LMFDB - Embedded newform 2009.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2009,2,Mod(1,2009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2009, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2009 = 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2009.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:	$16.0419457661$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 287)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	2009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.30278 q^{3} -2.00000 q^{4} -1.30278 q^{5} +2.30278 q^{9} +O(q^{10})$	$q-2.30278 q^{3} -2.00000 q^{4} -1.30278 q^{5} +2.30278 q^{9} -4.30278 q^{11} +4.60555 q^{12} +2.30278 q^{13} +3.00000 q^{15} +4.00000 q^{16} +3.90833 q^{17} +3.60555 q^{19} +2.60555 q^{20} -5.60555 q^{23} -3.30278 q^{25} +1.60555 q^{27} +5.60555 q^{29} +2.30278 q^{31} +9.90833 q^{33} -4.60555 q^{36} +7.21110 q^{37} -5.30278 q^{39} +1.00000 q^{41} -9.60555 q^{43} +8.60555 q^{44} -3.00000 q^{45} +9.00000 q^{47} -9.21110 q^{48} -9.00000 q^{51} -4.60555 q^{52} +9.90833 q^{53} +5.60555 q^{55} -8.30278 q^{57} -0.394449 q^{59} -6.00000 q^{60} -7.21110 q^{61} -8.00000 q^{64} -3.00000 q^{65} -6.60555 q^{67} -7.81665 q^{68} +12.9083 q^{69} +10.8167 q^{71} -9.30278 q^{73} +7.60555 q^{75} -7.21110 q^{76} -17.4222 q^{79} -5.21110 q^{80} -10.6056 q^{81} -11.6056 q^{83} -5.09167 q^{85} -12.9083 q^{87} -0.394449 q^{89} +11.2111 q^{92} -5.30278 q^{93} -4.69722 q^{95} -9.30278 q^{97} -9.90833 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 4 q^{4} + q^{5} + q^{9}+O(q^{10}) $ 2 * q - q^3 - 4 * q^4 + q^5 + q^9	$ 2 q - q^{3} - 4 q^{4} + q^{5} + q^{9} - 5 q^{11} + 2 q^{12} + q^{13} + 6 q^{15} + 8 q^{16} - 3 q^{17} - 2 q^{20} - 4 q^{23} - 3 q^{25} - 4 q^{27} + 4 q^{29} + q^{31} + 9 q^{33} - 2 q^{36} - 7 q^{39} + 2 q^{41} - 12 q^{43} + 10 q^{44} - 6 q^{45} + 18 q^{47} - 4 q^{48} - 18 q^{51} - 2 q^{52} + 9 q^{53} + 4 q^{55} - 13 q^{57} - 8 q^{59} - 12 q^{60} - 16 q^{64} - 6 q^{65} - 6 q^{67} + 6 q^{68} + 15 q^{69} - 15 q^{73} + 8 q^{75} - 6 q^{79} + 4 q^{80} - 14 q^{81} - 16 q^{83} - 21 q^{85} - 15 q^{87} - 8 q^{89} + 8 q^{92} - 7 q^{93} - 13 q^{95} - 15 q^{97} - 9 q^{99}+O(q^{100}) $ 2 * q - q^3 - 4 * q^4 + q^5 + q^9 - 5 * q^11 + 2 * q^12 + q^13 + 6 * q^15 + 8 * q^16 - 3 * q^17 - 2 * q^20 - 4 * q^23 - 3 * q^25 - 4 * q^27 + 4 * q^29 + q^31 + 9 * q^33 - 2 * q^36 - 7 * q^39 + 2 * q^41 - 12 * q^43 + 10 * q^44 - 6 * q^45 + 18 * q^47 - 4 * q^48 - 18 * q^51 - 2 * q^52 + 9 * q^53 + 4 * q^55 - 13 * q^57 - 8 * q^59 - 12 * q^60 - 16 * q^64 - 6 * q^65 - 6 * q^67 + 6 * q^68 + 15 * q^69 - 15 * q^73 + 8 * q^75 - 6 * q^79 + 4 * q^80 - 14 * q^81 - 16 * q^83 - 21 * q^85 - 15 * q^87 - 8 * q^89 + 8 * q^92 - 7 * q^93 - 13 * q^95 - 15 * q^97 - 9 * 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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	−2.30278	−1.32951	−0.664754	−	0.747062i	$-0.731464\pi$
$3$	−2.30278	−1.32951	−0.664754	+	0.747062i	$0.731464\pi$
$4$	−2.00000	−1.00000
$5$	−1.30278	−0.582619	−0.291309	−	0.956629i	$-0.594091\pi$
$5$	−1.30278	−0.582619	−0.291309	+	0.956629i	$0.594091\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.30278	0.767592
$10$	0	0
$11$	−4.30278	−1.29734	−0.648668	−	0.761072i	$-0.724674\pi$
$11$	−4.30278	−1.29734	−0.648668	+	0.761072i	$0.724674\pi$
$12$	4.60555	1.32951
$13$	2.30278	0.638675	0.319338	−	0.947641i	$-0.396540\pi$
$13$	2.30278	0.638675	0.319338	+	0.947641i	$0.396540\pi$
$14$	0	0
$15$	3.00000	0.774597
$16$	4.00000	1.00000
$17$	3.90833	0.947909	0.473954	−	0.880549i	$-0.342826\pi$
$17$	3.90833	0.947909	0.473954	+	0.880549i	$0.342826\pi$
$18$	0	0
$19$	3.60555	0.827170	0.413585	−	0.910465i	$-0.364276\pi$
$19$	3.60555	0.827170	0.413585	+	0.910465i	$0.364276\pi$
$20$	2.60555	0.582619
$21$	0	0
$22$	0	0
$23$	−5.60555	−1.16884	−0.584419	−	0.811452i	$-0.698678\pi$
$23$	−5.60555	−1.16884	−0.584419	+	0.811452i	$0.698678\pi$
$24$	0	0
$25$	−3.30278	−0.660555
$26$	0	0
$27$	1.60555	0.308988
$28$	0	0
$29$	5.60555	1.04092	0.520462	−	0.853885i	$-0.325760\pi$
$29$	5.60555	1.04092	0.520462	+	0.853885i	$0.325760\pi$
$30$	0	0
$31$	2.30278	0.413591	0.206795	−	0.978384i	$-0.433697\pi$
$31$	2.30278	0.413591	0.206795	+	0.978384i	$0.433697\pi$
$32$	0	0
$33$	9.90833	1.72482
$34$	0	0
$35$	0	0
$36$	−4.60555	−0.767592
$37$	7.21110	1.18550	0.592749	−	0.805387i	$-0.298043\pi$
$37$	7.21110	1.18550	0.592749	+	0.805387i	$0.298043\pi$
$38$	0	0
$39$	−5.30278	−0.849124
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−9.60555	−1.46483	−0.732416	−	0.680857i	$-0.761608\pi$
$43$	−9.60555	−1.46483	−0.732416	+	0.680857i	$0.761608\pi$
$44$	8.60555	1.29734
$45$	−3.00000	−0.447214
$46$	0	0
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	−9.21110	−1.32951
$49$	0	0
$50$	0	0
$51$	−9.00000	−1.26025
$52$	−4.60555	−0.638675
$53$	9.90833	1.36101	0.680507	−	0.732742i	$-0.261760\pi$
$53$	9.90833	1.36101	0.680507	+	0.732742i	$0.261760\pi$
$54$	0	0
$55$	5.60555	0.755852
$56$	0	0
$57$	−8.30278	−1.09973
$58$	0	0
$59$	−0.394449	−0.0513528	−0.0256764	−	0.999670i	$-0.508174\pi$
$59$	−0.394449	−0.0513528	−0.0256764	+	0.999670i	$0.508174\pi$
$60$	−6.00000	−0.774597
$61$	−7.21110	−0.923287	−0.461644	−	0.887066i	$-0.652740\pi$
$61$	−7.21110	−0.923287	−0.461644	+	0.887066i	$0.652740\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−3.00000	−0.372104
$66$	0	0
$67$	−6.60555	−0.806997	−0.403498	−	0.914980i	$-0.632206\pi$
$67$	−6.60555	−0.806997	−0.403498	+	0.914980i	$0.632206\pi$
$68$	−7.81665	−0.947909
$69$	12.9083	1.55398
$70$	0	0
$71$	10.8167	1.28370	0.641850	−	0.766830i	$-0.278167\pi$
$71$	10.8167	1.28370	0.641850	+	0.766830i	$0.278167\pi$
$72$	0	0
$73$	−9.30278	−1.08881	−0.544404	−	0.838823i	$-0.683244\pi$
$73$	−9.30278	−1.08881	−0.544404	+	0.838823i	$0.683244\pi$
$74$	0	0
$75$	7.60555	0.878213
$76$	−7.21110	−0.827170
$77$	0	0
$78$	0	0
$79$	−17.4222	−1.96015	−0.980076	−	0.198625i	$-0.936352\pi$
$79$	−17.4222	−1.96015	−0.980076	+	0.198625i	$0.936352\pi$
$80$	−5.21110	−0.582619
$81$	−10.6056	−1.17839
$82$	0	0
$83$	−11.6056	−1.27387	−0.636937	−	0.770916i	$-0.719799\pi$
$83$	−11.6056	−1.27387	−0.636937	+	0.770916i	$0.719799\pi$
$84$	0	0
$85$	−5.09167	−0.552269
$86$	0	0
$87$	−12.9083	−1.38392
$88$	0	0
$89$	−0.394449	−0.0418115	−0.0209057	−	0.999781i	$-0.506655\pi$
$89$	−0.394449	−0.0418115	−0.0209057	+	0.999781i	$0.506655\pi$
$90$	0	0
$91$	0	0
$92$	11.2111	1.16884
$93$	−5.30278	−0.549872
$94$	0	0
$95$	−4.69722	−0.481925
$96$	0	0
$97$	−9.30278	−0.944554	−0.472277	−	0.881450i	$-0.656568\pi$
$97$	−9.30278	−0.944554	−0.472277	+	0.881450i	$0.656568\pi$
$98$	0	0
$99$	−9.90833	−0.995824
$100$	6.60555	0.660555
$101$	11.6056	1.15480	0.577398	−	0.816463i	$-0.304068\pi$
$101$	11.6056	1.15480	0.577398	+	0.816463i	$0.304068\pi$
$102$	0	0
$103$	1.78890	0.176265	0.0881327	−	0.996109i	$-0.471910\pi$
$103$	1.78890	0.176265	0.0881327	+	0.996109i	$0.471910\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.90833	−0.667853	−0.333927	−	0.942599i	$-0.608374\pi$
$107$	−6.90833	−0.667853	−0.333927	+	0.942599i	$0.608374\pi$
$108$	−3.21110	−0.308988
$109$	6.30278	0.603696	0.301848	−	0.953356i	$-0.402396\pi$
$109$	6.30278	0.603696	0.301848	+	0.953356i	$0.402396\pi$
$110$	0	0
$111$	−16.6056	−1.57613
$112$	0	0
$113$	−7.81665	−0.735329	−0.367664	−	0.929959i	$-0.619843\pi$
$113$	−7.81665	−0.735329	−0.367664	+	0.929959i	$0.619843\pi$
$114$	0	0
$115$	7.30278	0.680987
$116$	−11.2111	−1.04092
$117$	5.30278	0.490242
$118$	0	0
$119$	0	0
$120$	0	0
$121$	7.51388	0.683080
$122$	0	0
$123$	−2.30278	−0.207634
$124$	−4.60555	−0.413591
$125$	10.8167	0.967471
$126$	0	0
$127$	13.7250	1.21790	0.608948	−	0.793210i	$-0.291592\pi$
$127$	13.7250	1.21790	0.608948	+	0.793210i	$0.291592\pi$
$128$	0	0
$129$	22.1194	1.94751
$130$	0	0
$131$	−1.69722	−0.148287	−0.0741436	−	0.997248i	$-0.523622\pi$
$131$	−1.69722	−0.148287	−0.0741436	+	0.997248i	$0.523622\pi$
$132$	−19.8167	−1.72482
$133$	0	0
$134$	0	0
$135$	−2.09167	−0.180023
$136$	0	0
$137$	−1.18335	−0.101100	−0.0505500	−	0.998722i	$-0.516097\pi$
$137$	−1.18335	−0.101100	−0.0505500	+	0.998722i	$0.516097\pi$
$138$	0	0
$139$	4.90833	0.416319	0.208159	−	0.978095i	$-0.433253\pi$
$139$	4.90833	0.416319	0.208159	+	0.978095i	$0.433253\pi$
$140$	0	0
$141$	−20.7250	−1.74536
$142$	0	0
$143$	−9.90833	−0.828576
$144$	9.21110	0.767592
$145$	−7.30278	−0.606463
$146$	0	0
$147$	0	0
$148$	−14.4222	−1.18550
$149$	−1.81665	−0.148826	−0.0744130	−	0.997228i	$-0.523708\pi$
$149$	−1.81665	−0.148826	−0.0744130	+	0.997228i	$0.523708\pi$
$150$	0	0
$151$	11.1194	0.904886	0.452443	−	0.891793i	$-0.350553\pi$
$151$	11.1194	0.904886	0.452443	+	0.891793i	$0.350553\pi$
$152$	0	0
$153$	9.00000	0.727607
$154$	0	0
$155$	−3.00000	−0.240966
$156$	10.6056	0.849124
$157$	1.78890	0.142770	0.0713848	−	0.997449i	$-0.477258\pi$
$157$	1.78890	0.142770	0.0713848	+	0.997449i	$0.477258\pi$
$158$	0	0
$159$	−22.8167	−1.80948
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−15.7250	−1.23168	−0.615838	−	0.787873i	$-0.711182\pi$
$163$	−15.7250	−1.23168	−0.615838	+	0.787873i	$0.711182\pi$
$164$	−2.00000	−0.156174
$165$	−12.9083	−1.00491
$166$	0	0
$167$	−4.81665	−0.372724	−0.186362	−	0.982481i	$-0.559670\pi$
$167$	−4.81665	−0.372724	−0.186362	+	0.982481i	$0.559670\pi$
$168$	0	0
$169$	−7.69722	−0.592094
$170$	0	0
$171$	8.30278	0.634929
$172$	19.2111	1.46483
$173$	15.5139	1.17950	0.589749	−	0.807586i	$-0.299227\pi$
$173$	15.5139	1.17950	0.589749	+	0.807586i	$0.299227\pi$
$174$	0	0
$175$	0	0
$176$	−17.2111	−1.29734
$177$	0.908327	0.0682740
$178$	0	0
$179$	−13.6972	−1.02378	−0.511889	−	0.859051i	$-0.671054\pi$
$179$	−13.6972	−1.02378	−0.511889	+	0.859051i	$0.671054\pi$
$180$	6.00000	0.447214
$181$	−21.0278	−1.56298	−0.781490	−	0.623917i	$-0.785540\pi$
$181$	−21.0278	−1.56298	−0.781490	+	0.623917i	$0.785540\pi$
$182$	0	0
$183$	16.6056	1.22752
$184$	0	0
$185$	−9.39445	−0.690694
$186$	0	0
$187$	−16.8167	−1.22976
$188$	−18.0000	−1.31278
$189$	0	0
$190$	0	0
$191$	−19.3028	−1.39670	−0.698350	−	0.715757i	$-0.746082\pi$
$191$	−19.3028	−1.39670	−0.698350	+	0.715757i	$0.746082\pi$
$192$	18.4222	1.32951
$193$	−7.39445	−0.532264	−0.266132	−	0.963937i	$-0.585746\pi$
$193$	−7.39445	−0.532264	−0.266132	+	0.963937i	$0.585746\pi$
$194$	0	0
$195$	6.90833	0.494716
$196$	0	0
$197$	3.90833	0.278457	0.139228	−	0.990260i	$-0.455538\pi$
$197$	3.90833	0.278457	0.139228	+	0.990260i	$0.455538\pi$
$198$	0	0
$199$	18.2111	1.29095	0.645475	−	0.763781i	$-0.276659\pi$
$199$	18.2111	1.29095	0.645475	+	0.763781i	$0.276659\pi$
$200$	0	0
$201$	15.2111	1.07291
$202$	0	0
$203$	0	0
$204$	18.0000	1.26025
$205$	−1.30278	−0.0909898
$206$	0	0
$207$	−12.9083	−0.897191
$208$	9.21110	0.638675
$209$	−15.5139	−1.07312
$210$	0	0
$211$	−10.1194	−0.696650	−0.348325	−	0.937374i	$-0.613249\pi$
$211$	−10.1194	−0.696650	−0.348325	+	0.937374i	$0.613249\pi$
$212$	−19.8167	−1.36101
$213$	−24.9083	−1.70669
$214$	0	0
$215$	12.5139	0.853439
$216$	0	0
$217$	0	0
$218$	0	0
$219$	21.4222	1.44758
$220$	−11.2111	−0.755852
$221$	9.00000	0.605406
$222$	0	0
$223$	−8.51388	−0.570131	−0.285066	−	0.958508i	$-0.592015\pi$
$223$	−8.51388	−0.570131	−0.285066	+	0.958508i	$0.592015\pi$
$224$	0	0
$225$	−7.60555	−0.507037
$226$	0	0
$227$	−12.7889	−0.848829	−0.424414	−	0.905468i	$-0.639520\pi$
$227$	−12.7889	−0.848829	−0.424414	+	0.905468i	$0.639520\pi$
$228$	16.6056	1.09973
$229$	3.72498	0.246154	0.123077	−	0.992397i	$-0.460724\pi$
$229$	3.72498	0.246154	0.123077	+	0.992397i	$0.460724\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	23.3305	1.52843	0.764217	−	0.644959i	$-0.223126\pi$
$233$	23.3305	1.52843	0.764217	+	0.644959i	$0.223126\pi$
$234$	0	0
$235$	−11.7250	−0.764853
$236$	0.788897	0.0513528
$237$	40.1194	2.60604
$238$	0	0
$239$	22.8167	1.47589	0.737943	−	0.674863i	$-0.235797\pi$
$239$	22.8167	1.47589	0.737943	+	0.674863i	$0.235797\pi$
$240$	12.0000	0.774597
$241$	−3.81665	−0.245852	−0.122926	−	0.992416i	$-0.539228\pi$
$241$	−3.81665	−0.245852	−0.122926	+	0.992416i	$0.539228\pi$
$242$	0	0
$243$	19.6056	1.25770
$244$	14.4222	0.923287
$245$	0	0
$246$	0	0
$247$	8.30278	0.528293
$248$	0	0
$249$	26.7250	1.69363
$250$	0	0
$251$	21.9083	1.38284	0.691421	−	0.722452i	$-0.256985\pi$
$251$	21.9083	1.38284	0.691421	+	0.722452i	$0.256985\pi$
$252$	0	0
$253$	24.1194	1.51638
$254$	0	0
$255$	11.7250	0.734247
$256$	16.0000	1.00000
$257$	−15.9083	−0.992334	−0.496167	−	0.868227i	$-0.665260\pi$
$257$	−15.9083	−0.992334	−0.496167	+	0.868227i	$0.665260\pi$
$258$	0	0
$259$	0	0
$260$	6.00000	0.372104
$261$	12.9083	0.799005
$262$	0	0
$263$	16.3028	1.00527	0.502636	−	0.864498i	$-0.332364\pi$
$263$	16.3028	1.00527	0.502636	+	0.864498i	$0.332364\pi$
$264$	0	0
$265$	−12.9083	−0.792952
$266$	0	0
$267$	0.908327	0.0555887
$268$	13.2111	0.806997
$269$	−7.30278	−0.445258	−0.222629	−	0.974903i	$-0.571464\pi$
$269$	−7.30278	−0.445258	−0.222629	+	0.974903i	$0.571464\pi$
$270$	0	0
$271$	−24.8167	−1.50750	−0.753752	−	0.657159i	$-0.771758\pi$
$271$	−24.8167	−1.50750	−0.753752	+	0.657159i	$0.771758\pi$
$272$	15.6333	0.947909
$273$	0	0
$274$	0	0
$275$	14.2111	0.856962
$276$	−25.8167	−1.55398
$277$	−6.72498	−0.404065	−0.202032	−	0.979379i	$-0.564755\pi$
$277$	−6.72498	−0.404065	−0.202032	+	0.979379i	$0.564755\pi$
$278$	0	0
$279$	5.30278	0.317469
$280$	0	0
$281$	−6.51388	−0.388585	−0.194293	−	0.980944i	$-0.562241\pi$
$281$	−6.51388	−0.388585	−0.194293	+	0.980944i	$0.562241\pi$
$282$	0	0
$283$	−9.42221	−0.560092	−0.280046	−	0.959987i	$-0.590350\pi$
$283$	−9.42221	−0.560092	−0.280046	+	0.959987i	$0.590350\pi$
$284$	−21.6333	−1.28370
$285$	10.8167	0.640723
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.72498	−0.101469
$290$	0	0
$291$	21.4222	1.25579
$292$	18.6056	1.08881
$293$	15.7889	0.922397	0.461199	−	0.887297i	$-0.347420\pi$
$293$	15.7889	0.922397	0.461199	+	0.887297i	$0.347420\pi$
$294$	0	0
$295$	0.513878	0.0299191
$296$	0	0
$297$	−6.90833	−0.400862
$298$	0	0
$299$	−12.9083	−0.746508
$300$	−15.2111	−0.878213
$301$	0	0
$302$	0	0
$303$	−26.7250	−1.53531
$304$	14.4222	0.827170
$305$	9.39445	0.537925
$306$	0	0
$307$	32.5416	1.85725	0.928625	−	0.371021i	$-0.120992\pi$
$307$	32.5416	1.85725	0.928625	+	0.371021i	$0.120992\pi$
$308$	0	0
$309$	−4.11943	−0.234346
$310$	0	0
$311$	−22.8167	−1.29381	−0.646907	−	0.762569i	$-0.723938\pi$
$311$	−22.8167	−1.29381	−0.646907	+	0.762569i	$0.723938\pi$
$312$	0	0
$313$	−19.7250	−1.11492	−0.557461	−	0.830203i	$-0.688224\pi$
$313$	−19.7250	−1.11492	−0.557461	+	0.830203i	$0.688224\pi$
$314$	0	0
$315$	0	0
$316$	34.8444	1.96015
$317$	−15.9083	−0.893501	−0.446750	−	0.894659i	$-0.647419\pi$
$317$	−15.9083	−0.893501	−0.446750	+	0.894659i	$0.647419\pi$
$318$	0	0
$319$	−24.1194	−1.35043
$320$	10.4222	0.582619
$321$	15.9083	0.887916
$322$	0	0
$323$	14.0917	0.784082
$324$	21.2111	1.17839
$325$	−7.60555	−0.421880
$326$	0	0
$327$	−14.5139	−0.802619
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0.0277564	0.00152563	0.000762814	−	1.00000i	$-0.499757\pi$
$331$	0.0277564	0.00152563	0.000762814		1.00000i	$0.499757\pi$
$332$	23.2111	1.27387
$333$	16.6056	0.909979
$334$	0	0
$335$	8.60555	0.470171
$336$	0	0
$337$	−15.7250	−0.856594	−0.428297	−	0.903638i	$-0.640886\pi$
$337$	−15.7250	−0.856594	−0.428297	+	0.903638i	$0.640886\pi$
$338$	0	0
$339$	18.0000	0.977626
$340$	10.1833	0.552269
$341$	−9.90833	−0.536566
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−16.8167	−0.905378
$346$	0	0
$347$	−10.1833	−0.546671	−0.273335	−	0.961919i	$-0.588127\pi$
$347$	−10.1833	−0.546671	−0.273335	+	0.961919i	$0.588127\pi$
$348$	25.8167	1.38392
$349$	5.18335	0.277458	0.138729	−	0.990330i	$-0.455698\pi$
$349$	5.18335	0.277458	0.138729	+	0.990330i	$0.455698\pi$
$350$	0	0
$351$	3.69722	0.197343
$352$	0	0
$353$	24.6333	1.31110	0.655549	−	0.755152i	$-0.272437\pi$
$353$	24.6333	1.31110	0.655549	+	0.755152i	$0.272437\pi$
$354$	0	0
$355$	−14.0917	−0.747908
$356$	0.788897	0.0418115
$357$	0	0
$358$	0	0
$359$	−27.6333	−1.45843	−0.729215	−	0.684285i	$-0.760115\pi$
$359$	−27.6333	−1.45843	−0.729215	+	0.684285i	$0.760115\pi$
$360$	0	0
$361$	−6.00000	−0.315789
$362$	0	0
$363$	−17.3028	−0.908160
$364$	0	0
$365$	12.1194	0.634360
$366$	0	0
$367$	16.7889	0.876373	0.438187	−	0.898884i	$-0.355621\pi$
$367$	16.7889	0.876373	0.438187	+	0.898884i	$0.355621\pi$
$368$	−22.4222	−1.16884
$369$	2.30278	0.119878
$370$	0	0
$371$	0	0
$372$	10.6056	0.549872
$373$	22.3305	1.15623	0.578116	−	0.815955i	$-0.303788\pi$
$373$	22.3305	1.15623	0.578116	+	0.815955i	$0.303788\pi$
$374$	0	0
$375$	−24.9083	−1.28626
$376$	0	0
$377$	12.9083	0.664813
$378$	0	0
$379$	−7.78890	−0.400089	−0.200044	−	0.979787i	$-0.564109\pi$
$379$	−7.78890	−0.400089	−0.200044	+	0.979787i	$0.564109\pi$
$380$	9.39445	0.481925
$381$	−31.6056	−1.61920
$382$	0	0
$383$	−5.21110	−0.266275	−0.133137	−	0.991098i	$-0.542505\pi$
$383$	−5.21110	−0.266275	−0.133137	+	0.991098i	$0.542505\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−22.1194	−1.12439
$388$	18.6056	0.944554
$389$	−0.119429	−0.00605531	−0.00302766	−	0.999995i	$-0.500964\pi$
$389$	−0.119429	−0.00605531	−0.00302766	+	0.999995i	$0.500964\pi$
$390$	0	0
$391$	−21.9083	−1.10795
$392$	0	0
$393$	3.90833	0.197149
$394$	0	0
$395$	22.6972	1.14202
$396$	19.8167	0.995824
$397$	36.7250	1.84317	0.921587	−	0.388172i	$-0.126893\pi$
$397$	36.7250	1.84317	0.921587	+	0.388172i	$0.126893\pi$
$398$	0	0
$399$	0	0
$400$	−13.2111	−0.660555
$401$	6.39445	0.319324	0.159662	−	0.987172i	$-0.448960\pi$
$401$	6.39445	0.319324	0.159662	+	0.987172i	$0.448960\pi$
$402$	0	0
$403$	5.30278	0.264150
$404$	−23.2111	−1.15480
$405$	13.8167	0.686555
$406$	0	0
$407$	−31.0278	−1.53799
$408$	0	0
$409$	28.0000	1.38451	0.692255	−	0.721653i	$-0.256617\pi$
$409$	28.0000	1.38451	0.692255	+	0.721653i	$0.256617\pi$
$410$	0	0
$411$	2.72498	0.134413
$412$	−3.57779	−0.176265
$413$	0	0
$414$	0	0
$415$	15.1194	0.742184
$416$	0	0
$417$	−11.3028	−0.553499
$418$	0	0
$419$	30.1194	1.47143	0.735715	−	0.677291i	$-0.236846\pi$
$419$	30.1194	1.47143	0.735715	+	0.677291i	$0.236846\pi$
$420$	0	0
$421$	4.09167	0.199416	0.0997080	−	0.995017i	$-0.468209\pi$
$421$	4.09167	0.199416	0.0997080	+	0.995017i	$0.468209\pi$
$422$	0	0
$423$	20.7250	1.00768
$424$	0	0
$425$	−12.9083	−0.626146
$426$	0	0
$427$	0	0
$428$	13.8167	0.667853
$429$	22.8167	1.10160
$430$	0	0
$431$	5.48612	0.264257	0.132129	−	0.991233i	$-0.457819\pi$
$431$	5.48612	0.264257	0.132129	+	0.991233i	$0.457819\pi$
$432$	6.42221	0.308988
$433$	−34.6056	−1.66304	−0.831518	−	0.555497i	$-0.812528\pi$
$433$	−34.6056	−1.66304	−0.831518	+	0.555497i	$0.812528\pi$
$434$	0	0
$435$	16.8167	0.806297
$436$	−12.6056	−0.603696
$437$	−20.2111	−0.966828
$438$	0	0
$439$	−37.2111	−1.77599	−0.887995	−	0.459854i	$-0.847902\pi$
$439$	−37.2111	−1.77599	−0.887995	+	0.459854i	$0.847902\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6.11943	0.290743	0.145371	−	0.989377i	$-0.453562\pi$
$443$	6.11943	0.290743	0.145371	+	0.989377i	$0.453562\pi$
$444$	33.2111	1.57613
$445$	0.513878	0.0243602
$446$	0	0
$447$	4.18335	0.197865
$448$	0	0
$449$	39.6333	1.87041	0.935206	−	0.354105i	$-0.115214\pi$
$449$	39.6333	1.87041	0.935206	+	0.354105i	$0.115214\pi$
$450$	0	0
$451$	−4.30278	−0.202610
$452$	15.6333	0.735329
$453$	−25.6056	−1.20305
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−20.0278	−0.936859	−0.468429	−	0.883501i	$-0.655180\pi$
$457$	−20.0278	−0.936859	−0.468429	+	0.883501i	$0.655180\pi$
$458$	0	0
$459$	6.27502	0.292893
$460$	−14.6056	−0.680987
$461$	9.00000	0.419172	0.209586	−	0.977790i	$-0.432788\pi$
$461$	9.00000	0.419172	0.209586	+	0.977790i	$0.432788\pi$
$462$	0	0
$463$	−14.6972	−0.683038	−0.341519	−	0.939875i	$-0.610941\pi$
$463$	−14.6972	−0.683038	−0.341519	+	0.939875i	$0.610941\pi$
$464$	22.4222	1.04092
$465$	6.90833	0.320366
$466$	0	0
$467$	−31.5416	−1.45957	−0.729786	−	0.683675i	$-0.760380\pi$
$467$	−31.5416	−1.45957	−0.729786	+	0.683675i	$0.760380\pi$
$468$	−10.6056	−0.490242
$469$	0	0
$470$	0	0
$471$	−4.11943	−0.189813
$472$	0	0
$473$	41.3305	1.90038
$474$	0	0
$475$	−11.9083	−0.546392
$476$	0	0
$477$	22.8167	1.04470
$478$	0	0
$479$	−30.7527	−1.40513	−0.702564	−	0.711620i	$-0.747962\pi$
$479$	−30.7527	−1.40513	−0.702564	+	0.711620i	$0.747962\pi$
$480$	0	0
$481$	16.6056	0.757148
$482$	0	0
$483$	0	0
$484$	−15.0278	−0.683080
$485$	12.1194	0.550315
$486$	0	0
$487$	−32.3028	−1.46378	−0.731889	−	0.681424i	$-0.761361\pi$
$487$	−32.3028	−1.46378	−0.731889	+	0.681424i	$0.761361\pi$
$488$	0	0
$489$	36.2111	1.63752
$490$	0	0
$491$	−12.1194	−0.546942	−0.273471	−	0.961880i	$-0.588172\pi$
$491$	−12.1194	−0.546942	−0.273471	+	0.961880i	$0.588172\pi$
$492$	4.60555	0.207634
$493$	21.9083	0.986701
$494$	0	0
$495$	12.9083	0.580186
$496$	9.21110	0.413591
$497$	0	0
$498$	0	0
$499$	16.8444	0.754059	0.377030	−	0.926201i	$-0.376945\pi$
$499$	16.8444	0.754059	0.377030	+	0.926201i	$0.376945\pi$
$500$	−21.6333	−0.967471
$501$	11.0917	0.495539
$502$	0	0
$503$	−28.8167	−1.28487	−0.642436	−	0.766340i	$-0.722076\pi$
$503$	−28.8167	−1.28487	−0.642436	+	0.766340i	$0.722076\pi$
$504$	0	0
$505$	−15.1194	−0.672806
$506$	0	0
$507$	17.7250	0.787194
$508$	−27.4500	−1.21790
$509$	−23.7250	−1.05159	−0.525796	−	0.850611i	$-0.676232\pi$
$509$	−23.7250	−1.05159	−0.525796	+	0.850611i	$0.676232\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	5.78890	0.255586
$514$	0	0
$515$	−2.33053	−0.102696
$516$	−44.2389	−1.94751
$517$	−38.7250	−1.70312
$518$	0	0
$519$	−35.7250	−1.56815
$520$	0	0
$521$	−44.0555	−1.93011	−0.965054	−	0.262053i	$-0.915601\pi$
$521$	−44.0555	−1.93011	−0.965054	+	0.262053i	$0.915601\pi$
$522$	0	0
$523$	−17.0000	−0.743358	−0.371679	−	0.928361i	$-0.621218\pi$
$523$	−17.0000	−0.743358	−0.371679	+	0.928361i	$0.621218\pi$
$524$	3.39445	0.148287
$525$	0	0
$526$	0	0
$527$	9.00000	0.392046
$528$	39.6333	1.72482
$529$	8.42221	0.366183
$530$	0	0
$531$	−0.908327	−0.0394180
$532$	0	0
$533$	2.30278	0.0997443
$534$	0	0
$535$	9.00000	0.389104
$536$	0	0
$537$	31.5416	1.36112
$538$	0	0
$539$	0	0
$540$	4.18335	0.180023
$541$	−22.7889	−0.979771	−0.489886	−	0.871787i	$-0.662961\pi$
$541$	−22.7889	−0.979771	−0.489886	+	0.871787i	$0.662961\pi$
$542$	0	0
$543$	48.4222	2.07800
$544$	0	0
$545$	−8.21110	−0.351725
$546$	0	0
$547$	31.7250	1.35646	0.678231	−	0.734849i	$-0.262747\pi$
$547$	31.7250	1.35646	0.678231	+	0.734849i	$0.262747\pi$
$548$	2.36669	0.101100
$549$	−16.6056	−0.708708
$550$	0	0
$551$	20.2111	0.861022
$552$	0	0
$553$	0	0
$554$	0	0
$555$	21.6333	0.918283
$556$	−9.81665	−0.416319
$557$	9.00000	0.381342	0.190671	−	0.981654i	$-0.438934\pi$
$557$	9.00000	0.381342	0.190671	+	0.981654i	$0.438934\pi$
$558$	0	0
$559$	−22.1194	−0.935552
$560$	0	0
$561$	38.7250	1.63497
$562$	0	0
$563$	−9.78890	−0.412553	−0.206276	−	0.978494i	$-0.566135\pi$
$563$	−9.78890	−0.412553	−0.206276	+	0.978494i	$0.566135\pi$
$564$	41.4500	1.74536
$565$	10.1833	0.428417
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−36.2389	−1.51921	−0.759606	−	0.650384i	$-0.774608\pi$
$569$	−36.2389	−1.51921	−0.759606	+	0.650384i	$0.774608\pi$
$570$	0	0
$571$	24.0278	1.00553	0.502765	−	0.864423i	$-0.332316\pi$
$571$	24.0278	1.00553	0.502765	+	0.864423i	$0.332316\pi$
$572$	19.8167	0.828576
$573$	44.4500	1.85692
$574$	0	0
$575$	18.5139	0.772082
$576$	−18.4222	−0.767592
$577$	30.0555	1.25123	0.625614	−	0.780133i	$-0.284849\pi$
$577$	30.0555	1.25123	0.625614	+	0.780133i	$0.284849\pi$
$578$	0	0
$579$	17.0278	0.707649
$580$	14.6056	0.606463
$581$	0	0
$582$	0	0
$583$	−42.6333	−1.76569
$584$	0	0
$585$	−6.90833	−0.285624
$586$	0	0
$587$	−35.4500	−1.46318	−0.731588	−	0.681747i	$-0.761221\pi$
$587$	−35.4500	−1.46318	−0.731588	+	0.681747i	$0.761221\pi$
$588$	0	0
$589$	8.30278	0.342110
$590$	0	0
$591$	−9.00000	−0.370211
$592$	28.8444	1.18550
$593$	−17.2111	−0.706775	−0.353388	−	0.935477i	$-0.614970\pi$
$593$	−17.2111	−0.706775	−0.353388	+	0.935477i	$0.614970\pi$
$594$	0	0
$595$	0	0
$596$	3.63331	0.148826
$597$	−41.9361	−1.71633
$598$	0	0
$599$	−6.39445	−0.261270	−0.130635	−	0.991431i	$-0.541702\pi$
$599$	−6.39445	−0.261270	−0.130635	+	0.991431i	$0.541702\pi$
$600$	0	0
$601$	−15.1472	−0.617867	−0.308933	−	0.951084i	$-0.599972\pi$
$601$	−15.1472	−0.617867	−0.308933	+	0.951084i	$0.599972\pi$
$602$	0	0
$603$	−15.2111	−0.619444
$604$	−22.2389	−0.904886
$605$	−9.78890	−0.397975
$606$	0	0
$607$	−3.81665	−0.154913	−0.0774566	−	0.996996i	$-0.524680\pi$
$607$	−3.81665	−0.154913	−0.0774566	+	0.996996i	$0.524680\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	20.7250	0.838443
$612$	−18.0000	−0.727607
$613$	−7.00000	−0.282727	−0.141364	−	0.989958i	$-0.545149\pi$
$613$	−7.00000	−0.282727	−0.141364	+	0.989958i	$0.545149\pi$
$614$	0	0
$615$	3.00000	0.120972
$616$	0	0
$617$	−16.5416	−0.665941	−0.332971	−	0.942937i	$-0.608051\pi$
$617$	−16.5416	−0.665941	−0.332971	+	0.942937i	$0.608051\pi$
$618$	0	0
$619$	−26.5139	−1.06568	−0.532841	−	0.846215i	$-0.678876\pi$
$619$	−26.5139	−1.06568	−0.532841	+	0.846215i	$0.678876\pi$
$620$	6.00000	0.240966
$621$	−9.00000	−0.361158
$622$	0	0
$623$	0	0
$624$	−21.2111	−0.849124
$625$	2.42221	0.0968882
$626$	0	0
$627$	35.7250	1.42672
$628$	−3.57779	−0.142770
$629$	28.1833	1.12374
$630$	0	0
$631$	−42.8444	−1.70561	−0.852805	−	0.522230i	$-0.825100\pi$
$631$	−42.8444	−1.70561	−0.852805	+	0.522230i	$0.825100\pi$
$632$	0	0
$633$	23.3028	0.926202
$634$	0	0
$635$	−17.8806	−0.709569
$636$	45.6333	1.80948
$637$	0	0
$638$	0	0
$639$	24.9083	0.985358
$640$	0	0
$641$	−33.0000	−1.30342	−0.651711	−	0.758468i	$-0.725948\pi$
$641$	−33.0000	−1.30342	−0.651711	+	0.758468i	$0.725948\pi$
$642$	0	0
$643$	−8.00000	−0.315489	−0.157745	−	0.987480i	$-0.550422\pi$
$643$	−8.00000	−0.315489	−0.157745	+	0.987480i	$0.550422\pi$
$644$	0	0
$645$	−28.8167	−1.13465
$646$	0	0
$647$	15.6333	0.614609	0.307304	−	0.951611i	$-0.400573\pi$
$647$	15.6333	0.614609	0.307304	+	0.951611i	$0.400573\pi$
$648$	0	0
$649$	1.69722	0.0666219
$650$	0	0
$651$	0	0
$652$	31.4500	1.23168
$653$	37.8167	1.47988	0.739940	−	0.672673i	$-0.234854\pi$
$653$	37.8167	1.47988	0.739940	+	0.672673i	$0.234854\pi$
$654$	0	0
$655$	2.21110	0.0863949
$656$	4.00000	0.156174
$657$	−21.4222	−0.835760
$658$	0	0
$659$	−30.2389	−1.17794	−0.588969	−	0.808155i	$-0.700466\pi$
$659$	−30.2389	−1.17794	−0.588969	+	0.808155i	$0.700466\pi$
$660$	25.8167	1.00491
$661$	−43.2111	−1.68072	−0.840359	−	0.542031i	$-0.817656\pi$
$661$	−43.2111	−1.68072	−0.840359	+	0.542031i	$0.817656\pi$
$662$	0	0
$663$	−20.7250	−0.804892
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−31.4222	−1.21667
$668$	9.63331	0.372724
$669$	19.6056	0.757994
$670$	0	0
$671$	31.0278	1.19781
$672$	0	0
$673$	−36.8444	−1.42025	−0.710124	−	0.704077i	$-0.751361\pi$
$673$	−36.8444	−1.42025	−0.710124	+	0.704077i	$0.751361\pi$
$674$	0	0
$675$	−5.30278	−0.204104
$676$	15.3944	0.592094
$677$	27.0000	1.03769	0.518847	−	0.854867i	$-0.326361\pi$
$677$	27.0000	1.03769	0.518847	+	0.854867i	$0.326361\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	29.4500	1.12852
$682$	0	0
$683$	27.7889	1.06331	0.531656	−	0.846960i	$-0.321570\pi$
$683$	27.7889	1.06331	0.531656	+	0.846960i	$0.321570\pi$
$684$	−16.6056	−0.634929
$685$	1.54163	0.0589028
$686$	0	0
$687$	−8.57779	−0.327263
$688$	−38.4222	−1.46483
$689$	22.8167	0.869245
$690$	0	0
$691$	17.6972	0.673234	0.336617	−	0.941642i	$-0.390717\pi$
$691$	17.6972	0.673234	0.336617	+	0.941642i	$0.390717\pi$
$692$	−31.0278	−1.17950
$693$	0	0
$694$	0	0
$695$	−6.39445	−0.242555
$696$	0	0
$697$	3.90833	0.148038
$698$	0	0
$699$	−53.7250	−2.03207
$700$	0	0
$701$	−14.8444	−0.560666	−0.280333	−	0.959903i	$-0.590445\pi$
$701$	−14.8444	−0.560666	−0.280333	+	0.959903i	$0.590445\pi$
$702$	0	0
$703$	26.0000	0.980609
$704$	34.4222	1.29734
$705$	27.0000	1.01688
$706$	0	0
$707$	0	0
$708$	−1.81665	−0.0682740
$709$	−22.3944	−0.841041	−0.420521	−	0.907283i	$-0.638153\pi$
$709$	−22.3944	−0.841041	−0.420521	+	0.907283i	$0.638153\pi$
$710$	0	0
$711$	−40.1194	−1.50460
$712$	0	0
$713$	−12.9083	−0.483421
$714$	0	0
$715$	12.9083	0.482744
$716$	27.3944	1.02378
$717$	−52.5416	−1.96220
$718$	0	0
$719$	30.1194	1.12327	0.561633	−	0.827387i	$-0.310173\pi$
$719$	30.1194	1.12327	0.561633	+	0.827387i	$0.310173\pi$
$720$	−12.0000	−0.447214
$721$	0	0
$722$	0	0
$723$	8.78890	0.326863
$724$	42.0555	1.56298
$725$	−18.5139	−0.687588
$726$	0	0
$727$	41.5416	1.54069	0.770347	−	0.637625i	$-0.220083\pi$
$727$	41.5416	1.54069	0.770347	+	0.637625i	$0.220083\pi$
$728$	0	0
$729$	−13.3305	−0.493723
$730$	0	0
$731$	−37.5416	−1.38853
$732$	−33.2111	−1.22752
$733$	49.2389	1.81868	0.909339	−	0.416055i	$-0.136588\pi$
$733$	49.2389	1.81868	0.909339	+	0.416055i	$0.136588\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	28.4222	1.04695
$738$	0	0
$739$	−1.78890	−0.0658057	−0.0329028	−	0.999459i	$-0.510475\pi$
$739$	−1.78890	−0.0658057	−0.0329028	+	0.999459i	$0.510475\pi$
$740$	18.7889	0.690694
$741$	−19.1194	−0.702370
$742$	0	0
$743$	33.3583	1.22380	0.611898	−	0.790936i	$-0.290406\pi$
$743$	33.3583	1.22380	0.611898	+	0.790936i	$0.290406\pi$
$744$	0	0
$745$	2.36669	0.0867089
$746$	0	0
$747$	−26.7250	−0.977816
$748$	33.6333	1.22976
$749$	0	0
$750$	0	0
$751$	−44.9361	−1.63974	−0.819870	−	0.572549i	$-0.805955\pi$
$751$	−44.9361	−1.63974	−0.819870	+	0.572549i	$0.805955\pi$
$752$	36.0000	1.31278
$753$	−50.4500	−1.83850
$754$	0	0
$755$	−14.4861	−0.527204
$756$	0	0
$757$	−42.0555	−1.52853	−0.764267	−	0.644900i	$-0.776899\pi$
$757$	−42.0555	−1.52853	−0.764267	+	0.644900i	$0.776899\pi$
$758$	0	0
$759$	−55.5416	−2.01603
$760$	0	0
$761$	−27.9083	−1.01168	−0.505838	−	0.862628i	$-0.668817\pi$
$761$	−27.9083	−1.01168	−0.505838	+	0.862628i	$0.668817\pi$
$762$	0	0
$763$	0	0
$764$	38.6056	1.39670
$765$	−11.7250	−0.423918
$766$	0	0
$767$	−0.908327	−0.0327978
$768$	−36.8444	−1.32951
$769$	−6.02776	−0.217366	−0.108683	−	0.994076i	$-0.534663\pi$
$769$	−6.02776	−0.217366	−0.108683	+	0.994076i	$0.534663\pi$
$770$	0	0
$771$	36.6333	1.31932
$772$	14.7889	0.532264
$773$	−9.78890	−0.352082	−0.176041	−	0.984383i	$-0.556329\pi$
$773$	−9.78890	−0.352082	−0.176041	+	0.984383i	$0.556329\pi$
$774$	0	0
$775$	−7.60555	−0.273199
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.60555	0.129182
$780$	−13.8167	−0.494716
$781$	−46.5416	−1.66539
$782$	0	0
$783$	9.00000	0.321634
$784$	0	0
$785$	−2.33053	−0.0831803
$786$	0	0
$787$	−24.8167	−0.884618	−0.442309	−	0.896863i	$-0.645841\pi$
$787$	−24.8167	−0.884618	−0.442309	+	0.896863i	$0.645841\pi$
$788$	−7.81665	−0.278457
$789$	−37.5416	−1.33652
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−16.6056	−0.589680
$794$	0	0
$795$	29.7250	1.05424
$796$	−36.4222	−1.29095
$797$	28.8167	1.02074	0.510369	−	0.859955i	$-0.329509\pi$
$797$	28.8167	1.02074	0.510369	+	0.859955i	$0.329509\pi$
$798$	0	0
$799$	35.1749	1.24440
$800$	0	0
$801$	−0.908327	−0.0320942
$802$	0	0
$803$	40.0278	1.41255
$804$	−30.4222	−1.07291
$805$	0	0
$806$	0	0
$807$	16.8167	0.591974
$808$	0	0
$809$	34.8167	1.22409	0.612044	−	0.790824i	$-0.290347\pi$
$809$	34.8167	1.22409	0.612044	+	0.790824i	$0.290347\pi$
$810$	0	0
$811$	−10.3305	−0.362754	−0.181377	−	0.983414i	$-0.558055\pi$
$811$	−10.3305	−0.362754	−0.181377	+	0.983414i	$0.558055\pi$
$812$	0	0
$813$	57.1472	2.00424
$814$	0	0
$815$	20.4861	0.717598
$816$	−36.0000	−1.26025
$817$	−34.6333	−1.21167
$818$	0	0
$819$	0	0
$820$	2.60555	0.0909898
$821$	27.7527	0.968577	0.484289	−	0.874908i	$-0.339078\pi$
$821$	27.7527	0.968577	0.484289	+	0.874908i	$0.339078\pi$
$822$	0	0
$823$	−12.4861	−0.435239	−0.217619	−	0.976034i	$-0.569829\pi$
$823$	−12.4861	−0.435239	−0.217619	+	0.976034i	$0.569829\pi$
$824$	0	0
$825$	−32.7250	−1.13934
$826$	0	0
$827$	−22.1833	−0.771391	−0.385695	−	0.922626i	$-0.626038\pi$
$827$	−22.1833	−0.771391	−0.385695	+	0.922626i	$0.626038\pi$
$828$	25.8167	0.897191
$829$	−0.816654	−0.0283636	−0.0141818	−	0.999899i	$-0.504514\pi$
$829$	−0.816654	−0.0283636	−0.0141818	+	0.999899i	$0.504514\pi$
$830$	0	0
$831$	15.4861	0.537208
$832$	−18.4222	−0.638675
$833$	0	0
$834$	0	0
$835$	6.27502	0.217156
$836$	31.0278	1.07312
$837$	3.69722	0.127795
$838$	0	0
$839$	5.60555	0.193525	0.0967626	−	0.995307i	$-0.469151\pi$
$839$	5.60555	0.193525	0.0967626	+	0.995307i	$0.469151\pi$
$840$	0	0
$841$	2.42221	0.0835243
$842$	0	0
$843$	15.0000	0.516627
$844$	20.2389	0.696650
$845$	10.0278	0.344965
$846$	0	0
$847$	0	0
$848$	39.6333	1.36101
$849$	21.6972	0.744647
$850$	0	0
$851$	−40.4222	−1.38566
$852$	49.8167	1.70669
$853$	2.02776	0.0694291	0.0347145	−	0.999397i	$-0.488948\pi$
$853$	2.02776	0.0694291	0.0347145	+	0.999397i	$0.488948\pi$
$854$	0	0
$855$	−10.8167	−0.369922
$856$	0	0
$857$	−48.5139	−1.65720	−0.828601	−	0.559839i	$-0.810863\pi$
$857$	−48.5139	−1.65720	−0.828601	+	0.559839i	$0.810863\pi$
$858$	0	0
$859$	−16.4861	−0.562499	−0.281250	−	0.959635i	$-0.590749\pi$
$859$	−16.4861	−0.562499	−0.281250	+	0.959635i	$0.590749\pi$
$860$	−25.0278	−0.853439
$861$	0	0
$862$	0	0
$863$	−17.8444	−0.607431	−0.303715	−	0.952763i	$-0.598227\pi$
$863$	−17.8444	−0.607431	−0.303715	+	0.952763i	$0.598227\pi$
$864$	0	0
$865$	−20.2111	−0.687198
$866$	0	0
$867$	3.97224	0.134904
$868$	0	0
$869$	74.9638	2.54297
$870$	0	0
$871$	−15.2111	−0.515409
$872$	0	0
$873$	−21.4222	−0.725032
$874$	0	0
$875$	0	0
$876$	−42.8444	−1.44758
$877$	−31.2389	−1.05486	−0.527431	−	0.849598i	$-0.676845\pi$
$877$	−31.2389	−1.05486	−0.527431	+	0.849598i	$0.676845\pi$
$878$	0	0
$879$	−36.3583	−1.22633
$880$	22.4222	0.755852
$881$	−45.9083	−1.54669	−0.773345	−	0.633985i	$-0.781418\pi$
$881$	−45.9083	−1.54669	−0.773345	+	0.633985i	$0.781418\pi$
$882$	0	0
$883$	14.7527	0.496469	0.248235	−	0.968700i	$-0.420150\pi$
$883$	14.7527	0.496469	0.248235	+	0.968700i	$0.420150\pi$
$884$	−18.0000	−0.605406
$885$	−1.18335	−0.0397777
$886$	0	0
$887$	−17.2111	−0.577892	−0.288946	−	0.957345i	$-0.593305\pi$
$887$	−17.2111	−0.577892	−0.288946	+	0.957345i	$0.593305\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	45.6333	1.52877
$892$	17.0278	0.570131
$893$	32.4500	1.08590
$894$	0	0
$895$	17.8444	0.596473
$896$	0	0
$897$	29.7250	0.992488
$898$	0	0
$899$	12.9083	0.430517
$900$	15.2111	0.507037
$901$	38.7250	1.29012
$902$	0	0
$903$	0	0
$904$	0	0
$905$	27.3944	0.910622
$906$	0	0
$907$	−17.9722	−0.596759	−0.298379	−	0.954447i	$-0.596446\pi$
$907$	−17.9722	−0.596759	−0.298379	+	0.954447i	$0.596446\pi$
$908$	25.5778	0.848829
$909$	26.7250	0.886412
$910$	0	0
$911$	−17.6056	−0.583298	−0.291649	−	0.956525i	$-0.594204\pi$
$911$	−17.6056	−0.583298	−0.291649	+	0.956525i	$0.594204\pi$
$912$	−33.2111	−1.09973
$913$	49.9361	1.65264
$914$	0	0
$915$	−21.6333	−0.715175
$916$	−7.44996	−0.246154
$917$	0	0
$918$	0	0
$919$	−19.3944	−0.639764	−0.319882	−	0.947457i	$-0.603643\pi$
$919$	−19.3944	−0.639764	−0.319882	+	0.947457i	$0.603643\pi$
$920$	0	0
$921$	−74.9361	−2.46923
$922$	0	0
$923$	24.9083	0.819868
$924$	0	0
$925$	−23.8167	−0.783087
$926$	0	0
$927$	4.11943	0.135300
$928$	0	0
$929$	55.0278	1.80540	0.902701	−	0.430268i	$-0.141581\pi$
$929$	55.0278	1.80540	0.902701	+	0.430268i	$0.141581\pi$
$930$	0	0
$931$	0	0
$932$	−46.6611	−1.52843
$933$	52.5416	1.72014
$934$	0	0
$935$	21.9083	0.716479
$936$	0	0
$937$	30.0555	0.981871	0.490935	−	0.871196i	$-0.336655\pi$
$937$	30.0555	0.981871	0.490935	+	0.871196i	$0.336655\pi$
$938$	0	0
$939$	45.4222	1.48230
$940$	23.4500	0.764853
$941$	41.4500	1.35123	0.675615	−	0.737255i	$-0.263878\pi$
$941$	41.4500	1.35123	0.675615	+	0.737255i	$0.263878\pi$
$942$	0	0
$943$	−5.60555	−0.182542
$944$	−1.57779	−0.0513528
$945$	0	0
$946$	0	0
$947$	−39.7889	−1.29297	−0.646483	−	0.762929i	$-0.723761\pi$
$947$	−39.7889	−1.29297	−0.646483	+	0.762929i	$0.723761\pi$
$948$	−80.2389	−2.60604
$949$	−21.4222	−0.695394
$950$	0	0
$951$	36.6333	1.18792
$952$	0	0
$953$	−19.5416	−0.633016	−0.316508	−	0.948590i	$-0.602510\pi$
$953$	−19.5416	−0.633016	−0.316508	+	0.948590i	$0.602510\pi$
$954$	0	0
$955$	25.1472	0.813744
$956$	−45.6333	−1.47589
$957$	55.5416	1.79541
$958$	0	0
$959$	0	0
$960$	−24.0000	−0.774597
$961$	−25.6972	−0.828943
$962$	0	0
$963$	−15.9083	−0.512639
$964$	7.63331	0.245852
$965$	9.63331	0.310107
$966$	0	0
$967$	−52.5139	−1.68873	−0.844366	−	0.535766i	$-0.820023\pi$
$967$	−52.5139	−1.68873	−0.844366	+	0.535766i	$0.820023\pi$
$968$	0	0
$969$	−32.4500	−1.04244
$970$	0	0
$971$	33.7527	1.08318	0.541588	−	0.840644i	$-0.317823\pi$
$971$	33.7527	1.08318	0.541588	+	0.840644i	$0.317823\pi$
$972$	−39.2111	−1.25770
$973$	0	0
$974$	0	0
$975$	17.5139	0.560893
$976$	−28.8444	−0.923287
$977$	18.6333	0.596132	0.298066	−	0.954545i	$-0.403658\pi$
$977$	18.6333	0.596132	0.298066	+	0.954545i	$0.403658\pi$
$978$	0	0
$979$	1.69722	0.0542435
$980$	0	0
$981$	14.5139	0.463392
$982$	0	0
$983$	1.45837	0.0465146	0.0232573	−	0.999730i	$-0.492596\pi$
$983$	1.45837	0.0465146	0.0232573	+	0.999730i	$0.492596\pi$
$984$	0	0
$985$	−5.09167	−0.162234
$986$	0	0
$987$	0	0
$988$	−16.6056	−0.528293
$989$	53.8444	1.71215
$990$	0	0
$991$	−6.88057	−0.218569	−0.109284	−	0.994011i	$-0.534856\pi$
$991$	−6.88057	−0.218569	−0.109284	+	0.994011i	$0.534856\pi$
$992$	0	0
$993$	−0.0639167	−0.00202834
$994$	0	0
$995$	−23.7250	−0.752132
$996$	−53.4500	−1.69363
$997$	54.0555	1.71196	0.855978	−	0.517013i	$-0.172956\pi$
$997$	54.0555	1.71196	0.855978	+	0.517013i	$0.172956\pi$
$998$	0	0
$999$	11.5778	0.366305

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2009.2.a.c.1.1		2
7.3	odd	6		287.2.e.b.247.1	yes	4
7.5	odd	6		287.2.e.b.165.1	✓	4
7.6	odd	2		2009.2.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.e.b.165.1	✓	4	7.5	odd	6
287.2.e.b.247.1	yes	4	7.3	odd	6
2009.2.a.c.1.1		2	1.1	even	1	trivial
2009.2.a.d.1.2		2	7.6	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:	\( N \)	\(=\)	\( 2009 = 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30278 q^{3} -2.00000 q^{4} -1.30278 q^{5} +2.30278 q^{9} +O(q^{10})\)	\(q-2.30278 q^{3} -2.00000 q^{4} -1.30278 q^{5} +2.30278 q^{9} -4.30278 q^{11} +4.60555 q^{12} +2.30278 q^{13} +3.00000 q^{15} +4.00000 q^{16} +3.90833 q^{17} +3.60555 q^{19} +2.60555 q^{20} -5.60555 q^{23} -3.30278 q^{25} +1.60555 q^{27} +5.60555 q^{29} +2.30278 q^{31} +9.90833 q^{33} -4.60555 q^{36} +7.21110 q^{37} -5.30278 q^{39} +1.00000 q^{41} -9.60555 q^{43} +8.60555 q^{44} -3.00000 q^{45} +9.00000 q^{47} -9.21110 q^{48} -9.00000 q^{51} -4.60555 q^{52} +9.90833 q^{53} +5.60555 q^{55} -8.30278 q^{57} -0.394449 q^{59} -6.00000 q^{60} -7.21110 q^{61} -8.00000 q^{64} -3.00000 q^{65} -6.60555 q^{67} -7.81665 q^{68} +12.9083 q^{69} +10.8167 q^{71} -9.30278 q^{73} +7.60555 q^{75} -7.21110 q^{76} -17.4222 q^{79} -5.21110 q^{80} -10.6056 q^{81} -11.6056 q^{83} -5.09167 q^{85} -12.9083 q^{87} -0.394449 q^{89} +11.2111 q^{92} -5.30278 q^{93} -4.69722 q^{95} -9.30278 q^{97} -9.90833 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 4 q^{4} + q^{5} + q^{9}+O(q^{10}) \) 2 * q - q^3 - 4 * q^4 + q^5 + q^9	\( 2 q - q^{3} - 4 q^{4} + q^{5} + q^{9} - 5 q^{11} + 2 q^{12} + q^{13} + 6 q^{15} + 8 q^{16} - 3 q^{17} - 2 q^{20} - 4 q^{23} - 3 q^{25} - 4 q^{27} + 4 q^{29} + q^{31} + 9 q^{33} - 2 q^{36} - 7 q^{39} + 2 q^{41} - 12 q^{43} + 10 q^{44} - 6 q^{45} + 18 q^{47} - 4 q^{48} - 18 q^{51} - 2 q^{52} + 9 q^{53} + 4 q^{55} - 13 q^{57} - 8 q^{59} - 12 q^{60} - 16 q^{64} - 6 q^{65} - 6 q^{67} + 6 q^{68} + 15 q^{69} - 15 q^{73} + 8 q^{75} - 6 q^{79} + 4 q^{80} - 14 q^{81} - 16 q^{83} - 21 q^{85} - 15 q^{87} - 8 q^{89} + 8 q^{92} - 7 q^{93} - 13 q^{95} - 15 q^{97} - 9 q^{99}+O(q^{100}) \) 2 * q - q^3 - 4 * q^4 + q^5 + q^9 - 5 * q^11 + 2 * q^12 + q^13 + 6 * q^15 + 8 * q^16 - 3 * q^17 - 2 * q^20 - 4 * q^23 - 3 * q^25 - 4 * q^27 + 4 * q^29 + q^31 + 9 * q^33 - 2 * q^36 - 7 * q^39 + 2 * q^41 - 12 * q^43 + 10 * q^44 - 6 * q^45 + 18 * q^47 - 4 * q^48 - 18 * q^51 - 2 * q^52 + 9 * q^53 + 4 * q^55 - 13 * q^57 - 8 * q^59 - 12 * q^60 - 16 * q^64 - 6 * q^65 - 6 * q^67 + 6 * q^68 + 15 * q^69 - 15 * q^73 + 8 * q^75 - 6 * q^79 + 4 * q^80 - 14 * q^81 - 16 * q^83 - 21 * q^85 - 15 * q^87 - 8 * q^89 + 8 * q^92 - 7 * q^93 - 13 * q^95 - 15 * q^97 - 9 * q^99

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