.59592	−0.307579	−0.153790	−	0.988104i	$-0.549148\pi$
$331$	−5.59592	−0.307579	−0.153790	+	0.988104i	$0.549148\pi$
$332$	24.8990	1.36651
$333$	0.101021	0.00553589
$334$	0	0
$335$	−2.20204	−0.120310
$336$	1.79796	0.0980867
$337$	−12.7980	−0.697149	−0.348575	−	0.937281i	$-0.613334\pi$
$337$	−12.7980	−0.697149	−0.348575	+	0.937281i	$0.613334\pi$
$338$	0	0
$339$	−7.10102	−0.385674
$340$	16.8990	0.916476
$341$	0	0
$342$	0	0
$343$	−6.20204	−0.334879
$344$	0	0
$345$	2.44949	0.131876
$346$	0	0
$347$	22.1464	1.18888	0.594441	−	0.804139i	$-0.297373\pi$
$347$	22.1464	1.18888	0.594441	+	0.804139i	$0.297373\pi$
$348$	−2.00000	−0.107211
$349$	−11.6969	−0.626123	−0.313061	−	0.949733i	$-0.601355\pi$
$349$	−11.6969	−0.626123	−0.313061	+	0.949733i	$0.601355\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−0.898979	−0.0478479	−0.0239239	−	0.999714i	$-0.507616\pi$
$353$	−0.898979	−0.0478479	−0.0239239	+	0.999714i	$0.507616\pi$
$354$	0	0
$355$	−5.75255	−0.305314
$356$	−1.10102	−0.0583540
$357$	1.55051	0.0820617
$358$	0	0
$359$	−29.0454	−1.53296	−0.766479	−	0.642269i	$-0.777993\pi$
$359$	−29.0454	−1.53296	−0.766479	+	0.642269i	$0.777993\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	−11.0000	−0.577350
$364$	−0.898979	−0.0471193
$365$	−25.1010	−1.31385
$366$	0	0
$367$	−14.0000	−0.730794	−0.365397	−	0.930852i	$-0.619067\pi$
$367$	−14.0000	−0.730794	−0.365397	+	0.930852i	$0.619067\pi$
$368$	−4.00000	−0.208514
$369$	5.34847	0.278430
$370$	0	0
$371$	−0.898979	−0.0466727
$372$	11.1010	0.575561
$373$	−19.3485	−1.00183	−0.500913	−	0.865498i	$-0.667002\pi$
$373$	−19.3485	−1.00183	−0.500913	+	0.865498i	$0.667002\pi$
$374$	0	0
$375$	9.79796	0.505964
$376$	0	0
$377$	1.00000	0.0515026
$378$	0	0
$379$	−5.00000	−0.256833	−0.128416	−	0.991720i	$-0.540989\pi$
$379$	−5.00000	−0.256833	−0.128416	+	0.991720i	$0.540989\pi$
$380$	−4.89898	−0.251312
$381$	−14.2474	−0.729919
$382$	0	0
$383$	−3.79796	−0.194067	−0.0970333	−	0.995281i	$-0.530935\pi$
$383$	−3.79796	−0.194067	−0.0970333	+	0.995281i	$0.530935\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−9.89898	−0.503193
$388$	−14.2020	−0.720999
$389$	15.6515	0.793564	0.396782	−	0.917913i	$-0.370127\pi$
$389$	15.6515	0.793564	0.396782	+	0.917913i	$0.370127\pi$
$390$	0	0
$391$	−3.44949	−0.174448
$392$	0	0
$393$	−10.6969	−0.539589
$394$	0	0
$395$	9.55051	0.480538
$396$	0	0
$397$	−22.6969	−1.13913	−0.569563	−	0.821947i	$-0.692888\pi$
$397$	−22.6969	−1.13913	−0.569563	+	0.821947i	$0.692888\pi$
$398$	0	0
$399$	−0.449490	−0.0225026
$400$	4.00000	0.200000
$401$	16.6969	0.833805	0.416903	−	0.908951i	$-0.363116\pi$
$401$	16.6969	0.833805	0.416903	+	0.908951i	$0.363116\pi$
$402$	0	0
$403$	−5.55051	−0.276491
$404$	−11.1010	−0.552296
$405$	−2.44949	−0.121716
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−11.1464	−0.551155	−0.275578	−	0.961279i	$-0.588869\pi$
$409$	−11.1464	−0.551155	−0.275578	+	0.961279i	$0.588869\pi$
$410$	0	0
$411$	−14.1464	−0.697792
$412$	35.5959	1.75369
$413$	−2.44949	−0.120532
$414$	0	0
$415$	30.4949	1.49694
$416$	0	0
$417$	6.00000	0.293821
$418$	0	0
$419$	31.5959	1.54356	0.771781	−	0.635889i	$-0.219366\pi$
$419$	31.5959	1.54356	0.771781	+	0.635889i	$0.219366\pi$
$420$	2.20204	0.107449
$421$	19.6969	0.959970	0.479985	−	0.877277i	$-0.340642\pi$
$421$	19.6969	0.959970	0.479985	+	0.877277i	$0.340642\pi$
$422$	0	0
$423$	−7.55051	−0.367118
$424$	0	0
$425$	3.44949	0.167325
$426$	0	0
$427$	−5.79796	−0.280583
$428$	−20.4949	−0.990658
$429$	0	0
$430$	0	0
$431$	14.4949	0.698195	0.349097	−	0.937086i	$-0.386488\pi$
$431$	14.4949	0.698195	0.349097	+	0.937086i	$0.386488\pi$
$432$	4.00000	0.192450
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	0	0
$435$	−2.44949	−0.117444
$436$	3.10102	0.148512
$437$	1.00000	0.0478365
$438$	0	0
$439$	37.6969	1.79918	0.899588	−	0.436739i	$-0.143867\pi$
$439$	37.6969	1.79918	0.899588	+	0.436739i	$0.143867\pi$
$440$	0	0
$441$	−6.79796	−0.323712
$442$	0	0
$443$	−23.1464	−1.09972	−0.549860	−	0.835257i	$-0.685319\pi$
$443$	−23.1464	−1.09972	−0.549860	+	0.835257i	$0.685319\pi$
$444$	−0.202041	−0.00958844
$445$	−1.34847	−0.0639236
$446$	0	0
$447$	8.89898	0.420907
$448$	−3.59592	−0.169891
$449$	33.7980	1.59502	0.797512	−	0.603303i	$-0.206149\pi$
$449$	33.7980	1.59502	0.797512	+	0.603303i	$0.206149\pi$
$450$	0	0
$451$	0	0
$452$	14.2020	0.668008
$453$	−20.5959	−0.967681
$454$	0	0
$455$	−1.10102	−0.0516166
$456$	0	0
$457$	11.5505	0.540310	0.270155	−	0.962817i	$-0.412925\pi$
$457$	11.5505	0.540310	0.270155	+	0.962817i	$0.412925\pi$
$458$	0	0
$459$	3.44949	0.161008
$460$	−4.89898	−0.228416
$461$	4.00000	0.186299	0.0931493	−	0.995652i	$-0.470307\pi$
$461$	4.00000	0.186299	0.0931493	+	0.995652i	$0.470307\pi$
$462$	0	0
$463$	13.8990	0.645940	0.322970	−	0.946409i	$-0.395319\pi$
$463$	13.8990	0.645940	0.322970	+	0.946409i	$0.395319\pi$
$464$	4.00000	0.185695
$465$	13.5959	0.630496
$466$	0	0
$467$	4.14643	0.191874	0.0959369	−	0.995387i	$-0.469415\pi$
$467$	4.14643	0.191874	0.0959369	+	0.995387i	$0.469415\pi$
$468$	−2.00000	−0.0924500
$469$	0.404082	0.0186588
$470$	0	0
$471$	18.5959	0.856855
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	−3.10102	−0.142135
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	7.10102	0.324454	0.162227	−	0.986753i	$-0.448132\pi$
$479$	7.10102	0.324454	0.162227	+	0.986753i	$0.448132\pi$
$480$	0	0
$481$	0.101021	0.00460614
$482$	0	0
$483$	−0.449490	−0.0204525
$484$	22.0000	1.00000
$485$	−17.3939	−0.789815
$486$	0	0
$487$	−4.69694	−0.212839	−0.106419	−	0.994321i	$-0.533939\pi$
$487$	−4.69694	−0.212839	−0.106419	+	0.994321i	$0.533939\pi$
$488$	0	0
$489$	3.10102	0.140233
$490$	0	0
$491$	−20.4495	−0.922873	−0.461436	−	0.887173i	$-0.652666\pi$
$491$	−20.4495	−0.922873	−0.461436	+	0.887173i	$0.652666\pi$
$492$	−10.6969	−0.482255
$493$	3.44949	0.155357
$494$	0	0
$495$	0	0
$496$	−22.2020	−0.996901
$497$	1.05561	0.0473507
$498$	0	0
$499$	−37.8990	−1.69659	−0.848296	−	0.529523i	$-0.822371\pi$
$499$	−37.8990	−1.69659	−0.848296	+	0.529523i	$0.822371\pi$
$500$	−19.5959	−0.876356
$501$	−8.34847	−0.372982
$502$	0	0
$503$	−25.2474	−1.12573	−0.562864	−	0.826549i	$-0.690301\pi$
$503$	−25.2474	−1.12573	−0.562864	+	0.826549i	$0.690301\pi$
$504$	0	0
$505$	−13.5959	−0.605010
$506$	0	0
$507$	−12.0000	−0.532939
$508$	28.4949	1.26426
$509$	36.1464	1.60216	0.801081	−	0.598556i	$-0.204259\pi$
$509$	36.1464	1.60216	0.801081	+	0.598556i	$0.204259\pi$
$510$	0	0
$511$	4.60612	0.203763
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	43.5959	1.92107
$516$	19.7980	0.871557
$517$	0	0
$518$	0	0
$519$	−1.24745	−0.0547569
$520$	0	0
$521$	18.0454	0.790584	0.395292	−	0.918556i	$-0.370643\pi$
$521$	18.0454	0.790584	0.395292	+	0.918556i	$0.370643\pi$
$522$	0	0
$523$	13.3485	0.583688	0.291844	−	0.956466i	$-0.405731\pi$
$523$	13.3485	0.583688	0.291844	+	0.956466i	$0.405731\pi$
$524$	21.3939	0.934596
$525$	0.449490	0.0196173
$526$	0	0
$527$	−19.1464	−0.834032
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−5.44949	−0.236488
$532$	0.898979	0.0389757
$533$	5.34847	0.231668
$534$	0	0
$535$	−25.1010	−1.08521
$536$	0	0
$537$	16.3485	0.705489
$538$	0	0
$539$	0	0
$540$	4.89898	0.210819
$541$	−7.10102	−0.305297	−0.152648	−	0.988281i	$-0.548780\pi$
$541$	−7.10102	−0.305297	−0.152648	+	0.988281i	$0.548780\pi$
$542$	0	0
$543$	3.79796	0.162986
$544$	0	0
$545$	3.79796	0.162687
$546$	0	0
$547$	15.6969	0.671153	0.335576	−	0.942013i	$-0.391069\pi$
$547$	15.6969	0.671153	0.335576	+	0.942013i	$0.391069\pi$
$548$	28.2929	1.20861
$549$	−12.8990	−0.550515
$550$	0	0
$551$	−1.00000	−0.0426014
$552$	0	0
$553$	−1.75255	−0.0745261
$554$	0	0
$555$	−0.247449	−0.0105036
$556$	−12.0000	−0.508913
$557$	−0.853572	−0.0361670	−0.0180835	−	0.999836i	$-0.505756\pi$
$557$	−0.853572	−0.0361670	−0.0180835	+	0.999836i	$0.505756\pi$
$558$	0	0
$559$	−9.89898	−0.418682
$560$	−4.40408	−0.186106
$561$	0	0
$562$	0	0
$563$	15.2474	0.642603	0.321302	−	0.946977i	$-0.395880\pi$
$563$	15.2474	0.642603	0.321302	+	0.946977i	$0.395880\pi$
$564$	15.1010	0.635868
$565$	17.3939	0.731766
$566$	0	0
$567$	0.449490	0.0188768
$568$	0	0
$569$	−35.7980	−1.50073	−0.750364	−	0.661025i	$-0.770122\pi$
$569$	−35.7980	−1.50073	−0.750364	+	0.661025i	$0.770122\pi$
$570$	0	0
$571$	−10.0454	−0.420387	−0.210194	−	0.977660i	$-0.567409\pi$
$571$	−10.0454	−0.420387	−0.210194	+	0.977660i	$0.567409\pi$
$572$	0	0
$573$	12.1464	0.507424
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	−8.00000	−0.333333
$577$	3.75255	0.156221	0.0781104	−	0.996945i	$-0.475111\pi$
$577$	3.75255	0.156221	0.0781104	+	0.996945i	$0.475111\pi$
$578$	0	0
$579$	3.79796	0.157838
$580$	4.89898	0.203419
$581$	−5.59592	−0.232158
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−2.44949	−0.101274
$586$	0	0
$587$	23.0454	0.951186	0.475593	−	0.879666i	$-0.342234\pi$
$587$	23.0454	0.951186	0.475593	+	0.879666i	$0.342234\pi$
$588$	13.5959	0.560686
$589$	5.55051	0.228705
$590$	0	0
$591$	−10.1464	−0.417368
$592$	0.404082	0.0166077
$593$	14.7526	0.605815	0.302907	−	0.953020i	$-0.402043\pi$
$593$	14.7526	0.605815	0.302907	+	0.953020i	$0.402043\pi$
$594$	0	0
$595$	−3.79796	−0.155701
$596$	−17.7980	−0.729033
$597$	4.24745	0.173837
$598$	0	0
$599$	−1.34847	−0.0550970	−0.0275485	−	0.999620i	$-0.508770\pi$
$599$	−1.34847	−0.0550970	−0.0275485	+	0.999620i	$0.508770\pi$
$600$	0	0
$601$	30.6969	1.25215	0.626077	−	0.779761i	$-0.284660\pi$
$601$	30.6969	1.25215	0.626077	+	0.779761i	$0.284660\pi$
$602$	0	0
$603$	0.898979	0.0366093
$604$	41.1918	1.67607
$605$	26.9444	1.09545
$606$	0	0
$607$	−9.34847	−0.379443	−0.189721	−	0.981838i	$-0.560758\pi$
$607$	−9.34847	−0.379443	−0.189721	+	0.981838i	$0.560758\pi$
$608$	0	0
$609$	0.449490	0.0182142
$610$	0	0
$611$	−7.55051	−0.305461
$612$	−6.89898	−0.278875
$613$	10.0000	0.403896	0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000	0.403896	0.201948	+	0.979396i	$0.435273\pi$
$614$	0	0
$615$	−13.1010	−0.528284
$616$	0	0
$617$	27.7980	1.11910	0.559552	−	0.828795i	$-0.310973\pi$
$617$	27.7980	1.11910	0.559552	+	0.828795i	$0.310973\pi$
$618$	0	0
$619$	−8.79796	−0.353620	−0.176810	−	0.984245i	$-0.556578\pi$
$619$	−8.79796	−0.353620	−0.176810	+	0.984245i	$0.556578\pi$
$620$	−27.1918	−1.09205
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	0.247449	0.00991382
$624$	4.00000	0.160128
$625$	−29.0000	−1.16000
$626$	0	0
$627$	0	0
$628$	−37.1918	−1.48412
$629$	0.348469	0.0138944
$630$	0	0
$631$	−3.30306	−0.131493	−0.0657464	−	0.997836i	$-0.520943\pi$
$631$	−3.30306	−0.131493	−0.0657464	+	0.997836i	$0.520943\pi$
$632$	0	0
$633$	26.0454	1.03521
$634$	0	0
$635$	34.8990	1.38492
$636$	4.00000	0.158610
$637$	−6.79796	−0.269345
$638$	0	0
$639$	2.34847	0.0929040
$640$	0	0
$641$	11.0454	0.436267	0.218134	−	0.975919i	$-0.430003\pi$
$641$	11.0454	0.436267	0.218134	+	0.975919i	$0.430003\pi$
$642$	0	0
$643$	−12.8990	−0.508686	−0.254343	−	0.967114i	$-0.581859\pi$
$643$	−12.8990	−0.508686	−0.254343	+	0.967114i	$0.581859\pi$
$644$	0.898979	0.0354248
$645$	24.2474	0.954742
$646$	0	0
$647$	23.2474	0.913952	0.456976	−	0.889479i	$-0.348933\pi$
$647$	23.2474	0.913952	0.456976	+	0.889479i	$0.348933\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−2.49490	−0.0977827
$652$	−6.20204	−0.242891
$653$	−30.7423	−1.20304	−0.601520	−	0.798857i	$-0.705438\pi$
$653$	−30.7423	−1.20304	−0.601520	+	0.798857i	$0.705438\pi$
$654$	0	0
$655$	26.2020	1.02380
$656$	21.3939	0.835291
$657$	10.2474	0.399791
$658$	0	0
$659$	−4.55051	−0.177263	−0.0886314	−	0.996064i	$-0.528249\pi$
$659$	−4.55051	−0.177263	−0.0886314	+	0.996064i	$0.528249\pi$
$660$	0	0
$661$	7.34847	0.285822	0.142911	−	0.989736i	$-0.454354\pi$
$661$	7.34847	0.285822	0.142911	+	0.989736i	$0.454354\pi$
$662$	0	0
$663$	3.44949	0.133967
$664$	0	0
$665$	1.10102	0.0426957
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	16.6969	0.646024
$669$	−12.1010	−0.467852
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−7.89898	−0.304483	−0.152242	−	0.988343i	$-0.548649\pi$
$673$	−7.89898	−0.304483	−0.152242	+	0.988343i	$0.548649\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	24.0000	0.923077
$677$	4.20204	0.161498	0.0807488	−	0.996734i	$-0.474269\pi$
$677$	4.20204	0.161498	0.0807488	+	0.996734i	$0.474269\pi$
$678$	0	0
$679$	3.19184	0.122491
$680$	0	0
$681$	14.0000	0.536481
$682$	0	0
$683$	−2.69694	−0.103195	−0.0515977	−	0.998668i	$-0.516431\pi$
$683$	−2.69694	−0.103195	−0.0515977	+	0.998668i	$0.516431\pi$
$684$	2.00000	0.0764719
$685$	34.6515	1.32397
$686$	0	0
$687$	−26.7980	−1.02241
$688$	−39.5959	−1.50958
$689$	−2.00000	−0.0761939
$690$	0	0
$691$	38.8990	1.47979	0.739893	−	0.672724i	$-0.234876\pi$
$691$	38.8990	1.47979	0.739893	+	0.672724i	$0.234876\pi$
$692$	2.49490	0.0948418
$693$	0	0
$694$	0	0
$695$	−14.6969	−0.557487
$696$	0	0
$697$	18.4495	0.698824
$698$	0	0
$699$	22.1464	0.837655
$700$	−0.898979	−0.0339782
$701$	−42.2474	−1.59566	−0.797832	−	0.602880i	$-0.794020\pi$
$701$	−42.2474	−1.59566	−0.797832	+	0.602880i	$0.794020\pi$
$702$	0	0
$703$	−0.101021	−0.00381006
$704$	0	0
$705$	18.4949	0.696558
$706$	0	0
$707$	2.49490	0.0938303
$708$	10.8990	0.409609
$709$	0.651531	0.0244688	0.0122344	−	0.999925i	$-0.496106\pi$
$709$	0.651531	0.0244688	0.0122344	+	0.999925i	$0.496106\pi$
$710$	0	0
$711$	−3.89898	−0.146223
$712$	0	0
$713$	5.55051	0.207868
$714$	0	0
$715$	0	0
$716$	−32.6969	−1.22194
$717$	13.4495	0.502280
$718$	0	0
$719$	−51.7423	−1.92966	−0.964832	−	0.262867i	$-0.915332\pi$
$719$	−51.7423	−1.92966	−0.964832	+	0.262867i	$0.915332\pi$
$720$	−9.79796	−0.365148
$721$	−8.00000	−0.297936
$722$	0	0
$723$	−8.69694	−0.323443
$724$	−7.59592	−0.282300
$725$	1.00000	0.0371391
$726$	0	0
$727$	3.59592	0.133365	0.0666826	−	0.997774i	$-0.478759\pi$
$727$	3.59592	0.133365	0.0666826	+	0.997774i	$0.478759\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−34.1464	−1.26295
$732$	25.7980	0.953520
$733$	−31.1010	−1.14874	−0.574371	−	0.818595i	$-0.694753\pi$
$733$	−31.1010	−1.14874	−0.574371	+	0.818595i	$0.694753\pi$
$734$	0	0
$735$	16.6515	0.614201
$736$	0	0
$737$	0	0
$738$	0	0
$739$	16.2020	0.596002	0.298001	−	0.954566i	$-0.403680\pi$
$739$	16.2020	0.596002	0.298001	+	0.954566i	$0.403680\pi$
$740$	0.494897	0.0181928
$741$	−1.00000	−0.0367359
$742$	0	0
$743$	41.2474	1.51322	0.756611	−	0.653865i	$-0.226854\pi$
$743$	41.2474	1.51322	0.756611	+	0.653865i	$0.226854\pi$
$744$	0	0
$745$	−21.7980	−0.798615
$746$	0	0
$747$	−12.4495	−0.455503
$748$	0	0
$749$	4.60612	0.168304
$750$	0	0
$751$	37.4949	1.36821	0.684104	−	0.729384i	$-0.260193\pi$
$751$	37.4949	1.36821	0.684104	+	0.729384i	$0.260193\pi$
$752$	−30.2020	−1.10136
$753$	−13.4495	−0.490127
$754$	0	0
$755$	50.4495	1.83604
$756$	−0.898979	−0.0326956
$757$	21.8990	0.795932	0.397966	−	0.917400i	$-0.369716\pi$
$757$	21.8990	0.795932	0.397966	+	0.917400i	$0.369716\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	21.2474	0.770219	0.385110	−	0.922871i	$-0.374164\pi$
$761$	21.2474	0.770219	0.385110	+	0.922871i	$0.374164\pi$
$762$	0	0
$763$	−0.696938	−0.0252309
$764$	−24.2929	−0.878885
$765$	−8.44949	−0.305492
$766$	0	0
$767$	−5.44949	−0.196770
$768$	16.0000	0.577350
$769$	−51.8990	−1.87153	−0.935763	−	0.352631i	$-0.885287\pi$
$769$	−51.8990	−1.87153	−0.935763	+	0.352631i	$0.885287\pi$
$770$	0	0
$771$	10.5505	0.379967
$772$	−7.59592	−0.273383
$773$	−7.30306	−0.262673	−0.131336	−	0.991338i	$-0.541927\pi$
$773$	−7.30306	−0.262673	−0.131336	+	0.991338i	$0.541927\pi$
$774$	0	0
$775$	−5.55051	−0.199380
$776$	0	0
$777$	0.0454077	0.00162899
$778$	0	0
$779$	−5.34847	−0.191629
$780$	4.89898	0.175412
$781$	0	0
$782$	0	0
$783$	1.00000	0.0357371
$784$	−27.1918	−0.971137
$785$	−45.5505	−1.62577
$786$	0	0
$787$	28.2929	1.00853	0.504266	−	0.863549i	$-0.331763\pi$
$787$	28.2929	1.00853	0.504266	+	0.863549i	$0.331763\pi$
$788$	20.2929	0.722903
$789$	−7.10102	−0.252803
$790$	0	0
$791$	−3.19184	−0.113489
$792$	0	0
$793$	−12.8990	−0.458056
$794$	0	0
$795$	4.89898	0.173749
$796$	−8.49490	−0.301094
$797$	3.30306	0.117000	0.0585002	−	0.998287i	$-0.481368\pi$
$797$	3.30306	0.117000	0.0585002	+	0.998287i	$0.481368\pi$
$798$	0	0
$799$	−26.0454	−0.921420
$800$	0	0
$801$	0.550510	0.0194513
$802$	0	0
$803$	0	0
$804$	−1.79796	−0.0634091
$805$	1.10102	0.0388059
$806$	0	0
$807$	15.5959	0.549002
$808$	0	0
$809$	30.8990	1.08635	0.543175	−	0.839619i	$-0.317222\pi$
$809$	30.8990	1.08635	0.543175	+	0.839619i	$0.317222\pi$
$810$	0	0
$811$	48.6969	1.70998	0.854990	−	0.518644i	$-0.173563\pi$
$811$	48.6969	1.70998	0.854990	+	0.518644i	$0.173563\pi$
$812$	−0.898979	−0.0315480
$813$	16.8990	0.592673
$814$	0	0
$815$	−7.59592	−0.266073
$816$	13.7980	0.483025
$817$	9.89898	0.346321
$818$	0	0
$819$	0.449490	0.0157064
$820$	26.2020	0.915015
$821$	−9.65153	−0.336841	−0.168420	−	0.985715i	$-0.553867\pi$
$821$	−9.65153	−0.336841	−0.168420	+	0.985715i	$0.553867\pi$
$822$	0	0
$823$	−51.1464	−1.78285	−0.891426	−	0.453166i	$-0.850295\pi$
$823$	−51.1464	−1.78285	−0.891426	+	0.453166i	$0.850295\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−7.65153	−0.266070	−0.133035	−	0.991111i	$-0.542472\pi$
$827$	−7.65153	−0.266070	−0.133035	+	0.991111i	$0.542472\pi$
$828$	2.00000	0.0695048
$829$	18.2929	0.635337	0.317669	−	0.948202i	$-0.397100\pi$
$829$	18.2929	0.635337	0.317669	+	0.948202i	$0.397100\pi$
$830$	0	0
$831$	−7.00000	−0.242827
$832$	−8.00000	−0.277350
$833$	−23.4495	−0.812477
$834$	0	0
$835$	20.4495	0.707684
$836$	0	0
$837$	−5.55051	−0.191854
$838$	0	0
$839$	−7.39388	−0.255265	−0.127632	−	0.991822i	$-0.540738\pi$
$839$	−7.39388	−0.255265	−0.127632	+	0.991822i	$0.540738\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−24.4949	−0.843649
$844$	−52.0908	−1.79304
$845$	29.3939	1.01118
$846$	0	0
$847$	−4.94439	−0.169891
$848$	−8.00000	−0.274721
$849$	−22.0454	−0.756596
$850$	0	0
$851$	−0.101021	−0.00346294
$852$	−4.69694	−0.160914
$853$	35.3939	1.21186	0.605932	−	0.795517i	$-0.292800\pi$
$853$	35.3939	1.21186	0.605932	+	0.795517i	$0.292800\pi$
$854$	0	0
$855$	2.44949	0.0837708
$856$	0	0
$857$	−28.2929	−0.966466	−0.483233	−	0.875492i	$-0.660538\pi$
$857$	−28.2929	−0.966466	−0.483233	+	0.875492i	$0.660538\pi$
$858$	0	0
$859$	−27.3485	−0.933118	−0.466559	−	0.884490i	$-0.654506\pi$
$859$	−27.3485	−0.933118	−0.466559	+	0.884490i	$0.654506\pi$
$860$	−48.4949	−1.65366
$861$	2.40408	0.0819309
$862$	0	0
$863$	−38.2929	−1.30350	−0.651752	−	0.758432i	$-0.725966\pi$
$863$	−38.2929	−1.30350	−0.651752	+	0.758432i	$0.725966\pi$
$864$	0	0
$865$	3.05561	0.103894
$866$	0	0
$867$	−5.10102	−0.173240
$868$	4.98979	0.169365
$869$	0	0
$870$	0	0
$871$	0.898979	0.0304608
$872$	0	0
$873$	7.10102	0.240333
$874$	0	0
$875$	4.40408	0.148885
$876$	−20.4949	−0.692458
$877$	−7.40408	−0.250018	−0.125009	−	0.992156i	$-0.539896\pi$
$877$	−7.40408	−0.250018	−0.125009	+	0.992156i	$0.539896\pi$
$878$	0	0
$879$	−27.2474	−0.919034
$880$	0	0
$881$	−14.3485	−0.483412	−0.241706	−	0.970350i	$-0.577707\pi$
$881$	−14.3485	−0.483412	−0.241706	+	0.970350i	$0.577707\pi$
$882$	0	0
$883$	53.0908	1.78665	0.893324	−	0.449413i	$-0.148367\pi$
$883$	53.0908	1.78665	0.893324	+	0.449413i	$0.148367\pi$
$884$	−6.89898	−0.232038
$885$	13.3485	0.448704
$886$	0	0
$887$	−50.5403	−1.69698	−0.848489	−	0.529214i	$-0.822487\pi$
$887$	−50.5403	−1.69698	−0.848489	+	0.529214i	$0.822487\pi$
$888$	0	0
$889$	−6.40408	−0.214786
$890$	0	0
$891$	0	0
$892$	24.2020	0.810344
$893$	7.55051	0.252668
$894$	0	0
$895$	−40.0454	−1.33857
$896$	0	0
$897$	−1.00000	−0.0333890
$898$	0	0
$899$	−5.55051	−0.185120
$900$	−2.00000	−0.0666667
$901$	−6.89898	−0.229838
$902$	0	0
$903$	−4.44949	−0.148070
$904$	0	0
$905$	−9.30306	−0.309244
$906$	0	0
$907$	12.5959	0.418241	0.209120	−	0.977890i	$-0.432940\pi$
$907$	12.5959	0.418241	0.209120	+	0.977890i	$0.432940\pi$
$908$	−28.0000	−0.929213
$909$	5.55051	0.184099
$910$	0	0
$911$	−42.5505	−1.40976	−0.704881	−	0.709326i	$-0.748999\pi$
$911$	−42.5505	−1.40976	−0.704881	+	0.709326i	$0.748999\pi$
$912$	−4.00000	−0.132453
$913$	0	0
$914$	0	0
$915$	31.5959	1.04453
$916$	53.5959	1.77086
$917$	−4.80816	−0.158780
$918$	0	0
$919$	0.696938	0.0229899	0.0114949	−	0.999934i	$-0.496341\pi$
$919$	0.696938	0.0229899	0.0114949	+	0.999934i	$0.496341\pi$
$920$	0	0
$921$	12.6515	0.416882
$922$	0	0
$923$	2.34847	0.0773008
$924$	0	0
$925$	0.101021	0.00332153
$926$	0	0
$927$	−17.7980	−0.584562
$928$	0	0
$929$	58.8434	1.93059	0.965294	−	0.261165i	$-0.0841067\pi$
$929$	58.8434	1.93059	0.965294	+	0.261165i	$0.0841067\pi$
$930$	0	0
$931$	6.79796	0.222794
$932$	−44.2929	−1.45086
$933$	−9.79796	−0.320771
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−49.1918	−1.60703	−0.803514	−	0.595286i	$-0.797039\pi$
$937$	−49.1918	−1.60703	−0.803514	+	0.595286i	$0.797039\pi$
$938$	0	0
$939$	0.651531	0.0212619
$940$	−36.9898	−1.20647
$941$	16.4949	0.537718	0.268859	−	0.963180i	$-0.413353\pi$
$941$	16.4949	0.537718	0.268859	+	0.963180i	$0.413353\pi$
$942$	0	0
$943$	−5.34847	−0.174170
$944$	−21.7980	−0.709463
$945$	−1.10102	−0.0358162
$946$	0	0
$947$	22.6969	0.737551	0.368776	−	0.929518i	$-0.379777\pi$
$947$	22.6969	0.737551	0.368776	+	0.929518i	$0.379777\pi$
$948$	7.79796	0.253266
$949$	10.2474	0.332646
$950$	0	0
$951$	18.2474	0.591714
$952$	0	0
$953$	−41.3485	−1.33941	−0.669704	−	0.742628i	$-0.733579\pi$
$953$	−41.3485	−1.33941	−0.669704	+	0.742628i	$0.733579\pi$
$954$	0	0
$955$	−29.7526	−0.962770
$956$	−26.8990	−0.869975
$957$	0	0
$958$	0	0
$959$	−6.35867	−0.205332
$960$	19.5959	0.632456
$961$	−0.191836	−0.00618825
$962$	0	0
$963$	10.2474	0.330219
$964$	17.3939	0.560219
$965$	−9.30306	−0.299476
$966$	0	0
$967$	−26.4495	−0.850558	−0.425279	−	0.905062i	$-0.639824\pi$
$967$	−26.4495	−0.850558	−0.425279	+	0.905062i	$0.639824\pi$
$968$	0	0
$969$	−3.44949	−0.110814
$970$	0	0
$971$	−46.2929	−1.48561	−0.742804	−	0.669509i	$-0.766505\pi$
$971$	−46.2929	−1.48561	−0.742804	+	0.669509i	$0.766505\pi$
$972$	−2.00000	−0.0641500
$973$	2.69694	0.0864599
$974$	0	0
$975$	1.00000	0.0320256
$976$	−51.5959	−1.65155
$977$	21.5505	0.689462	0.344731	−	0.938702i	$-0.387970\pi$
$977$	21.5505	0.689462	0.344731	+	0.938702i	$0.387970\pi$
$978$	0	0
$979$	0	0
$980$	−33.3031	−1.06383
$981$	−1.55051	−0.0495040
$982$	0	0
$983$	−61.7423	−1.96928	−0.984638	−	0.174611i	$-0.944133\pi$
$983$	−61.7423	−1.96928	−0.984638	+	0.174611i	$0.944133\pi$
$984$	0	0
$985$	24.8536	0.791901
$986$	0	0
$987$	−3.39388	−0.108028
$988$	2.00000	0.0636285
$989$	9.89898	0.314769
$990$	0	0
$991$	−12.4041	−0.394029	−0.197014	−	0.980401i	$-0.563125\pi$
$991$	−12.4041	−0.394029	−0.197014	+	0.980401i	$0.563125\pi$
$992$	0	0
$993$	−5.59592	−0.177581
$994$	0	0
$995$	−10.4041	−0.329832
$996$	24.8990	0.788954
$997$	36.9898	1.17148	0.585739	−	0.810500i	$-0.300804\pi$
$997$	36.9898	1.17148	0.585739	+	0.810500i	$0.300804\pi$
$998$	0	0
$999$	0.101021	0.00319615

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2001.2.a.e.1.1	✓	2
3.2	odd	2		6003.2.a.e.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.e.1.1	✓	2	1.1	even	1	trivial
6003.2.a.e.1.2		2	3.2	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:	\( N \)	\(=\)	\( 2001 = 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2001.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.00000 q^{4} -2.44949 q^{5} +0.449490 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.00000 q^{4} -2.44949 q^{5} +0.449490 q^{7} +1.00000 q^{9} -2.00000 q^{12} +1.00000 q^{13} -2.44949 q^{15} +4.00000 q^{16} +3.44949 q^{17} -1.00000 q^{19} +4.89898 q^{20} +0.449490 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -0.898979 q^{28} +1.00000 q^{29} -5.55051 q^{31} -1.10102 q^{35} -2.00000 q^{36} +0.101021 q^{37} +1.00000 q^{39} +5.34847 q^{41} -9.89898 q^{43} -2.44949 q^{45} -7.55051 q^{47} +4.00000 q^{48} -6.79796 q^{49} +3.44949 q^{51} -2.00000 q^{52} -2.00000 q^{53} -1.00000 q^{57} -5.44949 q^{59} +4.89898 q^{60} -12.8990 q^{61} +0.449490 q^{63} -8.00000 q^{64} -2.44949 q^{65} +0.898979 q^{67} -6.89898 q^{68} -1.00000 q^{69} +2.34847 q^{71} +10.2474 q^{73} +1.00000 q^{75} +2.00000 q^{76} -3.89898 q^{79} -9.79796 q^{80} +1.00000 q^{81} -12.4495 q^{83} -0.898979 q^{84} -8.44949 q^{85} +1.00000 q^{87} +0.550510 q^{89} +0.449490 q^{91} +2.00000 q^{92} -5.55051 q^{93} +2.44949 q^{95} +7.10102 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{4} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 4 * q^4 - 4 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 4 q^{4} - 4 q^{7} + 2 q^{9} - 4 q^{12} + 2 q^{13} + 8 q^{16} + 2 q^{17} - 2 q^{19} - 4 q^{21} - 2 q^{23} + 2 q^{25} + 2 q^{27} + 8 q^{28} + 2 q^{29} - 16 q^{31} - 12 q^{35} - 4 q^{36} + 10 q^{37} + 2 q^{39} - 4 q^{41} - 10 q^{43} - 20 q^{47} + 8 q^{48} + 6 q^{49} + 2 q^{51} - 4 q^{52} - 4 q^{53} - 2 q^{57} - 6 q^{59} - 16 q^{61} - 4 q^{63} - 16 q^{64} - 8 q^{67} - 4 q^{68} - 2 q^{69} - 10 q^{71} - 4 q^{73} + 2 q^{75} + 4 q^{76} + 2 q^{79} + 2 q^{81} - 20 q^{83} + 8 q^{84} - 12 q^{85} + 2 q^{87} + 6 q^{89} - 4 q^{91} + 4 q^{92} - 16 q^{93} + 24 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 - 4 * q^4 - 4 * q^7 + 2 * q^9 - 4 * q^12 + 2 * q^13 + 8 * q^16 + 2 * q^17 - 2 * q^19 - 4 * q^21 - 2 * q^23 + 2 * q^25 + 2 * q^27 + 8 * q^28 + 2 * q^29 - 16 * q^31 - 12 * q^35 - 4 * q^36 + 10 * q^37 + 2 * q^39 - 4 * q^41 - 10 * q^43 - 20 * q^47 + 8 * q^48 + 6 * q^49 + 2 * q^51 - 4 * q^52 - 4 * q^53 - 2 * q^57 - 6 * q^59 - 16 * q^61 - 4 * q^63 - 16 * q^64 - 8 * q^67 - 4 * q^68 - 2 * q^69 - 10 * q^71 - 4 * q^73 + 2 * q^75 + 4 * q^76 + 2 * q^79 + 2 * q^81 - 20 * q^83 + 8 * q^84 - 12 * q^85 + 2 * q^87 + 6 * q^89 - 4 * q^91 + 4 * q^92 - 16 * q^93 + 24 * q^97

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