LMFDB - Embedded newform 2005.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2005,2,Mod(1,2005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2005, 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("2005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2005 = 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2005.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:	$16.0100056053$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} +3.00000 q^{8} -3.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} +3.00000 q^{8} -3.00000 q^{9} -1.00000 q^{10} -4.00000 q^{11} +4.00000 q^{13} -1.00000 q^{16} +4.00000 q^{17} +3.00000 q^{18} -6.00000 q^{19} -1.00000 q^{20} +4.00000 q^{22} +8.00000 q^{23} +1.00000 q^{25} -4.00000 q^{26} -6.00000 q^{29} -2.00000 q^{31} -5.00000 q^{32} -4.00000 q^{34} +3.00000 q^{36} +4.00000 q^{37} +6.00000 q^{38} +3.00000 q^{40} +2.00000 q^{41} -4.00000 q^{43} +4.00000 q^{44} -3.00000 q^{45} -8.00000 q^{46} -7.00000 q^{49} -1.00000 q^{50} -4.00000 q^{52} -4.00000 q^{53} -4.00000 q^{55} +6.00000 q^{58} -6.00000 q^{59} +14.0000 q^{61} +2.00000 q^{62} +7.00000 q^{64} +4.00000 q^{65} -12.0000 q^{67} -4.00000 q^{68} -10.0000 q^{71} -9.00000 q^{72} -6.00000 q^{73} -4.00000 q^{74} +6.00000 q^{76} +6.00000 q^{79} -1.00000 q^{80} +9.00000 q^{81} -2.00000 q^{82} +4.00000 q^{83} +4.00000 q^{85} +4.00000 q^{86} -12.0000 q^{88} +6.00000 q^{89} +3.00000 q^{90} -8.00000 q^{92} -6.00000 q^{95} +7.00000 q^{98} +12.0000 q^{99} +O(q^{100})$

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 $

$2$	−1.00000	−0.707107	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	−1.00000	−0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	3.00000	1.06066
$9$	−3.00000	−1.00000
$10$	−1.00000	−0.316228
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	3.00000	0.707107
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	4.00000	0.852803
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−4.00000	−0.784465
$27$	0	0
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	−4.00000	−0.685994
$35$	0	0
$36$	3.00000	0.500000
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	6.00000	0.973329
$39$	0	0
$40$	3.00000	0.474342
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	4.00000	0.603023
$45$	−3.00000	−0.447214
$46$	−8.00000	−1.17954
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−4.00000	−0.554700
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	0	0
$58$	6.00000	0.787839
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	2.00000	0.254000
$63$	0	0
$64$	7.00000	0.875000
$65$	4.00000	0.496139
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	−4.00000	−0.485071
$69$	0	0
$70$	0	0
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	−9.00000	−1.06066
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	6.00000	0.688247
$77$	0	0
$78$	0	0
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	−1.00000	−0.111803
$81$	9.00000	1.00000
$82$	−2.00000	−0.220863
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	4.00000	0.431331
$87$	0	0
$88$	−12.0000	−1.27920
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	3.00000	0.316228
$91$	0	0
$92$	−8.00000	−0.834058
$93$	0	0
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	7.00000	0.707107
$99$	12.0000	1.20605
$100$	−1.00000	−0.100000
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	12.0000	1.17670
$105$	0	0
$106$	4.00000	0.388514
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	4.00000	0.381385
$111$	0	0
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	8.00000	0.746004
$116$	6.00000	0.557086
$117$	−12.0000	−1.10940
$118$	6.00000	0.552345
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	−14.0000	−1.26750
$123$	0	0
$124$	2.00000	0.179605
$125$	1.00000	0.0894427
$126$	0	0
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	3.00000	0.265165
$129$	0	0
$130$	−4.00000	−0.350823
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	0	0
$133$	0	0
$134$	12.0000	1.03664
$135$	0	0
$136$	12.0000	1.02899
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	0	0
$141$	0	0
$142$	10.0000	0.839181
$143$	−16.0000	−1.33799
$144$	3.00000	0.250000
$145$	−6.00000	−0.498273
$146$	6.00000	0.496564
$147$	0	0
$148$	−4.00000	−0.328798
$149$	−22.0000	−1.80231	−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000	−1.80231	−0.901155	+	0.433497i	$0.857280\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	−18.0000	−1.45999
$153$	−12.0000	−0.970143
$154$	0	0
$155$	−2.00000	−0.160644
$156$	0	0
$157$	−12.0000	−0.957704	−0.478852	−	0.877896i	$-0.658947\pi$
$157$	−12.0000	−0.957704	−0.478852	+	0.877896i	$0.658947\pi$
$158$	−6.00000	−0.477334
$159$	0	0
$160$	−5.00000	−0.395285
$161$	0	0
$162$	−9.00000	−0.707107
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	−4.00000	−0.310460
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	−4.00000	−0.306786
$171$	18.0000	1.37649
$172$	4.00000	0.304997
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	0	0
$176$	4.00000	0.301511
$177$	0	0
$178$	−6.00000	−0.449719
$179$	8.00000	0.597948	0.298974	−	0.954261i	$-0.403356\pi$
$179$	8.00000	0.597948	0.298974	+	0.954261i	$0.403356\pi$
$180$	3.00000	0.223607
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	0	0
$184$	24.0000	1.76930
$185$	4.00000	0.294086
$186$	0	0
$187$	−16.0000	−1.17004
$188$	0	0
$189$	0	0
$190$	6.00000	0.435286
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	0	0
$193$	24.0000	1.72756	0.863779	−	0.503871i	$-0.168091\pi$
$193$	24.0000	1.72756	0.863779	+	0.503871i	$0.168091\pi$
$194$	0	0
$195$	0	0
$196$	7.00000	0.500000
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−12.0000	−0.852803
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	3.00000	0.212132
$201$	0	0
$202$	14.0000	0.985037
$203$	0	0
$204$	0	0
$205$	2.00000	0.139686
$206$	0	0
$207$	−24.0000	−1.66812
$208$	−4.00000	−0.277350
$209$	24.0000	1.66011
$210$	0	0
$211$	−2.00000	−0.137686	−0.0688428	−	0.997628i	$-0.521931\pi$
$211$	−2.00000	−0.137686	−0.0688428	+	0.997628i	$0.521931\pi$
$212$	4.00000	0.274721
$213$	0	0
$214$	0	0
$215$	−4.00000	−0.272798
$216$	0	0
$217$	0	0
$218$	14.0000	0.948200
$219$	0	0
$220$	4.00000	0.269680
$221$	16.0000	1.07628
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	−3.00000	−0.200000
$226$	2.00000	0.133038
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	−8.00000	−0.527504
$231$	0	0
$232$	−18.0000	−1.18176
$233$	12.0000	0.786146	0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000	0.786146	0.393073	+	0.919507i	$0.371412\pi$
$234$	12.0000	0.784465
$235$	0	0
$236$	6.00000	0.390567
$237$	0	0
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	−5.00000	−0.321412
$243$	0	0
$244$	−14.0000	−0.896258
$245$	−7.00000	−0.447214
$246$	0	0
$247$	−24.0000	−1.52708
$248$	−6.00000	−0.381000
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−30.0000	−1.89358	−0.946792	−	0.321847i	$-0.895696\pi$
$251$	−30.0000	−1.89358	−0.946792	+	0.321847i	$0.895696\pi$
$252$	0	0
$253$	−32.0000	−2.01182
$254$	12.0000	0.752947
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	0	0
$260$	−4.00000	−0.248069
$261$	18.0000	1.11417
$262$	18.0000	1.11204
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	−4.00000	−0.245718
$266$	0	0
$267$	0	0
$268$	12.0000	0.733017
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	0	0
$271$	26.0000	1.57939	0.789694	−	0.613501i	$-0.210239\pi$
$271$	26.0000	1.57939	0.789694	+	0.613501i	$0.210239\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	−12.0000	−0.724947
$275$	−4.00000	−0.241209
$276$	0	0
$277$	16.0000	0.961347	0.480673	−	0.876900i	$-0.340392\pi$
$277$	16.0000	0.961347	0.480673	+	0.876900i	$0.340392\pi$
$278$	10.0000	0.599760
$279$	6.00000	0.359211
$280$	0	0
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	0	0
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	10.0000	0.593391
$285$	0	0
$286$	16.0000	0.946100
$287$	0	0
$288$	15.0000	0.883883
$289$	−1.00000	−0.0588235
$290$	6.00000	0.352332
$291$	0	0
$292$	6.00000	0.351123
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	−6.00000	−0.349334
$296$	12.0000	0.697486
$297$	0	0
$298$	22.0000	1.27443
$299$	32.0000	1.85061
$300$	0	0
$301$	0	0
$302$	8.00000	0.460348
$303$	0	0
$304$	6.00000	0.344124
$305$	14.0000	0.801638
$306$	12.0000	0.685994
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	0	0
$309$	0	0
$310$	2.00000	0.113592
$311$	−16.0000	−0.907277	−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000	−0.907277	−0.453638	+	0.891186i	$0.649874\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	12.0000	0.677199
$315$	0	0
$316$	−6.00000	−0.337526
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	0	0
$319$	24.0000	1.34374
$320$	7.00000	0.391312
$321$	0	0
$322$	0	0
$323$	−24.0000	−1.33540
$324$	−9.00000	−0.500000
$325$	4.00000	0.221880
$326$	4.00000	0.221540
$327$	0	0
$328$	6.00000	0.331295
$329$	0	0
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	−4.00000	−0.219529
$333$	−12.0000	−0.657596
$334$	16.0000	0.875481
$335$	−12.0000	−0.655630
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	−3.00000	−0.163178
$339$	0	0
$340$	−4.00000	−0.216930
$341$	8.00000	0.433224
$342$	−18.0000	−0.973329
$343$	0	0
$344$	−12.0000	−0.646997
$345$	0	0
$346$	−2.00000	−0.107521
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	22.0000	1.17763	0.588817	−	0.808267i	$-0.299594\pi$
$349$	22.0000	1.17763	0.588817	+	0.808267i	$0.299594\pi$
$350$	0	0
$351$	0	0
$352$	20.0000	1.06600
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	0	0
$355$	−10.0000	−0.530745
$356$	−6.00000	−0.317999
$357$	0	0
$358$	−8.00000	−0.422813
$359$	2.00000	0.105556	0.0527780	−	0.998606i	$-0.483192\pi$
$359$	2.00000	0.105556	0.0527780	+	0.998606i	$0.483192\pi$
$360$	−9.00000	−0.474342
$361$	17.0000	0.894737
$362$	22.0000	1.15629
$363$	0	0
$364$	0	0
$365$	−6.00000	−0.314054
$366$	0	0
$367$	20.0000	1.04399	0.521996	−	0.852948i	$-0.325188\pi$
$367$	20.0000	1.04399	0.521996	+	0.852948i	$0.325188\pi$
$368$	−8.00000	−0.417029
$369$	−6.00000	−0.312348
$370$	−4.00000	−0.207950
$371$	0	0
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	16.0000	0.827340
$375$	0	0
$376$	0	0
$377$	−24.0000	−1.23606
$378$	0	0
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	6.00000	0.307794
$381$	0	0
$382$	6.00000	0.306987
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	0	0
$385$	0	0
$386$	−24.0000	−1.22157
$387$	12.0000	0.609994
$388$	0	0
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	32.0000	1.61831
$392$	−21.0000	−1.06066
$393$	0	0
$394$	6.00000	0.302276
$395$	6.00000	0.301893
$396$	−12.0000	−0.603023
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	−2.00000	−0.100251
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	−8.00000	−0.398508
$404$	14.0000	0.696526
$405$	9.00000	0.447214
$406$	0	0
$407$	−16.0000	−0.793091
$408$	0	0
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	−2.00000	−0.0987730
$411$	0	0
$412$	0	0
$413$	0	0
$414$	24.0000	1.17954
$415$	4.00000	0.196352
$416$	−20.0000	−0.980581
$417$	0	0
$418$	−24.0000	−1.17388
$419$	−16.0000	−0.781651	−0.390826	−	0.920465i	$-0.627810\pi$
$419$	−16.0000	−0.781651	−0.390826	+	0.920465i	$0.627810\pi$
$420$	0	0
$421$	38.0000	1.85201	0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000	1.85201	0.926003	+	0.377515i	$0.123221\pi$
$422$	2.00000	0.0973585
$423$	0	0
$424$	−12.0000	−0.582772
$425$	4.00000	0.194029
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	4.00000	0.192897
$431$	38.0000	1.83040	0.915198	−	0.403005i	$-0.132034\pi$
$431$	38.0000	1.83040	0.915198	+	0.403005i	$0.132034\pi$
$432$	0	0
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	0	0
$435$	0	0
$436$	14.0000	0.670478
$437$	−48.0000	−2.29615
$438$	0	0
$439$	−6.00000	−0.286364	−0.143182	−	0.989696i	$-0.545733\pi$
$439$	−6.00000	−0.286364	−0.143182	+	0.989696i	$0.545733\pi$
$440$	−12.0000	−0.572078
$441$	21.0000	1.00000
$442$	−16.0000	−0.761042
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	−16.0000	−0.757622
$447$	0	0
$448$	0	0
$449$	−22.0000	−1.03824	−0.519122	−	0.854700i	$-0.673741\pi$
$449$	−22.0000	−1.03824	−0.519122	+	0.854700i	$0.673741\pi$
$450$	3.00000	0.141421
$451$	−8.00000	−0.376705
$452$	2.00000	0.0940721
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	14.0000	0.654892	0.327446	−	0.944870i	$-0.393812\pi$
$457$	14.0000	0.654892	0.327446	+	0.944870i	$0.393812\pi$
$458$	10.0000	0.467269
$459$	0	0
$460$	−8.00000	−0.373002
$461$	−22.0000	−1.02464	−0.512321	−	0.858794i	$-0.671214\pi$
$461$	−22.0000	−1.02464	−0.512321	+	0.858794i	$0.671214\pi$
$462$	0	0
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−12.0000	−0.555889
$467$	−16.0000	−0.740392	−0.370196	−	0.928954i	$-0.620709\pi$
$467$	−16.0000	−0.740392	−0.370196	+	0.928954i	$0.620709\pi$
$468$	12.0000	0.554700
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−18.0000	−0.828517
$473$	16.0000	0.735681
$474$	0	0
$475$	−6.00000	−0.275299
$476$	0	0
$477$	12.0000	0.549442
$478$	24.0000	1.09773
$479$	−20.0000	−0.913823	−0.456912	−	0.889512i	$-0.651044\pi$
$479$	−20.0000	−0.913823	−0.456912	+	0.889512i	$0.651044\pi$
$480$	0	0
$481$	16.0000	0.729537
$482$	2.00000	0.0910975
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	0	0
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	42.0000	1.90125
$489$	0	0
$490$	7.00000	0.316228
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	0	0
$493$	−24.0000	−1.08091
$494$	24.0000	1.07981
$495$	12.0000	0.539360
$496$	2.00000	0.0898027
$497$	0	0
$498$	0	0
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	30.0000	1.33897
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	−14.0000	−0.622992
$506$	32.0000	1.42257
$507$	0	0
$508$	12.0000	0.532414
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	0	0
$512$	11.0000	0.486136
$513$	0	0
$514$	2.00000	0.0882162
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	12.0000	0.526235
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	−18.0000	−0.787839
$523$	12.0000	0.524723	0.262362	−	0.964970i	$-0.415499\pi$
$523$	12.0000	0.524723	0.262362	+	0.964970i	$0.415499\pi$
$524$	18.0000	0.786334
$525$	0	0
$526$	16.0000	0.697633
$527$	−8.00000	−0.348485
$528$	0	0
$529$	41.0000	1.78261
$530$	4.00000	0.173749
$531$	18.0000	0.781133
$532$	0	0
$533$	8.00000	0.346518
$534$	0	0
$535$	0	0
$536$	−36.0000	−1.55496
$537$	0	0
$538$	−10.0000	−0.431131
$539$	28.0000	1.20605
$540$	0	0
$541$	−26.0000	−1.11783	−0.558914	−	0.829226i	$-0.688782\pi$
$541$	−26.0000	−1.11783	−0.558914	+	0.829226i	$0.688782\pi$
$542$	−26.0000	−1.11680
$543$	0	0
$544$	−20.0000	−0.857493
$545$	−14.0000	−0.599694
$546$	0	0
$547$	36.0000	1.53925	0.769624	−	0.638497i	$-0.220443\pi$
$547$	36.0000	1.53925	0.769624	+	0.638497i	$0.220443\pi$
$548$	−12.0000	−0.512615
$549$	−42.0000	−1.79252
$550$	4.00000	0.170561
$551$	36.0000	1.53365
$552$	0	0
$553$	0	0
$554$	−16.0000	−0.679775
$555$	0	0
$556$	10.0000	0.424094
$557$	38.0000	1.61011	0.805056	−	0.593199i	$-0.202135\pi$
$557$	38.0000	1.61011	0.805056	+	0.593199i	$0.202135\pi$
$558$	−6.00000	−0.254000
$559$	−16.0000	−0.676728
$560$	0	0
$561$	0	0
$562$	2.00000	0.0843649
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	0	0
$565$	−2.00000	−0.0841406
$566$	−12.0000	−0.504398
$567$	0	0
$568$	−30.0000	−1.25877
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	−10.0000	−0.418487	−0.209243	−	0.977864i	$-0.567100\pi$
$571$	−10.0000	−0.418487	−0.209243	+	0.977864i	$0.567100\pi$
$572$	16.0000	0.668994
$573$	0	0
$574$	0	0
$575$	8.00000	0.333623
$576$	−21.0000	−0.875000
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	1.00000	0.0415945
$579$	0	0
$580$	6.00000	0.249136
$581$	0	0
$582$	0	0
$583$	16.0000	0.662652
$584$	−18.0000	−0.744845
$585$	−12.0000	−0.496139
$586$	0	0
$587$	4.00000	0.165098	0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000	0.165098	0.0825488	+	0.996587i	$0.473694\pi$
$588$	0	0
$589$	12.0000	0.494451
$590$	6.00000	0.247016
$591$	0	0
$592$	−4.00000	−0.164399
$593$	36.0000	1.47834	0.739171	−	0.673517i	$-0.235217\pi$
$593$	36.0000	1.47834	0.739171	+	0.673517i	$0.235217\pi$
$594$	0	0
$595$	0	0
$596$	22.0000	0.901155
$597$	0	0
$598$	−32.0000	−1.30858
$599$	−8.00000	−0.326871	−0.163436	−	0.986554i	$-0.552258\pi$
$599$	−8.00000	−0.326871	−0.163436	+	0.986554i	$0.552258\pi$
$600$	0	0
$601$	38.0000	1.55005	0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000	1.55005	0.775026	+	0.631929i	$0.217737\pi$
$602$	0	0
$603$	36.0000	1.46603
$604$	8.00000	0.325515
$605$	5.00000	0.203279
$606$	0	0
$607$	−24.0000	−0.974130	−0.487065	−	0.873366i	$-0.661933\pi$
$607$	−24.0000	−0.974130	−0.487065	+	0.873366i	$0.661933\pi$
$608$	30.0000	1.21666
$609$	0	0
$610$	−14.0000	−0.566843
$611$	0	0
$612$	12.0000	0.485071
$613$	28.0000	1.13091	0.565455	−	0.824779i	$-0.308701\pi$
$613$	28.0000	1.13091	0.565455	+	0.824779i	$0.308701\pi$
$614$	−20.0000	−0.807134
$615$	0	0
$616$	0	0
$617$	−20.0000	−0.805170	−0.402585	−	0.915383i	$-0.631888\pi$
$617$	−20.0000	−0.805170	−0.402585	+	0.915383i	$0.631888\pi$
$618$	0	0
$619$	8.00000	0.321547	0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000	0.321547	0.160774	+	0.986991i	$0.448601\pi$
$620$	2.00000	0.0803219
$621$	0	0
$622$	16.0000	0.641542
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	−14.0000	−0.559553
$627$	0	0
$628$	12.0000	0.478852
$629$	16.0000	0.637962
$630$	0	0
$631$	26.0000	1.03504	0.517522	−	0.855670i	$-0.326855\pi$
$631$	26.0000	1.03504	0.517522	+	0.855670i	$0.326855\pi$
$632$	18.0000	0.716002
$633$	0	0
$634$	−12.0000	−0.476581
$635$	−12.0000	−0.476205
$636$	0	0
$637$	−28.0000	−1.10940
$638$	−24.0000	−0.950169
$639$	30.0000	1.18678
$640$	3.00000	0.118585
$641$	−26.0000	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$642$	0	0
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	0	0
$645$	0	0
$646$	24.0000	0.944267
$647$	−36.0000	−1.41531	−0.707653	−	0.706560i	$-0.750246\pi$
$647$	−36.0000	−1.41531	−0.707653	+	0.706560i	$0.750246\pi$
$648$	27.0000	1.06066
$649$	24.0000	0.942082
$650$	−4.00000	−0.156893
$651$	0	0
$652$	4.00000	0.156652
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	0	0
$655$	−18.0000	−0.703318
$656$	−2.00000	−0.0780869
$657$	18.0000	0.702247
$658$	0	0
$659$	−26.0000	−1.01282	−0.506408	−	0.862294i	$-0.669027\pi$
$659$	−26.0000	−1.01282	−0.506408	+	0.862294i	$0.669027\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	−8.00000	−0.310929
$663$	0	0
$664$	12.0000	0.465690
$665$	0	0
$666$	12.0000	0.464991
$667$	−48.0000	−1.85857
$668$	16.0000	0.619059
$669$	0	0
$670$	12.0000	0.463600
$671$	−56.0000	−2.16186
$672$	0	0
$673$	−36.0000	−1.38770	−0.693849	−	0.720121i	$-0.744086\pi$
$673$	−36.0000	−1.38770	−0.693849	+	0.720121i	$0.744086\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	−3.00000	−0.115385
$677$	42.0000	1.61419	0.807096	−	0.590421i	$-0.201038\pi$
$677$	42.0000	1.61419	0.807096	+	0.590421i	$0.201038\pi$
$678$	0	0
$679$	0	0
$680$	12.0000	0.460179
$681$	0	0
$682$	−8.00000	−0.306336
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	−18.0000	−0.688247
$685$	12.0000	0.458496
$686$	0	0
$687$	0	0
$688$	4.00000	0.152499
$689$	−16.0000	−0.609551
$690$	0	0
$691$	−40.0000	−1.52167	−0.760836	−	0.648944i	$-0.775211\pi$
$691$	−40.0000	−1.52167	−0.760836	+	0.648944i	$0.775211\pi$
$692$	−2.00000	−0.0760286
$693$	0	0
$694$	12.0000	0.455514
$695$	−10.0000	−0.379322
$696$	0	0
$697$	8.00000	0.303022
$698$	−22.0000	−0.832712
$699$	0	0
$700$	0	0
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	0	0
$703$	−24.0000	−0.905177
$704$	−28.0000	−1.05529
$705$	0	0
$706$	−24.0000	−0.903252
$707$	0	0
$708$	0	0
$709$	42.0000	1.57734	0.788672	−	0.614815i	$-0.210769\pi$
$709$	42.0000	1.57734	0.788672	+	0.614815i	$0.210769\pi$
$710$	10.0000	0.375293
$711$	−18.0000	−0.675053
$712$	18.0000	0.674579
$713$	−16.0000	−0.599205
$714$	0	0
$715$	−16.0000	−0.598366
$716$	−8.00000	−0.298974
$717$	0	0
$718$	−2.00000	−0.0746393
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	3.00000	0.111803
$721$	0	0
$722$	−17.0000	−0.632674
$723$	0	0
$724$	22.0000	0.817624
$725$	−6.00000	−0.222834
$726$	0	0
$727$	28.0000	1.03846	0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	6.00000	0.222070
$731$	−16.0000	−0.591781
$732$	0	0
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	−20.0000	−0.738213
$735$	0	0
$736$	−40.0000	−1.47442
$737$	48.0000	1.76810
$738$	6.00000	0.220863
$739$	24.0000	0.882854	0.441427	−	0.897297i	$-0.354472\pi$
$739$	24.0000	0.882854	0.441427	+	0.897297i	$0.354472\pi$
$740$	−4.00000	−0.147043
$741$	0	0
$742$	0	0
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	−22.0000	−0.806018
$746$	10.0000	0.366126
$747$	−12.0000	−0.439057
$748$	16.0000	0.585018
$749$	0	0
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	0	0
$753$	0	0
$754$	24.0000	0.874028
$755$	−8.00000	−0.291150
$756$	0	0
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	8.00000	0.290573
$759$	0	0
$760$	−18.0000	−0.652929
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	0	0
$763$	0	0
$764$	6.00000	0.217072
$765$	−12.0000	−0.433861
$766$	−16.0000	−0.578103
$767$	−24.0000	−0.866590
$768$	0	0
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	0	0
$772$	−24.0000	−0.863779
$773$	−42.0000	−1.51064	−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000	−1.51064	−0.755318	+	0.655359i	$0.772517\pi$
$774$	−12.0000	−0.431331
$775$	−2.00000	−0.0718421
$776$	0	0
$777$	0	0
$778$	30.0000	1.07555
$779$	−12.0000	−0.429945
$780$	0	0
$781$	40.0000	1.43131
$782$	−32.0000	−1.14432
$783$	0	0
$784$	7.00000	0.250000
$785$	−12.0000	−0.428298
$786$	0	0
$787$	−32.0000	−1.14068	−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000	−1.14068	−0.570338	+	0.821410i	$0.693188\pi$
$788$	6.00000	0.213741
$789$	0	0
$790$	−6.00000	−0.213470
$791$	0	0
$792$	36.0000	1.27920
$793$	56.0000	1.98862
$794$	22.0000	0.780751
$795$	0	0
$796$	−2.00000	−0.0708881
$797$	−38.0000	−1.34603	−0.673015	−	0.739629i	$-0.735001\pi$
$797$	−38.0000	−1.34603	−0.673015	+	0.739629i	$0.735001\pi$
$798$	0	0
$799$	0	0
$800$	−5.00000	−0.176777
$801$	−18.0000	−0.635999
$802$	−1.00000	−0.0353112
$803$	24.0000	0.846942
$804$	0	0
$805$	0	0
$806$	8.00000	0.281788
$807$	0	0
$808$	−42.0000	−1.47755
$809$	50.0000	1.75791	0.878953	−	0.476908i	$-0.158243\pi$
$809$	50.0000	1.75791	0.878953	+	0.476908i	$0.158243\pi$
$810$	−9.00000	−0.316228
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	16.0000	0.560800
$815$	−4.00000	−0.140114
$816$	0	0
$817$	24.0000	0.839654
$818$	−6.00000	−0.209785
$819$	0	0
$820$	−2.00000	−0.0698430
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	0	0
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	24.0000	0.834058
$829$	−30.0000	−1.04194	−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000	−1.04194	−0.520972	+	0.853574i	$0.674430\pi$
$830$	−4.00000	−0.138842
$831$	0	0
$832$	28.0000	0.970725
$833$	−28.0000	−0.970143
$834$	0	0
$835$	−16.0000	−0.553703
$836$	−24.0000	−0.830057
$837$	0	0
$838$	16.0000	0.552711
$839$	−18.0000	−0.621429	−0.310715	−	0.950503i	$-0.600568\pi$
$839$	−18.0000	−0.621429	−0.310715	+	0.950503i	$0.600568\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−38.0000	−1.30957
$843$	0	0
$844$	2.00000	0.0688428
$845$	3.00000	0.103203
$846$	0	0
$847$	0	0
$848$	4.00000	0.137361
$849$	0	0
$850$	−4.00000	−0.137199
$851$	32.0000	1.09695
$852$	0	0
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	0	0
$855$	18.0000	0.615587
$856$	0	0
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	16.0000	0.545913	0.272956	−	0.962026i	$-0.411998\pi$
$859$	16.0000	0.545913	0.272956	+	0.962026i	$0.411998\pi$
$860$	4.00000	0.136399
$861$	0	0
$862$	−38.0000	−1.29429
$863$	−36.0000	−1.22545	−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000	−1.22545	−0.612727	+	0.790295i	$0.709928\pi$
$864$	0	0
$865$	2.00000	0.0680020
$866$	26.0000	0.883516
$867$	0	0
$868$	0	0
$869$	−24.0000	−0.814144
$870$	0	0
$871$	−48.0000	−1.62642
$872$	−42.0000	−1.42230
$873$	0	0
$874$	48.0000	1.62362
$875$	0	0
$876$	0	0
$877$	−32.0000	−1.08056	−0.540282	−	0.841484i	$-0.681682\pi$
$877$	−32.0000	−1.08056	−0.540282	+	0.841484i	$0.681682\pi$
$878$	6.00000	0.202490
$879$	0	0
$880$	4.00000	0.134840
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	−21.0000	−0.707107
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	−16.0000	−0.538138
$885$	0	0
$886$	−4.00000	−0.134383
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	0	0
$889$	0	0
$890$	−6.00000	−0.201120
$891$	−36.0000	−1.20605
$892$	−16.0000	−0.535720
$893$	0	0
$894$	0	0
$895$	8.00000	0.267411
$896$	0	0
$897$	0	0
$898$	22.0000	0.734150
$899$	12.0000	0.400222
$900$	3.00000	0.100000
$901$	−16.0000	−0.533037
$902$	8.00000	0.266371
$903$	0	0
$904$	−6.00000	−0.199557
$905$	−22.0000	−0.731305
$906$	0	0
$907$	32.0000	1.06254	0.531271	−	0.847202i	$-0.321714\pi$
$907$	32.0000	1.06254	0.531271	+	0.847202i	$0.321714\pi$
$908$	0	0
$909$	42.0000	1.39305
$910$	0	0
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	0	0
$913$	−16.0000	−0.529523
$914$	−14.0000	−0.463079
$915$	0	0
$916$	10.0000	0.330409
$917$	0	0
$918$	0	0
$919$	34.0000	1.12156	0.560778	−	0.827966i	$-0.310502\pi$
$919$	34.0000	1.12156	0.560778	+	0.827966i	$0.310502\pi$
$920$	24.0000	0.791257
$921$	0	0
$922$	22.0000	0.724531
$923$	−40.0000	−1.31662
$924$	0	0
$925$	4.00000	0.131519
$926$	−24.0000	−0.788689
$927$	0	0
$928$	30.0000	0.984798
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	42.0000	1.37649
$932$	−12.0000	−0.393073
$933$	0	0
$934$	16.0000	0.523536
$935$	−16.0000	−0.523256
$936$	−36.0000	−1.17670
$937$	32.0000	1.04539	0.522697	−	0.852518i	$-0.324926\pi$
$937$	32.0000	1.04539	0.522697	+	0.852518i	$0.324926\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−46.0000	−1.49956	−0.749779	−	0.661689i	$-0.769840\pi$
$941$	−46.0000	−1.49956	−0.749779	+	0.661689i	$0.769840\pi$
$942$	0	0
$943$	16.0000	0.521032
$944$	6.00000	0.195283
$945$	0	0
$946$	−16.0000	−0.520205
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	0	0
$949$	−24.0000	−0.779073
$950$	6.00000	0.194666
$951$	0	0
$952$	0	0
$953$	14.0000	0.453504	0.226752	−	0.973952i	$-0.427189\pi$
$953$	14.0000	0.453504	0.226752	+	0.973952i	$0.427189\pi$
$954$	−12.0000	−0.388514
$955$	−6.00000	−0.194155
$956$	24.0000	0.776215
$957$	0	0
$958$	20.0000	0.646171
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−16.0000	−0.515861
$963$	0	0
$964$	2.00000	0.0644157
$965$	24.0000	0.772587
$966$	0	0
$967$	16.0000	0.514525	0.257263	−	0.966342i	$-0.417179\pi$
$967$	16.0000	0.514525	0.257263	+	0.966342i	$0.417179\pi$
$968$	15.0000	0.482118
$969$	0	0
$970$	0	0
$971$	24.0000	0.770197	0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000	0.770197	0.385098	+	0.922876i	$0.374168\pi$
$972$	0	0
$973$	0	0
$974$	−32.0000	−1.02535
$975$	0	0
$976$	−14.0000	−0.448129
$977$	−2.00000	−0.0639857	−0.0319928	−	0.999488i	$-0.510185\pi$
$977$	−2.00000	−0.0639857	−0.0319928	+	0.999488i	$0.510185\pi$
$978$	0	0
$979$	−24.0000	−0.767043
$980$	7.00000	0.223607
$981$	42.0000	1.34096
$982$	−20.0000	−0.638226
$983$	−32.0000	−1.02064	−0.510321	−	0.859984i	$-0.670473\pi$
$983$	−32.0000	−1.02064	−0.510321	+	0.859984i	$0.670473\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	24.0000	0.764316
$987$	0	0
$988$	24.0000	0.763542
$989$	−32.0000	−1.01754
$990$	−12.0000	−0.381385
$991$	−42.0000	−1.33417	−0.667087	−	0.744980i	$-0.732459\pi$
$991$	−42.0000	−1.33417	−0.667087	+	0.744980i	$0.732459\pi$
$992$	10.0000	0.317500
$993$	0	0
$994$	0	0
$995$	2.00000	0.0634043
$996$	0	0
$997$	22.0000	0.696747	0.348373	−	0.937356i	$-0.386734\pi$
$997$	22.0000	0.696747	0.348373	+	0.937356i	$0.386734\pi$
$998$	−16.0000	−0.506471
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2005.2.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2005.2.a.a.1.1	✓	1	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:	\( N \)	\(=\)	\( 2005 = 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2005.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more