LMFDB - Embedded newform 2009.2.a.i.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:	$3$
Coefficient field:	3.3.257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 3 $ x^3 - x^2 - 4*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.19869$ 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)

$f(q)$	$=$	$q-1.83424 q^{2} +2.83424 q^{3} +1.36445 q^{4} -5.19869 q^{6} +1.16576 q^{8} +5.03293 q^{9} +O(q^{10})$	\(q-1.83424 q^{2} +2.83424 q^{3} +1.36445 q^{4} -5.19869 q^{6} +1.16576 q^{8} +5.03293 q^{9} -6.39738 q^{11} +3.86718 q^{12} +3.86718 q^{13} -4.86718 q^{16} -2.36445 q^{17} -9.23163 q^{18} -8.03293 q^{19} +11.7344 q^{22} +1.56314 q^{23} +3.30404 q^{24} -5.00000 q^{25} -7.09334 q^{26} +5.76183 q^{27} -3.66849 q^{29} -6.79476 q^{31} +6.59607 q^{32} -18.1317 q^{33} +4.33697 q^{34} +6.86718 q^{36} -9.86718 q^{37} +14.7344 q^{38} +10.9605 q^{39} +1.00000 q^{41} -0.364448 q^{43} -8.72890 q^{44} -2.86718 q^{46} +8.59607 q^{47} -13.7948 q^{48} +9.17122 q^{50} -6.70142 q^{51} +5.27656 q^{52} -6.39738 q^{53} -10.5686 q^{54} -22.7673 q^{57} +6.72890 q^{58} +12.4633 q^{59} +5.85517 q^{61} +12.4633 q^{62} -2.36445 q^{64} +33.2580 q^{66} -8.46325 q^{67} -3.22617 q^{68} +4.43032 q^{69} +4.06587 q^{71} +5.86718 q^{72} -7.73436 q^{73} +18.0988 q^{74} -14.1712 q^{75} -10.9605 q^{76} -20.1043 q^{78} -6.00000 q^{79} +1.23163 q^{81} -1.83424 q^{82} -4.72890 q^{83} +0.668486 q^{86} -10.3974 q^{87} -7.45779 q^{88} +3.86718 q^{89} +2.13282 q^{92} -19.2580 q^{93} -15.7673 q^{94} +18.6949 q^{96} -0.768374 q^{97} -32.1976 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 4 q^{4} - 10 q^{6} + 9 q^{8} + 4 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 4 * q^4 - 10 * q^6 + 9 * q^8 + 4 * q^9	$ 3 q + 3 q^{3} + 4 q^{4} - 10 q^{6} + 9 q^{8} + 4 q^{9} - 8 q^{11} - 5 q^{12} - 5 q^{13} + 2 q^{16} - 7 q^{17} - 11 q^{18} - 13 q^{19} + 2 q^{22} - q^{23} - q^{24} - 15 q^{25} - 21 q^{26} + 6 q^{27} + 2 q^{31} + 3 q^{32} - 10 q^{33} - 9 q^{34} + 4 q^{36} - 13 q^{37} + 11 q^{38} + 16 q^{39} + 3 q^{41} - q^{43} - 26 q^{44} + 8 q^{46} + 9 q^{47} - 19 q^{48} + 2 q^{51} - 17 q^{52} - 8 q^{53} + 7 q^{54} - 24 q^{57} + 20 q^{58} + 4 q^{59} + 6 q^{61} + 4 q^{62} - 7 q^{64} + 44 q^{66} + 8 q^{67} - 26 q^{68} - 9 q^{69} - 10 q^{71} + q^{72} + 10 q^{73} + 21 q^{74} - 15 q^{75} - 16 q^{76} + 6 q^{78} - 18 q^{79} - 13 q^{81} - 14 q^{83} - 9 q^{86} - 20 q^{87} - 22 q^{88} - 5 q^{89} + 23 q^{92} - 2 q^{93} - 3 q^{94} + 6 q^{96} - 19 q^{97} - 30 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 4 * q^4 - 10 * q^6 + 9 * q^8 + 4 * q^9 - 8 * q^11 - 5 * q^12 - 5 * q^13 + 2 * q^16 - 7 * q^17 - 11 * q^18 - 13 * q^19 + 2 * q^22 - q^23 - q^24 - 15 * q^25 - 21 * q^26 + 6 * q^27 + 2 * q^31 + 3 * q^32 - 10 * q^33 - 9 * q^34 + 4 * q^36 - 13 * q^37 + 11 * q^38 + 16 * q^39 + 3 * q^41 - q^43 - 26 * q^44 + 8 * q^46 + 9 * q^47 - 19 * q^48 + 2 * q^51 - 17 * q^52 - 8 * q^53 + 7 * q^54 - 24 * q^57 + 20 * q^58 + 4 * q^59 + 6 * q^61 + 4 * q^62 - 7 * q^64 + 44 * q^66 + 8 * q^67 - 26 * q^68 - 9 * q^69 - 10 * q^71 + q^72 + 10 * q^73 + 21 * q^74 - 15 * q^75 - 16 * q^76 + 6 * q^78 - 18 * q^79 - 13 * q^81 - 14 * q^83 - 9 * q^86 - 20 * q^87 - 22 * q^88 - 5 * q^89 + 23 * q^92 - 2 * q^93 - 3 * q^94 + 6 * q^96 - 19 * q^97 - 30 * 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 $

$2$	−1.83424	−1.29701	−0.648503	−	0.761212i	$-0.724605\pi$
$2$	−1.83424	−1.29701	−0.648503	+	0.761212i	$0.724605\pi$
$3$	2.83424	1.63635	0.818176	−	0.574969i	$-0.194986\pi$
$3$	2.83424	1.63635	0.818176	+	0.574969i	$0.194986\pi$
$4$	1.36445	0.682224
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	−5.19869	−2.12236
$7$	0	0
$7$	0	0
$8$	1.16576	0.412157
$9$	5.03293	1.67764
$10$	0	0
$11$	−6.39738	−1.92888	−0.964442	−	0.264296i	$-0.914861\pi$
$11$	−6.39738	−1.92888	−0.964442	+	0.264296i	$0.914861\pi$
$12$	3.86718	1.11636
$13$	3.86718	1.07256	0.536281	−	0.844039i	$-0.319829\pi$
$13$	3.86718	1.07256	0.536281	+	0.844039i	$0.319829\pi$
$14$	0	0
$15$	0	0
$16$	−4.86718	−1.21679
$17$	−2.36445	−0.573463	−0.286731	−	0.958011i	$-0.592569\pi$
$17$	−2.36445	−0.573463	−0.286731	+	0.958011i	$0.592569\pi$
$18$	−9.23163	−2.17592
$19$	−8.03293	−1.84288	−0.921441	−	0.388519i	$-0.872987\pi$
$19$	−8.03293	−1.84288	−0.921441	+	0.388519i	$0.872987\pi$
$20$	0	0
$21$	0	0
$22$	11.7344	2.50177
$23$	1.56314	0.325937	0.162969	−	0.986631i	$-0.447893\pi$
$23$	1.56314	0.325937	0.162969	+	0.986631i	$0.447893\pi$
$24$	3.30404	0.674434
$25$	−5.00000	−1.00000
$26$	−7.09334	−1.39112
$27$	5.76183	1.10886
$28$	0	0
$29$	−3.66849	−0.681221	−0.340610	−	0.940205i	$-0.610634\pi$
$29$	−3.66849	−0.681221	−0.340610	+	0.940205i	$0.610634\pi$
$30$	0	0
$31$	−6.79476	−1.22038	−0.610188	−	0.792257i	$-0.708906\pi$
$31$	−6.79476	−1.22038	−0.610188	+	0.792257i	$0.708906\pi$
$32$	6.59607	1.16603
$33$	−18.1317	−3.15633
$34$	4.33697	0.743785
$35$	0	0
$36$	6.86718	1.14453
$37$	−9.86718	−1.62215	−0.811077	−	0.584939i	$-0.801118\pi$
$37$	−9.86718	−1.62215	−0.811077	+	0.584939i	$0.801118\pi$
$38$	14.7344	2.39023
$39$	10.9605	1.75509
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−0.364448	−0.0555778	−0.0277889	−	0.999614i	$-0.508847\pi$
$43$	−0.364448	−0.0555778	−0.0277889	+	0.999614i	$0.508847\pi$
$44$	−8.72890	−1.31593
$45$	0	0
$46$	−2.86718	−0.422742
$47$	8.59607	1.25387	0.626933	−	0.779073i	$-0.284310\pi$
$47$	8.59607	1.25387	0.626933	+	0.779073i	$0.284310\pi$
$48$	−13.7948	−1.99110
$49$	0	0
$50$	9.17122	1.29701
$51$	−6.70142	−0.938387
$52$	5.27656	0.731728
$53$	−6.39738	−0.878748	−0.439374	−	0.898304i	$-0.644800\pi$
$53$	−6.39738	−0.878748	−0.439374	+	0.898304i	$0.644800\pi$
$54$	−10.5686	−1.43820
$55$	0	0
$56$	0	0
$57$	−22.7673	−3.01560
$58$	6.72890	0.883547
$59$	12.4633	1.62258	0.811289	−	0.584646i	$-0.198767\pi$
$59$	12.4633	1.62258	0.811289	+	0.584646i	$0.198767\pi$
$60$	0	0
$61$	5.85517	0.749678	0.374839	−	0.927090i	$-0.377698\pi$
$61$	5.85517	0.749678	0.374839	+	0.927090i	$0.377698\pi$
$62$	12.4633	1.58283
$63$	0	0
$64$	−2.36445	−0.295556
$65$	0	0
$66$	33.2580	4.09378
$67$	−8.46325	−1.03395	−0.516975	−	0.856000i	$-0.672942\pi$
$67$	−8.46325	−1.03395	−0.516975	+	0.856000i	$0.672942\pi$
$68$	−3.22617	−0.391230
$69$	4.43032	0.533347
$70$	0	0
$71$	4.06587	0.482530	0.241265	−	0.970459i	$-0.422438\pi$
$71$	4.06587	0.482530	0.241265	+	0.970459i	$0.422438\pi$
$72$	5.86718	0.691454
$73$	−7.73436	−0.905238	−0.452619	−	0.891704i	$-0.649510\pi$
$73$	−7.73436	−0.905238	−0.452619	+	0.891704i	$0.649510\pi$
$74$	18.0988	2.10394
$75$	−14.1712	−1.63635
$76$	−10.9605	−1.25726
$77$	0	0
$78$	−20.1043	−2.27636
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	0	0
$81$	1.23163	0.136847
$82$	−1.83424	−0.202558
$83$	−4.72890	−0.519064	−0.259532	−	0.965735i	$-0.583568\pi$
$83$	−4.72890	−0.519064	−0.259532	+	0.965735i	$0.583568\pi$
$84$	0	0
$85$	0	0
$86$	0.668486	0.0720847
$87$	−10.3974	−1.11472
$88$	−7.45779	−0.795003
$89$	3.86718	0.409920	0.204960	−	0.978770i	$-0.434294\pi$
$89$	3.86718	0.409920	0.204960	+	0.978770i	$0.434294\pi$
$90$	0	0
$91$	0	0
$92$	2.13282	0.222362
$93$	−19.2580	−1.99696
$94$	−15.7673	−1.62627
$95$	0	0
$96$	18.6949	1.90804
$97$	−0.768374	−0.0780166	−0.0390083	−	0.999239i	$-0.512420\pi$
$97$	−0.768374	−0.0780166	−0.0390083	+	0.999239i	$0.512420\pi$
$98$	0	0
$99$	−32.1976	−3.23598
$100$	−6.82224	−0.682224
$101$	8.68942	0.864629	0.432315	−	0.901723i	$-0.357697\pi$
$101$	8.68942	0.864629	0.432315	+	0.901723i	$0.357697\pi$
$102$	12.2920	1.21709
$103$	−0.939590	−0.0925806	−0.0462903	−	0.998928i	$-0.514740\pi$
$103$	−0.939590	−0.0925806	−0.0462903	+	0.998928i	$0.514740\pi$
$104$	4.50819	0.442064
$105$	0	0
$106$	11.7344	1.13974
$107$	1.09334	0.105698	0.0528488	−	0.998603i	$-0.483170\pi$
$107$	1.09334	0.105698	0.0528488	+	0.998603i	$0.483170\pi$
$108$	7.86172	0.756494
$109$	11.7344	1.12395	0.561974	−	0.827155i	$-0.310042\pi$
$109$	11.7344	1.12395	0.561974	+	0.827155i	$0.310042\pi$
$110$	0	0
$111$	−27.9660	−2.65441
$112$	0	0
$113$	−3.49727	−0.328996	−0.164498	−	0.986377i	$-0.552600\pi$
$113$	−3.49727	−0.328996	−0.164498	+	0.986377i	$0.552600\pi$
$114$	41.7607	3.91125
$115$	0	0
$116$	−5.00546	−0.464745
$117$	19.4633	1.79938
$118$	−22.8606	−2.10449
$119$	0	0
$120$	0	0
$121$	29.9265	2.72059
$122$	−10.7398	−0.972337
$123$	2.83424	0.255555
$124$	−9.27110	−0.832570
$125$	0	0
$126$	0	0
$127$	10.5961	0.940249	0.470125	−	0.882600i	$-0.344209\pi$
$127$	10.5961	0.940249	0.470125	+	0.882600i	$0.344209\pi$
$128$	−8.85517	−0.782694
$129$	−1.03293	−0.0909448
$130$	0	0
$131$	−9.27110	−0.810020	−0.405010	−	0.914312i	$-0.632732\pi$
$131$	−9.27110	−0.810020	−0.405010	+	0.914312i	$0.632732\pi$
$132$	−24.7398	−2.15332
$133$	0	0
$134$	15.5237	1.34104
$135$	0	0
$136$	−2.75637	−0.236357
$137$	−5.45779	−0.466291	−0.233145	−	0.972442i	$-0.574902\pi$
$137$	−5.45779	−0.466291	−0.233145	+	0.972442i	$0.574902\pi$
$138$	−8.12628	−0.691755
$139$	−15.1263	−1.28299	−0.641497	−	0.767125i	$-0.721686\pi$
$139$	−15.1263	−1.28299	−0.641497	+	0.767125i	$0.721686\pi$
$140$	0	0
$141$	24.3634	2.05177
$142$	−7.45779	−0.625844
$143$	−24.7398	−2.06885
$144$	−24.4962	−2.04135
$145$	0	0
$146$	14.1867	1.17410
$147$	0	0
$148$	−13.4633	−1.10667
$149$	11.0055	0.901602	0.450801	−	0.892624i	$-0.351138\pi$
$149$	11.0055	0.901602	0.450801	+	0.892624i	$0.351138\pi$
$150$	25.9935	2.12236
$151$	21.0055	1.70940	0.854700	−	0.519122i	$-0.173741\pi$
$151$	21.0055	1.70940	0.854700	+	0.519122i	$0.173741\pi$
$152$	−9.36445	−0.759557
$153$	−11.9001	−0.962067
$154$	0	0
$155$	0	0
$156$	14.9551	1.19736
$157$	22.5621	1.80065	0.900324	−	0.435220i	$-0.143330\pi$
$157$	22.5621	1.80065	0.900324	+	0.435220i	$0.143330\pi$
$158$	11.0055	0.875547
$159$	−18.1317	−1.43794
$160$	0	0
$161$	0	0
$162$	−2.25910	−0.177492
$163$	−17.6949	−1.38597	−0.692985	−	0.720952i	$-0.743705\pi$
$163$	−17.6949	−1.38597	−0.692985	+	0.720952i	$0.743705\pi$
$164$	1.36445	0.106545
$165$	0	0
$166$	8.67395	0.673229
$167$	0.827699	0.0640493	0.0320247	−	0.999487i	$-0.489804\pi$
$167$	0.827699	0.0640493	0.0320247	+	0.999487i	$0.489804\pi$
$168$	0	0
$169$	1.95506	0.150389
$170$	0	0
$171$	−40.4292	−3.09170
$172$	−0.497270	−0.0379165
$173$	9.27110	0.704869	0.352434	−	0.935836i	$-0.385354\pi$
$173$	9.27110	0.704869	0.352434	+	0.935836i	$0.385354\pi$
$174$	19.0713	1.44579
$175$	0	0
$176$	31.1372	2.34705
$177$	35.3239	2.65511
$178$	−7.09334	−0.531669
$179$	18.3424	1.37098	0.685489	−	0.728083i	$-0.259589\pi$
$179$	18.3424	1.37098	0.685489	+	0.728083i	$0.259589\pi$
$180$	0	0
$181$	−0.768374	−0.0571128	−0.0285564	−	0.999592i	$-0.509091\pi$
$181$	−0.768374	−0.0571128	−0.0285564	+	0.999592i	$0.509091\pi$
$182$	0	0
$183$	16.5950	1.22674
$184$	1.82224	0.134337
$185$	0	0
$186$	35.3239	2.59007
$187$	15.1263	1.10614
$188$	11.7289	0.855418
$189$	0	0
$190$	0	0
$191$	−12.1976	−0.882588	−0.441294	−	0.897363i	$-0.645480\pi$
$191$	−12.1976	−0.882588	−0.441294	+	0.897363i	$0.645480\pi$
$192$	−6.70142	−0.483633
$193$	−13.0055	−0.936153	−0.468077	−	0.883688i	$-0.655053\pi$
$193$	−13.0055	−0.936153	−0.468077	+	0.883688i	$0.655053\pi$
$194$	1.40939	0.101188
$195$	0	0
$196$	0	0
$197$	−10.1647	−0.724203	−0.362101	−	0.932139i	$-0.617941\pi$
$197$	−10.1647	−0.724203	−0.362101	+	0.932139i	$0.617941\pi$
$198$	59.0582	4.19709
$199$	−4.46434	−0.316468	−0.158234	−	0.987402i	$-0.550580\pi$
$199$	−4.46434	−0.316468	−0.158234	+	0.987402i	$0.550580\pi$
$200$	−5.82878	−0.412157
$201$	−23.9869	−1.69191
$202$	−15.9385	−1.12143
$203$	0	0
$204$	−9.14374	−0.640190
$205$	0	0
$206$	1.72344	0.120078
$207$	7.86718	0.546807
$208$	−18.8222	−1.30509
$209$	51.3898	3.55470
$210$	0	0
$211$	8.26564	0.569030	0.284515	−	0.958672i	$-0.408167\pi$
$211$	8.26564	0.569030	0.284515	+	0.958672i	$0.408167\pi$
$212$	−8.72890	−0.599503
$213$	11.5237	0.789588
$214$	−2.00546	−0.137090
$215$	0	0
$216$	6.71689	0.457027
$217$	0	0
$218$	−21.5237	−1.45777
$219$	−21.9210	−1.48129
$220$	0	0
$221$	−9.14374	−0.615075
$222$	51.2964	3.44279
$223$	−22.8606	−1.53086	−0.765431	−	0.643518i	$-0.777474\pi$
$223$	−22.8606	−1.53086	−0.765431	+	0.643518i	$0.777474\pi$
$224$	0	0
$225$	−25.1647	−1.67764
$226$	6.41484	0.426709
$227$	15.4687	1.02669	0.513347	−	0.858181i	$-0.328405\pi$
$227$	15.4687	1.02669	0.513347	+	0.858181i	$0.328405\pi$
$228$	−31.0648	−2.05732
$229$	−16.2371	−1.07298	−0.536488	−	0.843908i	$-0.680249\pi$
$229$	−16.2371	−1.07298	−0.536488	+	0.843908i	$0.680249\pi$
$230$	0	0
$231$	0	0
$232$	−4.27656	−0.280770
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	−35.7003	−2.33380
$235$	0	0
$236$	17.0055	1.10696
$237$	−17.0055	−1.10462
$238$	0	0
$239$	−9.52366	−0.616034	−0.308017	−	0.951381i	$-0.599665\pi$
$239$	−9.52366	−0.616034	−0.308017	+	0.951381i	$0.599665\pi$
$240$	0	0
$241$	−22.8606	−1.47258	−0.736291	−	0.676665i	$-0.763425\pi$
$241$	−22.8606	−1.47258	−0.736291	+	0.676665i	$0.763425\pi$
$242$	−54.8925	−3.52862
$243$	−13.7948	−0.884935
$244$	7.98908	0.511449
$245$	0	0
$246$	−5.19869	−0.331456
$247$	−31.0648	−1.97660
$248$	−7.92104	−0.502987
$249$	−13.4028	−0.849371
$250$	0	0
$251$	−15.6554	−0.988160	−0.494080	−	0.869416i	$-0.664495\pi$
$251$	−15.6554	−0.988160	−0.494080	+	0.869416i	$0.664495\pi$
$252$	0	0
$253$	−10.0000	−0.628695
$254$	−19.4358	−1.21951
$255$	0	0
$256$	20.9714	1.31072
$257$	27.2316	1.69866	0.849331	−	0.527861i	$-0.177006\pi$
$257$	27.2316	1.69866	0.849331	+	0.527861i	$0.177006\pi$
$258$	1.89465	0.117956
$259$	0	0
$260$	0	0
$261$	−18.4633	−1.14285
$262$	17.0055	1.05060
$263$	7.78931	0.480309	0.240155	−	0.970735i	$-0.422802\pi$
$263$	7.78931	0.480309	0.240155	+	0.970735i	$0.422802\pi$
$264$	−21.1372	−1.30090
$265$	0	0
$266$	0	0
$267$	10.9605	0.670773
$268$	−11.5477	−0.705386
$269$	−20.0109	−1.22009	−0.610044	−	0.792368i	$-0.708848\pi$
$269$	−20.0109	−1.22009	−0.610044	+	0.792368i	$0.708848\pi$
$270$	0	0
$271$	−21.9210	−1.33161	−0.665804	−	0.746126i	$-0.731911\pi$
$271$	−21.9210	−1.33161	−0.665804	+	0.746126i	$0.731911\pi$
$272$	11.5082	0.697786
$273$	0	0
$274$	10.0109	0.604782
$275$	31.9869	1.92888
$276$	6.04494	0.363862
$277$	21.9605	1.31948	0.659740	−	0.751494i	$-0.270666\pi$
$277$	21.9605	1.31948	0.659740	+	0.751494i	$0.270666\pi$
$278$	27.7453	1.66405
$279$	−34.1976	−2.04736
$280$	0	0
$281$	4.66303	0.278173	0.139086	−	0.990280i	$-0.455583\pi$
$281$	4.66303	0.278173	0.139086	+	0.990280i	$0.455583\pi$
$282$	−44.6883	−2.66115
$283$	−17.6026	−1.04637	−0.523183	−	0.852220i	$-0.675256\pi$
$283$	−17.6026	−1.04637	−0.523183	+	0.852220i	$0.675256\pi$
$284$	5.54767	0.329193
$285$	0	0
$286$	45.3788	2.68331
$287$	0	0
$288$	33.1976	1.95619
$289$	−11.4094	−0.671140
$290$	0	0
$291$	−2.17776	−0.127663
$292$	−10.5531	−0.617575
$293$	−29.4687	−1.72158	−0.860790	−	0.508960i	$-0.830030\pi$
$293$	−29.4687	−1.72158	−0.860790	+	0.508960i	$0.830030\pi$
$294$	0	0
$295$	0	0
$296$	−11.5027	−0.668583
$297$	−36.8606	−2.13887
$298$	−20.1867	−1.16938
$299$	6.04494	0.349588
$300$	−19.3359	−1.11636
$301$	0	0
$302$	−38.5291	−2.21710
$303$	24.6279	1.41484
$304$	39.0977	2.24241
$305$	0	0
$306$	21.8277	1.24781
$307$	−7.73436	−0.441423	−0.220711	−	0.975339i	$-0.570838\pi$
$307$	−7.73436	−0.441423	−0.220711	+	0.975339i	$0.570838\pi$
$308$	0	0
$309$	−2.66303	−0.151494
$310$	0	0
$311$	−9.91211	−0.562064	−0.281032	−	0.959698i	$-0.590677\pi$
$311$	−9.91211	−0.562064	−0.281032	+	0.959698i	$0.590677\pi$
$312$	12.7773	0.723372
$313$	6.49619	0.367186	0.183593	−	0.983002i	$-0.441227\pi$
$313$	6.49619	0.367186	0.183593	+	0.983002i	$0.441227\pi$
$314$	−41.3843	−2.33545
$315$	0	0
$316$	−8.18669	−0.460537
$317$	0.794765	0.0446384	0.0223192	−	0.999751i	$-0.492895\pi$
$317$	0.794765	0.0446384	0.0223192	+	0.999751i	$0.492895\pi$
$318$	33.2580	1.86502
$319$	23.4687	1.31400
$320$	0	0
$321$	3.09880	0.172958
$322$	0	0
$323$	18.9935	1.05682
$324$	1.68049	0.0933605
$325$	−19.3359	−1.07256
$326$	32.4567	1.79761
$327$	33.2580	1.83917
$328$	1.16576	0.0643682
$329$	0	0
$330$	0	0
$331$	−15.4578	−0.849637	−0.424819	−	0.905279i	$-0.639662\pi$
$331$	−15.4578	−0.849637	−0.424819	+	0.905279i	$0.639662\pi$
$332$	−6.45233	−0.354118
$333$	−49.6609	−2.72140
$334$	−1.51820	−0.0830723
$335$	0	0
$336$	0	0
$337$	5.96052	0.324690	0.162345	−	0.986734i	$-0.448094\pi$
$337$	5.96052	0.324690	0.162345	+	0.986734i	$0.448094\pi$
$338$	−3.58606	−0.195056
$339$	−9.91211	−0.538352
$340$	0	0
$341$	43.4687	2.35396
$342$	74.1570	4.00995
$343$	0	0
$344$	−0.424858	−0.0229068
$345$	0	0
$346$	−17.0055	−0.914219
$347$	27.6135	1.48237	0.741186	−	0.671300i	$-0.234264\pi$
$347$	27.6135	1.48237	0.741186	+	0.671300i	$0.234264\pi$
$348$	−14.1867	−0.760486
$349$	25.6794	1.37459	0.687294	−	0.726380i	$-0.258799\pi$
$349$	25.6794	1.37459	0.687294	+	0.726380i	$0.258799\pi$
$350$	0	0
$351$	22.2820	1.18933
$352$	−42.1976	−2.24914
$353$	−12.6499	−0.673288	−0.336644	−	0.941632i	$-0.609292\pi$
$353$	−12.6499	−0.673288	−0.336644	+	0.941632i	$0.609292\pi$
$354$	−64.7926	−3.44369
$355$	0	0
$356$	5.27656	0.279657
$357$	0	0
$358$	−33.6445	−1.77817
$359$	−10.4643	−0.552287	−0.276143	−	0.961116i	$-0.589056\pi$
$359$	−10.4643	−0.552287	−0.276143	+	0.961116i	$0.589056\pi$
$360$	0	0
$361$	45.5280	2.39621
$362$	1.40939	0.0740757
$363$	84.8190	4.45184
$364$	0	0
$365$	0	0
$366$	−30.4392	−1.59108
$367$	−24.3974	−1.27353	−0.636766	−	0.771057i	$-0.719728\pi$
$367$	−24.3974	−1.27353	−0.636766	+	0.771057i	$0.719728\pi$
$368$	−7.60808	−0.396598
$369$	5.03293	0.262004
$370$	0	0
$371$	0	0
$372$	−26.2766	−1.36238
$373$	32.2855	1.67168	0.835840	−	0.548974i	$-0.184981\pi$
$373$	32.2855	1.67168	0.835840	+	0.548974i	$0.184981\pi$
$374$	−27.7453	−1.43467
$375$	0	0
$376$	10.0209	0.516790
$377$	−14.1867	−0.730652
$378$	0	0
$379$	7.31058	0.375519	0.187760	−	0.982215i	$-0.439877\pi$
$379$	7.31058	0.375519	0.187760	+	0.982215i	$0.439877\pi$
$380$	0	0
$381$	30.0318	1.53858
$382$	22.3734	1.14472
$383$	10.0395	0.512993	0.256497	−	0.966545i	$-0.417432\pi$
$383$	10.0395	0.512993	0.256497	+	0.966545i	$0.417432\pi$
$384$	−25.0977	−1.28076
$385$	0	0
$386$	23.8552	1.21420
$387$	−1.83424	−0.0932398
$388$	−1.04841	−0.0532248
$389$	9.18014	0.465452	0.232726	−	0.972542i	$-0.425235\pi$
$389$	9.18014	0.465452	0.232726	+	0.972542i	$0.425235\pi$
$390$	0	0
$391$	−3.69596	−0.186913
$392$	0	0
$393$	−26.2766	−1.32548
$394$	18.6445	0.939295
$395$	0	0
$396$	−43.9320	−2.20766
$397$	−27.2162	−1.36594	−0.682970	−	0.730447i	$-0.739312\pi$
$397$	−27.2162	−1.36594	−0.682970	+	0.730447i	$0.739312\pi$
$398$	8.18868	0.410461
$399$	0	0
$400$	24.3359	1.21679
$401$	−29.3119	−1.46377	−0.731883	−	0.681431i	$-0.761358\pi$
$401$	−29.3119	−1.46377	−0.731883	+	0.681431i	$0.761358\pi$
$402$	43.9978	2.19441
$403$	−26.2766	−1.30893
$404$	11.8563	0.589871
$405$	0	0
$406$	0	0
$407$	63.1241	3.12895
$408$	−7.81223	−0.386763
$409$	−0.597158	−0.0295276	−0.0147638	−	0.999891i	$-0.504700\pi$
$409$	−0.597158	−0.0295276	−0.0147638	+	0.999891i	$0.504700\pi$
$410$	0	0
$411$	−15.4687	−0.763015
$412$	−1.28202	−0.0631607
$413$	0	0
$414$	−14.4303	−0.709211
$415$	0	0
$416$	25.5082	1.25064
$417$	−42.8716	−2.09943
$418$	−94.2613	−4.61047
$419$	12.2766	0.599749	0.299875	−	0.953979i	$-0.403055\pi$
$419$	12.2766	0.599749	0.299875	+	0.953979i	$0.403055\pi$
$420$	0	0
$421$	−8.92650	−0.435051	−0.217526	−	0.976055i	$-0.569799\pi$
$421$	−8.92650	−0.435051	−0.217526	+	0.976055i	$0.569799\pi$
$422$	−15.1612	−0.738036
$423$	43.2635	2.10354
$424$	−7.45779	−0.362182
$425$	11.8222	0.573463
$426$	−21.1372	−1.02410
$427$	0	0
$428$	1.49181	0.0721094
$429$	−70.1187	−3.38536
$430$	0	0
$431$	−19.0473	−0.917477	−0.458739	−	0.888571i	$-0.651699\pi$
$431$	−19.0473	−0.917477	−0.458739	+	0.888571i	$0.651699\pi$
$432$	−28.0439	−1.34926
$433$	18.5422	0.891082	0.445541	−	0.895262i	$-0.353011\pi$
$433$	18.5422	0.891082	0.445541	+	0.895262i	$0.353011\pi$
$434$	0	0
$435$	0	0
$436$	16.0109	0.766784
$437$	−12.5566	−0.600663
$438$	40.2085	1.92124
$439$	−27.0449	−1.29078	−0.645392	−	0.763851i	$-0.723306\pi$
$439$	−27.0449	−1.29078	−0.645392	+	0.763851i	$0.723306\pi$
$440$	0	0
$441$	0	0
$442$	16.7718	0.797755
$443$	4.31951	0.205226	0.102613	−	0.994721i	$-0.467280\pi$
$443$	4.31951	0.205226	0.102613	+	0.994721i	$0.467280\pi$
$444$	−38.1581	−1.81090
$445$	0	0
$446$	41.9320	1.98554
$447$	31.1921	1.47534
$448$	0	0
$449$	−3.49727	−0.165046	−0.0825232	−	0.996589i	$-0.526298\pi$
$449$	−3.49727	−0.165046	−0.0825232	+	0.996589i	$0.526298\pi$
$450$	46.1581	2.17592
$451$	−6.39738	−0.301241
$452$	−4.77184	−0.224449
$453$	59.5346	2.79718
$454$	−28.3734	−1.33163
$455$	0	0
$456$	−26.5411	−1.24290
$457$	29.5477	1.38218	0.691091	−	0.722768i	$-0.257130\pi$
$457$	29.5477	1.38218	0.691091	+	0.722768i	$0.257130\pi$
$458$	29.7828	1.39166
$459$	−13.6235	−0.635893
$460$	0	0
$461$	31.1921	1.45276	0.726382	−	0.687292i	$-0.241201\pi$
$461$	31.1921	1.45276	0.726382	+	0.687292i	$0.241201\pi$
$462$	0	0
$463$	−10.6499	−0.494945	−0.247472	−	0.968895i	$-0.579600\pi$
$463$	−10.6499	−0.494945	−0.247472	+	0.968895i	$0.579600\pi$
$464$	17.8552	0.828906
$465$	0	0
$466$	11.0055	0.509818
$467$	−12.6499	−0.585369	−0.292685	−	0.956209i	$-0.594549\pi$
$467$	−12.6499	−0.585369	−0.292685	+	0.956209i	$0.594549\pi$
$468$	26.5566	1.22758
$469$	0	0
$470$	0	0
$471$	63.9463	2.94649
$472$	14.5291	0.668757
$473$	2.33151	0.107203
$474$	31.1921	1.43270
$475$	40.1647	1.84288
$476$	0	0
$477$	−32.1976	−1.47423
$478$	17.4687	0.799000
$479$	−27.0449	−1.23571	−0.617857	−	0.786290i	$-0.711999\pi$
$479$	−27.0449	−1.23571	−0.617857	+	0.786290i	$0.711999\pi$
$480$	0	0
$481$	−38.1581	−1.73986
$482$	41.9320	1.90995
$483$	0	0
$484$	40.8332	1.85605
$485$	0	0
$486$	25.3030	1.14777
$487$	−28.2065	−1.27816	−0.639080	−	0.769140i	$-0.720685\pi$
$487$	−28.2065	−1.27816	−0.639080	+	0.769140i	$0.720685\pi$
$488$	6.82571	0.308985
$489$	−50.1516	−2.26793
$490$	0	0
$491$	20.9001	0.943209	0.471604	−	0.881810i	$-0.343675\pi$
$491$	20.9001	0.943209	0.471604	+	0.881810i	$0.343675\pi$
$492$	3.86718	0.174346
$493$	8.67395	0.390655
$494$	56.9804	2.56367
$495$	0	0
$496$	33.0713	1.48495
$497$	0	0
$498$	24.5841	1.10164
$499$	6.27656	0.280978	0.140489	−	0.990082i	$-0.455133\pi$
$499$	6.27656	0.280978	0.140489	+	0.990082i	$0.455133\pi$
$500$	0	0
$501$	2.34590	0.104807
$502$	28.7158	1.28165
$503$	17.5871	0.784172	0.392086	−	0.919929i	$-0.371754\pi$
$503$	17.5871	0.784172	0.392086	+	0.919929i	$0.371754\pi$
$504$	0	0
$505$	0	0
$506$	18.3424	0.815421
$507$	5.54112	0.246090
$508$	14.4578	0.641461
$509$	−0.793680	−0.0351793	−0.0175896	−	0.999845i	$-0.505599\pi$
$509$	−0.793680	−0.0351793	−0.0175896	+	0.999845i	$0.505599\pi$
$510$	0	0
$511$	0	0
$512$	−20.7564	−0.917311
$513$	−46.2844	−2.04351
$514$	−49.9494	−2.20317
$515$	0	0
$516$	−1.40939	−0.0620447
$517$	−54.9924	−2.41856
$518$	0	0
$519$	26.2766	1.15341
$520$	0	0
$521$	22.7828	0.998131	0.499065	−	0.866564i	$-0.333677\pi$
$521$	22.7828	0.998131	0.499065	+	0.866564i	$0.333677\pi$
$522$	33.8661	1.48228
$523$	37.3898	1.63494	0.817470	−	0.575971i	$-0.195376\pi$
$523$	37.3898	1.63494	0.817470	+	0.575971i	$0.195376\pi$
$524$	−12.6499	−0.552615
$525$	0	0
$526$	−14.2875	−0.622964
$527$	16.0659	0.699840
$528$	88.2504	3.84061
$529$	−20.5566	−0.893765
$530$	0	0
$531$	62.7267	2.72211
$532$	0	0
$533$	3.86718	0.167506
$534$	−20.1043	−0.869997
$535$	0	0
$536$	−9.86609	−0.426150
$537$	51.9869	2.24340
$538$	36.7049	1.58246
$539$	0	0
$540$	0	0
$541$	31.9121	1.37201	0.686004	−	0.727597i	$-0.259363\pi$
$541$	31.9121	1.37201	0.686004	+	0.727597i	$0.259363\pi$
$542$	40.2085	1.72710
$543$	−2.17776	−0.0934566
$544$	−15.5961	−0.668676
$545$	0	0
$546$	0	0
$547$	−2.54221	−0.108697	−0.0543485	−	0.998522i	$-0.517308\pi$
$547$	−2.54221	−0.108697	−0.0543485	+	0.998522i	$0.517308\pi$
$548$	−7.44687	−0.318115
$549$	29.4687	1.25769
$550$	−58.6718	−2.50177
$551$	29.4687	1.25541
$552$	5.16467	0.219823
$553$	0	0
$554$	−40.2809	−1.71137
$555$	0	0
$556$	−20.6390	−0.875289
$557$	−16.0109	−0.678404	−0.339202	−	0.940713i	$-0.610157\pi$
$557$	−16.0109	−0.678404	−0.339202	+	0.940713i	$0.610157\pi$
$558$	62.7267	2.65543
$559$	−1.40939	−0.0596106
$560$	0	0
$561$	42.8716	1.81004
$562$	−8.55313	−0.360792
$563$	41.2569	1.73877	0.869386	−	0.494133i	$-0.164515\pi$
$563$	41.2569	1.73877	0.869386	+	0.494133i	$0.164515\pi$
$564$	33.2425	1.39976
$565$	0	0
$566$	32.2875	1.35714
$567$	0	0
$568$	4.73981	0.198878
$569$	38.5202	1.61485	0.807425	−	0.589970i	$-0.200860\pi$
$569$	38.5202	1.61485	0.807425	+	0.589970i	$0.200860\pi$
$570$	0	0
$571$	−30.5710	−1.27936	−0.639678	−	0.768643i	$-0.720932\pi$
$571$	−30.5710	−1.27936	−0.639678	+	0.768643i	$0.720932\pi$
$572$	−33.7562	−1.41142
$573$	−34.5710	−1.44422
$574$	0	0
$575$	−7.81570	−0.325937
$576$	−11.9001	−0.495838
$577$	31.3634	1.30567	0.652837	−	0.757498i	$-0.273579\pi$
$577$	31.3634	1.30567	0.652837	+	0.757498i	$0.273579\pi$
$578$	20.9276	0.870473
$579$	−36.8606	−1.53188
$580$	0	0
$581$	0	0
$582$	3.99454	0.165579
$583$	40.9265	1.69500
$584$	−9.01638	−0.373100
$585$	0	0
$586$	54.0528	2.23290
$587$	−5.18322	−0.213934	−0.106967	−	0.994263i	$-0.534114\pi$
$587$	−5.18322	−0.213934	−0.106967	+	0.994263i	$0.534114\pi$
$588$	0	0
$589$	54.5819	2.24901
$590$	0	0
$591$	−28.8092	−1.18505
$592$	48.0253	1.97383
$593$	−4.24055	−0.174139	−0.0870693	−	0.996202i	$-0.527750\pi$
$593$	−4.24055	−0.174139	−0.0870693	+	0.996202i	$0.527750\pi$
$594$	67.6114	2.77413
$595$	0	0
$596$	15.0164	0.615095
$597$	−12.6530	−0.517853
$598$	−11.0879	−0.453417
$599$	19.3908	0.792288	0.396144	−	0.918188i	$-0.370348\pi$
$599$	19.3908	0.792288	0.396144	+	0.918188i	$0.370348\pi$
$600$	−16.5202	−0.674434
$601$	−25.6356	−1.04570	−0.522848	−	0.852426i	$-0.675130\pi$
$601$	−25.6356	−1.04570	−0.522848	+	0.852426i	$0.675130\pi$
$602$	0	0
$603$	−42.5950	−1.73460
$604$	28.6609	1.16619
$605$	0	0
$606$	−45.1736	−1.83505
$607$	−32.7289	−1.32842	−0.664212	−	0.747544i	$-0.731233\pi$
$607$	−32.7289	−1.32842	−0.664212	+	0.747544i	$0.731233\pi$
$608$	−52.9858	−2.14886
$609$	0	0
$610$	0	0
$611$	33.2425	1.34485
$612$	−16.2371	−0.656345
$613$	3.04494	0.122984	0.0614919	−	0.998108i	$-0.480414\pi$
$613$	3.04494	0.122984	0.0614919	+	0.998108i	$0.480414\pi$
$614$	14.1867	0.572528
$615$	0	0
$616$	0	0
$617$	0.948519	0.0381859	0.0190930	−	0.999818i	$-0.493922\pi$
$617$	0.948519	0.0381859	0.0190930	+	0.999818i	$0.493922\pi$
$618$	4.88464	0.196489
$619$	19.4447	0.781549	0.390774	−	0.920487i	$-0.372207\pi$
$619$	19.4447	0.781549	0.390774	+	0.920487i	$0.372207\pi$
$620$	0	0
$621$	9.00654	0.361420
$622$	18.1812	0.729001
$623$	0	0
$624$	−53.3468	−2.13558
$625$	25.0000	1.00000
$626$	−11.9156	−0.476243
$627$	145.651	5.81674
$628$	30.7848	1.22845
$629$	23.3304	0.930245
$630$	0	0
$631$	8.87264	0.353214	0.176607	−	0.984281i	$-0.443488\pi$
$631$	8.87264	0.353214	0.176607	+	0.984281i	$0.443488\pi$
$632$	−6.99454	−0.278228
$633$	23.4268	0.931134
$634$	−1.45779	−0.0578963
$635$	0	0
$636$	−24.7398	−0.980997
$637$	0	0
$638$	−43.0473	−1.70426
$639$	20.4633	0.809514
$640$	0	0
$641$	28.5160	1.12632	0.563158	−	0.826349i	$-0.309586\pi$
$641$	28.5160	1.12632	0.563158	+	0.826349i	$0.309586\pi$
$642$	−5.68396	−0.224328
$643$	0.488342	0.0192583	0.00962916	−	0.999954i	$-0.496935\pi$
$643$	0.488342	0.0192583	0.00962916	+	0.999954i	$0.496935\pi$
$644$	0	0
$645$	0	0
$646$	−34.8386	−1.37071
$647$	−6.38429	−0.250993	−0.125496	−	0.992094i	$-0.540052\pi$
$647$	−6.38429	−0.250993	−0.125496	+	0.992094i	$0.540052\pi$
$648$	1.43578	0.0564026
$649$	−79.7322	−3.12976
$650$	35.4667	1.39112
$651$	0	0
$652$	−24.1437	−0.945542
$653$	−27.3239	−1.06927	−0.534633	−	0.845084i	$-0.679550\pi$
$653$	−27.3239	−1.06927	−0.534633	+	0.845084i	$0.679550\pi$
$654$	−61.0033	−2.38542
$655$	0	0
$656$	−4.86718	−0.190031
$657$	−38.9265	−1.51867
$658$	0	0
$659$	10.2635	0.399808	0.199904	−	0.979815i	$-0.435937\pi$
$659$	10.2635	0.399808	0.199904	+	0.979815i	$0.435937\pi$
$660$	0	0
$661$	−20.0790	−0.780981	−0.390490	−	0.920607i	$-0.627695\pi$
$661$	−20.0790	−0.780981	−0.390490	+	0.920607i	$0.627695\pi$
$662$	28.3534	1.10198
$663$	−25.9156	−1.00648
$664$	−5.51274	−0.213936
$665$	0	0
$666$	91.0901	3.52967
$667$	−5.73436	−0.222035
$668$	1.12935	0.0436960
$669$	−64.7926	−2.50503
$670$	0	0
$671$	−37.4578	−1.44604
$672$	0	0
$673$	−30.7289	−1.18451	−0.592256	−	0.805750i	$-0.701763\pi$
$673$	−30.7289	−1.18451	−0.592256	+	0.805750i	$0.701763\pi$
$674$	−10.9330	−0.421125
$675$	−28.8092	−1.10886
$676$	2.66758	0.102599
$677$	34.1976	1.31432	0.657160	−	0.753751i	$-0.271757\pi$
$677$	34.1976	1.31432	0.657160	+	0.753751i	$0.271757\pi$
$678$	18.1812	0.698246
$679$	0	0
$680$	0	0
$681$	43.8421	1.68003
$682$	−79.7322	−3.05310
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	−55.1636	−2.10923
$685$	0	0
$686$	0	0
$687$	−46.0198	−1.75577
$688$	1.77383	0.0676268
$689$	−24.7398	−0.942512
$690$	0	0
$691$	15.9571	0.607035	0.303517	−	0.952826i	$-0.401839\pi$
$691$	15.9571	0.607035	0.303517	+	0.952826i	$0.401839\pi$
$692$	12.6499	0.480879
$693$	0	0
$694$	−50.6499	−1.92265
$695$	0	0
$696$	−12.1208	−0.459438
$697$	−2.36445	−0.0895599
$698$	−47.1023	−1.78285
$699$	−17.0055	−0.643206
$700$	0	0
$701$	−4.11973	−0.155600	−0.0778001	−	0.996969i	$-0.524790\pi$
$701$	−4.11973	−0.155600	−0.0778001	+	0.996969i	$0.524790\pi$
$702$	−40.8706	−1.54256
$703$	79.2624	2.98944
$704$	15.1263	0.570093
$705$	0	0
$706$	23.2031	0.873259
$707$	0	0
$708$	48.1976	1.81138
$709$	8.37338	0.314469	0.157234	−	0.987561i	$-0.449742\pi$
$709$	8.37338	0.314469	0.157234	+	0.987561i	$0.449742\pi$
$710$	0	0
$711$	−30.1976	−1.13250
$712$	4.50819	0.168952
$713$	−10.6212	−0.397766
$714$	0	0
$715$	0	0
$716$	25.0273	0.935314
$717$	−26.9924	−1.00805
$718$	19.1941	0.716319
$719$	−18.5422	−0.691508	−0.345754	−	0.938325i	$-0.612377\pi$
$719$	−18.5422	−0.691508	−0.345754	+	0.938325i	$0.612377\pi$
$720$	0	0
$721$	0	0
$722$	−83.5095	−3.10790
$723$	−64.7926	−2.40966
$724$	−1.04841	−0.0389637
$725$	18.3424	0.681221
$726$	−155.579	−5.77407
$727$	−9.91211	−0.367620	−0.183810	−	0.982962i	$-0.558843\pi$
$727$	−9.91211	−0.367620	−0.183810	+	0.982962i	$0.558843\pi$
$728$	0	0
$729$	−42.7926	−1.58491
$730$	0	0
$731$	0.861719	0.0318718
$732$	22.6430	0.836909
$733$	−29.3130	−1.08270	−0.541350	−	0.840798i	$-0.682086\pi$
$733$	−29.3130	−1.08270	−0.541350	+	0.840798i	$0.682086\pi$
$734$	44.7507	1.65178
$735$	0	0
$736$	10.3106	0.380053
$737$	54.1427	1.99437
$738$	−9.23163	−0.339821
$739$	51.5226	1.89529	0.947644	−	0.319328i	$-0.103457\pi$
$739$	51.5226	1.89529	0.947644	+	0.319328i	$0.103457\pi$
$740$	0	0
$741$	−88.0452	−3.23442
$742$	0	0
$743$	−47.5915	−1.74596	−0.872982	−	0.487753i	$-0.837817\pi$
$743$	−47.5915	−1.74596	−0.872982	+	0.487753i	$0.837817\pi$
$744$	−22.4502	−0.823063
$745$	0	0
$746$	−59.2194	−2.16818
$747$	−23.8002	−0.870805
$748$	20.6390	0.754637
$749$	0	0
$750$	0	0
$751$	−30.9265	−1.12852	−0.564262	−	0.825596i	$-0.690839\pi$
$751$	−30.9265	−1.12852	−0.564262	+	0.825596i	$0.690839\pi$
$752$	−41.8386	−1.52570
$753$	−44.3712	−1.61698
$754$	26.0218	0.947659
$755$	0	0
$756$	0	0
$757$	8.19761	0.297947	0.148974	−	0.988841i	$-0.452403\pi$
$757$	8.19761	0.297947	0.148974	+	0.988841i	$0.452403\pi$
$758$	−13.4094	−0.487051
$759$	−28.3424	−1.02877
$760$	0	0
$761$	43.2820	1.56897	0.784486	−	0.620146i	$-0.212927\pi$
$761$	43.2820	1.56897	0.784486	+	0.620146i	$0.212927\pi$
$762$	−55.0857	−1.99554
$763$	0	0
$764$	−16.6430	−0.602123
$765$	0	0
$766$	−18.4148	−0.665355
$767$	48.1976	1.74031
$768$	59.4382	2.14479
$769$	6.45233	0.232677	0.116339	−	0.993210i	$-0.462884\pi$
$769$	6.45233	0.232677	0.116339	+	0.993210i	$0.462884\pi$
$770$	0	0
$771$	77.1810	2.77961
$772$	−17.7453	−0.638666
$773$	−18.2065	−0.654844	−0.327422	−	0.944878i	$-0.606180\pi$
$773$	−18.2065	−0.654844	−0.327422	+	0.944878i	$0.606180\pi$
$774$	3.36445	0.120933
$775$	33.9738	1.22038
$776$	−0.895738	−0.0321551
$777$	0	0
$778$	−16.8386	−0.603694
$779$	−8.03293	−0.287810
$780$	0	0
$781$	−26.0109	−0.930744
$782$	6.77929	0.242427
$783$	−21.1372	−0.755382
$784$	0	0
$785$	0	0
$786$	48.1976	1.71915
$787$	−17.6026	−0.627466	−0.313733	−	0.949511i	$-0.601580\pi$
$787$	−17.6026	−0.627466	−0.313733	+	0.949511i	$0.601580\pi$
$788$	−13.8692	−0.494069
$789$	22.0768	0.785954
$790$	0	0
$791$	0	0
$792$	−37.5346	−1.33373
$793$	22.6430	0.804077
$794$	49.9210	1.77163
$795$	0	0
$796$	−6.09135	−0.215902
$797$	22.1077	0.783096	0.391548	−	0.920158i	$-0.371940\pi$
$797$	22.1077	0.783096	0.391548	+	0.920158i	$0.371940\pi$
$798$	0	0
$799$	−20.3250	−0.719046
$800$	−32.9804	−1.16603
$801$	19.4633	0.687700
$802$	53.7651	1.89851
$803$	49.4796	1.74610
$804$	−32.7289	−1.15426
$805$	0	0
$806$	48.1976	1.69769
$807$	−56.7158	−1.99649
$808$	10.1297	0.356363
$809$	−2.38429	−0.0838273	−0.0419137	−	0.999121i	$-0.513345\pi$
$809$	−2.38429	−0.0838273	−0.0419137	+	0.999121i	$0.513345\pi$
$810$	0	0
$811$	34.5710	1.21395	0.606976	−	0.794720i	$-0.292383\pi$
$811$	34.5710	1.21395	0.606976	+	0.794720i	$0.292383\pi$
$812$	0	0
$813$	−62.1296	−2.17898
$814$	−115.785	−4.05826
$815$	0	0
$816$	32.6170	1.14182
$817$	2.92759	0.102423
$818$	1.09533	0.0382974
$819$	0	0
$820$	0	0
$821$	45.2779	1.58021	0.790104	−	0.612973i	$-0.210026\pi$
$821$	45.2779	1.58021	0.790104	+	0.612973i	$0.210026\pi$
$822$	28.3734	0.989635
$823$	11.9210	0.415541	0.207771	−	0.978178i	$-0.433379\pi$
$823$	11.9210	0.415541	0.207771	+	0.978178i	$0.433379\pi$
$824$	−1.09533	−0.0381578
$825$	90.6587	3.15633
$826$	0	0
$827$	6.54221	0.227495	0.113747	−	0.993510i	$-0.463715\pi$
$827$	6.54221	0.227495	0.113747	+	0.993510i	$0.463715\pi$
$828$	10.7344	0.373045
$829$	41.4028	1.43798	0.718990	−	0.695020i	$-0.244605\pi$
$829$	41.4028	1.43798	0.718990	+	0.695020i	$0.244605\pi$
$830$	0	0
$831$	62.2415	2.15913
$832$	−9.14374	−0.317002
$833$	0	0
$834$	78.6369	2.72297
$835$	0	0
$836$	70.1187	2.42510
$837$	−39.1503	−1.35323
$838$	−22.5182	−0.777878
$839$	−23.9714	−0.827586	−0.413793	−	0.910371i	$-0.635796\pi$
$839$	−23.9714	−0.827586	−0.413793	+	0.910371i	$0.635796\pi$
$840$	0	0
$841$	−15.5422	−0.535938
$842$	16.3734	0.564264
$843$	13.2162	0.455188
$844$	11.2780	0.388206
$845$	0	0
$846$	−79.3557	−2.72831
$847$	0	0
$848$	31.1372	1.06926
$849$	−49.8901	−1.71222
$850$	−21.6849	−0.743785
$851$	−15.4238	−0.528720
$852$	15.7234	0.538676
$853$	−50.6739	−1.73504	−0.867521	−	0.497400i	$-0.834288\pi$
$853$	−50.6739	−1.73504	−0.867521	+	0.497400i	$0.834288\pi$
$854$	0	0
$855$	0	0
$856$	1.27457	0.0435640
$857$	−38.9265	−1.32970	−0.664852	−	0.746975i	$-0.731505\pi$
$857$	−38.9265	−1.32970	−0.664852	+	0.746975i	$0.731505\pi$
$858$	128.615	4.39083
$859$	−6.23470	−0.212725	−0.106363	−	0.994327i	$-0.533920\pi$
$859$	−6.23470	−0.212725	−0.106363	+	0.994327i	$0.533920\pi$
$860$	0	0
$861$	0	0
$862$	34.9374	1.18997
$863$	32.9265	1.12083	0.560416	−	0.828212i	$-0.310641\pi$
$863$	32.9265	1.12083	0.560416	+	0.828212i	$0.310641\pi$
$864$	38.0055	1.29297
$865$	0	0
$866$	−34.0109	−1.15574
$867$	−32.3370	−1.09822
$868$	0	0
$869$	38.3843	1.30210
$870$	0	0
$871$	−32.7289	−1.10898
$872$	13.6794	0.463243
$873$	−3.86718	−0.130884
$874$	23.0318	0.779064
$875$	0	0
$876$	−29.9101	−1.01057
$877$	32.3448	1.09221	0.546103	−	0.837718i	$-0.316111\pi$
$877$	32.3448	1.09221	0.546103	+	0.837718i	$0.316111\pi$
$878$	49.6070	1.67415
$879$	−83.5215	−2.81711
$880$	0	0
$881$	−54.3952	−1.83262	−0.916311	−	0.400468i	$-0.868847\pi$
$881$	−54.3952	−1.83262	−0.916311	+	0.400468i	$0.868847\pi$
$882$	0	0
$883$	−33.8530	−1.13924	−0.569622	−	0.821907i	$-0.692910\pi$
$883$	−33.8530	−1.13924	−0.569622	+	0.821907i	$0.692910\pi$
$884$	−12.4762	−0.419619
$885$	0	0
$886$	−7.92303	−0.266180
$887$	−19.7773	−0.664057	−0.332028	−	0.943269i	$-0.607733\pi$
$887$	−19.7773	−0.664057	−0.332028	+	0.943269i	$0.607733\pi$
$888$	−32.6015	−1.09404
$889$	0	0
$890$	0	0
$891$	−7.87918	−0.263962
$892$	−31.1921	−1.04439
$893$	−69.0517	−2.31073
$894$	−57.2140	−1.91352
$895$	0	0
$896$	0	0
$897$	17.1328	0.572048
$898$	6.41484	0.214066
$899$	24.9265	0.831345
$900$	−34.3359	−1.14453
$901$	15.1263	0.503929
$902$	11.7344	0.390711
$903$	0	0
$904$	−4.07697	−0.135598
$905$	0	0
$906$	−109.201	−3.62796
$907$	0.319511	0.0106092	0.00530459	−	0.999986i	$-0.498311\pi$
$907$	0.319511	0.0106092	0.00530459	+	0.999986i	$0.498311\pi$
$908$	21.1063	0.700436
$909$	43.7333	1.45054
$910$	0	0
$911$	−27.1701	−0.900187	−0.450093	−	0.892982i	$-0.648609\pi$
$911$	−27.1701	−0.900187	−0.450093	+	0.892982i	$0.648609\pi$
$912$	110.812	3.66937
$913$	30.2526	1.00121
$914$	−54.1976	−1.79270
$915$	0	0
$916$	−22.1547	−0.732011
$917$	0	0
$918$	24.9889	0.824757
$919$	11.3788	0.375353	0.187677	−	0.982231i	$-0.439904\pi$
$919$	11.3788	0.375353	0.187677	+	0.982231i	$0.439904\pi$
$920$	0	0
$921$	−21.9210	−0.722323
$922$	−57.2140	−1.88424
$923$	15.7234	0.517543
$924$	0	0
$925$	49.3359	1.62215
$926$	19.5346	0.641946
$927$	−4.72890	−0.155317
$928$	−24.1976	−0.794325
$929$	17.7738	0.583141	0.291570	−	0.956549i	$-0.405822\pi$
$929$	17.7738	0.583141	0.291570	+	0.956549i	$0.405822\pi$
$930$	0	0
$931$	0	0
$932$	−8.18669	−0.268164
$933$	−28.0933	−0.919735
$934$	23.2031	0.759228
$935$	0	0
$936$	22.6894	0.741627
$937$	−57.3228	−1.87265	−0.936327	−	0.351129i	$-0.885798\pi$
$937$	−57.3228	−1.87265	−0.936327	+	0.351129i	$0.885798\pi$
$938$	0	0
$939$	18.4118	0.600845
$940$	0	0
$941$	−6.07896	−0.198168	−0.0990842	−	0.995079i	$-0.531591\pi$
$941$	−6.07896	−0.198168	−0.0990842	+	0.995079i	$0.531591\pi$
$942$	−117.293	−3.82162
$943$	1.56314	0.0509028
$944$	−60.6609	−1.97434
$945$	0	0
$946$	−4.27656	−0.139043
$947$	−29.0318	−0.943408	−0.471704	−	0.881757i	$-0.656361\pi$
$947$	−29.0318	−0.943408	−0.471704	+	0.881757i	$0.656361\pi$
$948$	−23.2031	−0.753601
$949$	−29.9101	−0.970924
$950$	−73.6718	−2.39023
$951$	2.25256	0.0730442
$952$	0	0
$953$	22.1568	0.717730	0.358865	−	0.933389i	$-0.383164\pi$
$953$	22.1568	0.717730	0.358865	+	0.933389i	$0.383164\pi$
$954$	59.0582	1.91208
$955$	0	0
$956$	−12.9945	−0.420273
$957$	66.5160	2.15016
$958$	49.6070	1.60273
$959$	0	0
$960$	0	0
$961$	15.1688	0.489317
$962$	69.9913	2.25661
$963$	5.50273	0.177323
$964$	−31.1921	−1.00463
$965$	0	0
$966$	0	0
$967$	−19.3610	−0.622607	−0.311304	−	0.950311i	$-0.600766\pi$
$967$	−19.3610	−0.622607	−0.311304	+	0.950311i	$0.600766\pi$
$968$	34.8870	1.12131
$969$	53.8321	1.72934
$970$	0	0
$971$	−33.1832	−1.06490	−0.532450	−	0.846461i	$-0.678729\pi$
$971$	−33.1832	−1.06490	−0.532450	+	0.846461i	$0.678729\pi$
$972$	−18.8222	−0.603724
$973$	0	0
$974$	51.7376	1.65778
$975$	−54.8026	−1.75509
$976$	−28.4982	−0.912204
$977$	−18.4873	−0.591460	−0.295730	−	0.955272i	$-0.595563\pi$
$977$	−18.4873	−0.591460	−0.295730	+	0.955272i	$0.595563\pi$
$978$	91.9902	2.94152
$979$	−24.7398	−0.790688
$980$	0	0
$981$	59.0582	1.88558
$982$	−38.3359	−1.22335
$983$	−56.1187	−1.78991	−0.894953	−	0.446159i	$-0.852791\pi$
$983$	−56.1187	−1.78991	−0.894953	+	0.446159i	$0.852791\pi$
$984$	3.30404	0.105329
$985$	0	0
$986$	−15.9101	−0.506682
$987$	0	0
$988$	−42.3863	−1.34849
$989$	−0.569683	−0.0181149
$990$	0	0
$991$	−46.8297	−1.48759	−0.743797	−	0.668406i	$-0.766977\pi$
$991$	−46.8297	−1.48759	−0.743797	+	0.668406i	$0.766977\pi$
$992$	−44.8188	−1.42300
$993$	−43.8111	−1.39030
$994$	0	0
$995$	0	0
$996$	−18.2875	−0.579461
$997$	32.5577	1.03111	0.515556	−	0.856856i	$-0.327585\pi$
$997$	32.5577	1.03111	0.515556	+	0.856856i	$0.327585\pi$
$998$	−11.5127	−0.364430
$999$	−56.8530	−1.79875

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2009.2.a.i.1.1	yes	3
7.6	odd	2		2009.2.a.h.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2009.2.a.h.1.1	✓	3	7.6	odd	2
2009.2.a.i.1.1	yes	3	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:	\( N \)	\(=\)	\( 2009 = 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2009.a (trivial)

\(f(q)\)	\(=\)	\(q-1.83424 q^{2} +2.83424 q^{3} +1.36445 q^{4} -5.19869 q^{6} +1.16576 q^{8} +5.03293 q^{9} +O(q^{10})\)	\(q-1.83424 q^{2} +2.83424 q^{3} +1.36445 q^{4} -5.19869 q^{6} +1.16576 q^{8} +5.03293 q^{9} -6.39738 q^{11} +3.86718 q^{12} +3.86718 q^{13} -4.86718 q^{16} -2.36445 q^{17} -9.23163 q^{18} -8.03293 q^{19} +11.7344 q^{22} +1.56314 q^{23} +3.30404 q^{24} -5.00000 q^{25} -7.09334 q^{26} +5.76183 q^{27} -3.66849 q^{29} -6.79476 q^{31} +6.59607 q^{32} -18.1317 q^{33} +4.33697 q^{34} +6.86718 q^{36} -9.86718 q^{37} +14.7344 q^{38} +10.9605 q^{39} +1.00000 q^{41} -0.364448 q^{43} -8.72890 q^{44} -2.86718 q^{46} +8.59607 q^{47} -13.7948 q^{48} +9.17122 q^{50} -6.70142 q^{51} +5.27656 q^{52} -6.39738 q^{53} -10.5686 q^{54} -22.7673 q^{57} +6.72890 q^{58} +12.4633 q^{59} +5.85517 q^{61} +12.4633 q^{62} -2.36445 q^{64} +33.2580 q^{66} -8.46325 q^{67} -3.22617 q^{68} +4.43032 q^{69} +4.06587 q^{71} +5.86718 q^{72} -7.73436 q^{73} +18.0988 q^{74} -14.1712 q^{75} -10.9605 q^{76} -20.1043 q^{78} -6.00000 q^{79} +1.23163 q^{81} -1.83424 q^{82} -4.72890 q^{83} +0.668486 q^{86} -10.3974 q^{87} -7.45779 q^{88} +3.86718 q^{89} +2.13282 q^{92} -19.2580 q^{93} -15.7673 q^{94} +18.6949 q^{96} -0.768374 q^{97} -32.1976 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 4 q^{4} - 10 q^{6} + 9 q^{8} + 4 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 4 * q^4 - 10 * q^6 + 9 * q^8 + 4 * q^9	\( 3 q + 3 q^{3} + 4 q^{4} - 10 q^{6} + 9 q^{8} + 4 q^{9} - 8 q^{11} - 5 q^{12} - 5 q^{13} + 2 q^{16} - 7 q^{17} - 11 q^{18} - 13 q^{19} + 2 q^{22} - q^{23} - q^{24} - 15 q^{25} - 21 q^{26} + 6 q^{27} + 2 q^{31} + 3 q^{32} - 10 q^{33} - 9 q^{34} + 4 q^{36} - 13 q^{37} + 11 q^{38} + 16 q^{39} + 3 q^{41} - q^{43} - 26 q^{44} + 8 q^{46} + 9 q^{47} - 19 q^{48} + 2 q^{51} - 17 q^{52} - 8 q^{53} + 7 q^{54} - 24 q^{57} + 20 q^{58} + 4 q^{59} + 6 q^{61} + 4 q^{62} - 7 q^{64} + 44 q^{66} + 8 q^{67} - 26 q^{68} - 9 q^{69} - 10 q^{71} + q^{72} + 10 q^{73} + 21 q^{74} - 15 q^{75} - 16 q^{76} + 6 q^{78} - 18 q^{79} - 13 q^{81} - 14 q^{83} - 9 q^{86} - 20 q^{87} - 22 q^{88} - 5 q^{89} + 23 q^{92} - 2 q^{93} - 3 q^{94} + 6 q^{96} - 19 q^{97} - 30 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 4 * q^4 - 10 * q^6 + 9 * q^8 + 4 * q^9 - 8 * q^11 - 5 * q^12 - 5 * q^13 + 2 * q^16 - 7 * q^17 - 11 * q^18 - 13 * q^19 + 2 * q^22 - q^23 - q^24 - 15 * q^25 - 21 * q^26 + 6 * q^27 + 2 * q^31 + 3 * q^32 - 10 * q^33 - 9 * q^34 + 4 * q^36 - 13 * q^37 + 11 * q^38 + 16 * q^39 + 3 * q^41 - q^43 - 26 * q^44 + 8 * q^46 + 9 * q^47 - 19 * q^48 + 2 * q^51 - 17 * q^52 - 8 * q^53 + 7 * q^54 - 24 * q^57 + 20 * q^58 + 4 * q^59 + 6 * q^61 + 4 * q^62 - 7 * q^64 + 44 * q^66 + 8 * q^67 - 26 * q^68 - 9 * q^69 - 10 * q^71 + q^72 + 10 * q^73 + 21 * q^74 - 15 * q^75 - 16 * q^76 + 6 * q^78 - 18 * q^79 - 13 * q^81 - 14 * q^83 - 9 * q^86 - 20 * q^87 - 22 * q^88 - 5 * q^89 + 23 * q^92 - 2 * q^93 - 3 * q^94 + 6 * q^96 - 19 * q^97 - 30 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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