LMFDB - Embedded newform 4005.2.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4005,2,Mod(1,4005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4005, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4005 = 3^{2} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4005.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:	$31.9800860095$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 445)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	4005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.732051 q^{7} -1.73205 q^{8} +O(q^{10})$	$q+1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.732051 q^{7} -1.73205 q^{8} -1.73205 q^{10} +2.00000 q^{13} +1.26795 q^{14} -5.00000 q^{16} -2.73205 q^{19} -1.00000 q^{20} -4.73205 q^{23} +1.00000 q^{25} +3.46410 q^{26} +0.732051 q^{28} -3.46410 q^{29} +4.19615 q^{31} -5.19615 q^{32} -0.732051 q^{35} -0.535898 q^{37} -4.73205 q^{38} +1.73205 q^{40} -3.46410 q^{41} +12.7321 q^{43} -8.19615 q^{46} -12.9282 q^{47} -6.46410 q^{49} +1.73205 q^{50} +2.00000 q^{52} -6.92820 q^{53} -1.26795 q^{56} -6.00000 q^{58} -3.80385 q^{59} +2.00000 q^{61} +7.26795 q^{62} +1.00000 q^{64} -2.00000 q^{65} -14.3923 q^{67} -1.26795 q^{70} -9.46410 q^{71} -4.92820 q^{73} -0.928203 q^{74} -2.73205 q^{76} -8.39230 q^{79} +5.00000 q^{80} -6.00000 q^{82} -2.19615 q^{83} +22.0526 q^{86} +1.00000 q^{89} +1.46410 q^{91} -4.73205 q^{92} -22.3923 q^{94} +2.73205 q^{95} -4.00000 q^{97} -11.1962 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} - 2 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^4 - 2 * q^5 - 2 * q^7	$ 2 q + 2 q^{4} - 2 q^{5} - 2 q^{7} + 4 q^{13} + 6 q^{14} - 10 q^{16} - 2 q^{19} - 2 q^{20} - 6 q^{23} + 2 q^{25} - 2 q^{28} - 2 q^{31} + 2 q^{35} - 8 q^{37} - 6 q^{38} + 22 q^{43} - 6 q^{46} - 12 q^{47} - 6 q^{49} + 4 q^{52} - 6 q^{56} - 12 q^{58} - 18 q^{59} + 4 q^{61} + 18 q^{62} + 2 q^{64} - 4 q^{65} - 8 q^{67} - 6 q^{70} - 12 q^{71} + 4 q^{73} + 12 q^{74} - 2 q^{76} + 4 q^{79} + 10 q^{80} - 12 q^{82} + 6 q^{83} + 6 q^{86} + 2 q^{89} - 4 q^{91} - 6 q^{92} - 24 q^{94} + 2 q^{95} - 8 q^{97} - 12 q^{98}+O(q^{100}) $ 2 * q + 2 * q^4 - 2 * q^5 - 2 * q^7 + 4 * q^13 + 6 * q^14 - 10 * q^16 - 2 * q^19 - 2 * q^20 - 6 * q^23 + 2 * q^25 - 2 * q^28 - 2 * q^31 + 2 * q^35 - 8 * q^37 - 6 * q^38 + 22 * q^43 - 6 * q^46 - 12 * q^47 - 6 * q^49 + 4 * q^52 - 6 * q^56 - 12 * q^58 - 18 * q^59 + 4 * q^61 + 18 * q^62 + 2 * q^64 - 4 * q^65 - 8 * q^67 - 6 * q^70 - 12 * q^71 + 4 * q^73 + 12 * q^74 - 2 * q^76 + 4 * q^79 + 10 * q^80 - 12 * q^82 + 6 * q^83 + 6 * q^86 + 2 * q^89 - 4 * q^91 - 6 * q^92 - 24 * q^94 + 2 * q^95 - 8 * q^97 - 12 * q^98

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 $

$2$	1.73205	1.22474	0.612372	−	0.790569i	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.732051	0.276689	0.138345	−	0.990384i	$-0.455822\pi$
$7$	0.732051	0.276689	0.138345	+	0.990384i	$0.455822\pi$
$8$	−1.73205	−0.612372
$9$	0	0
$10$	−1.73205	−0.547723
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	1.26795	0.338874
$15$	0	0
$16$	−5.00000	−1.25000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−2.73205	−0.626775	−0.313388	−	0.949625i	$-0.601464\pi$
$19$	−2.73205	−0.626775	−0.313388	+	0.949625i	$0.601464\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	0	0
$23$	−4.73205	−0.986701	−0.493350	−	0.869831i	$-0.664228\pi$
$23$	−4.73205	−0.986701	−0.493350	+	0.869831i	$0.664228\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	3.46410	0.679366
$27$	0	0
$28$	0.732051	0.138345
$29$	−3.46410	−0.643268	−0.321634	−	0.946864i	$-0.604232\pi$
$29$	−3.46410	−0.643268	−0.321634	+	0.946864i	$0.604232\pi$
$30$	0	0
$31$	4.19615	0.753651	0.376826	−	0.926284i	$-0.377016\pi$
$31$	4.19615	0.753651	0.376826	+	0.926284i	$0.377016\pi$
$32$	−5.19615	−0.918559
$33$	0	0
$34$	0	0
$35$	−0.732051	−0.123739
$36$	0	0
$37$	−0.535898	−0.0881012	−0.0440506	−	0.999029i	$-0.514026\pi$
$37$	−0.535898	−0.0881012	−0.0440506	+	0.999029i	$0.514026\pi$
$38$	−4.73205	−0.767640
$39$	0	0
$40$	1.73205	0.273861
$41$	−3.46410	−0.541002	−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410	−0.541002	−0.270501	+	0.962720i	$0.587189\pi$
$42$	0	0
$43$	12.7321	1.94162	0.970810	−	0.239851i	$-0.0770985\pi$
$43$	12.7321	1.94162	0.970810	+	0.239851i	$0.0770985\pi$
$44$	0	0
$45$	0	0
$46$	−8.19615	−1.20846
$47$	−12.9282	−1.88577	−0.942886	−	0.333115i	$-0.891900\pi$
$47$	−12.9282	−1.88577	−0.942886	+	0.333115i	$0.891900\pi$
$48$	0	0
$49$	−6.46410	−0.923443
$50$	1.73205	0.244949
$51$	0	0
$52$	2.00000	0.277350
$53$	−6.92820	−0.951662	−0.475831	−	0.879537i	$-0.657853\pi$
$53$	−6.92820	−0.951662	−0.475831	+	0.879537i	$0.657853\pi$
$54$	0	0
$55$	0	0
$56$	−1.26795	−0.169437
$57$	0	0
$58$	−6.00000	−0.787839
$59$	−3.80385	−0.495219	−0.247609	−	0.968860i	$-0.579645\pi$
$59$	−3.80385	−0.495219	−0.247609	+	0.968860i	$0.579645\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	7.26795	0.923030
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−14.3923	−1.75830	−0.879150	−	0.476545i	$-0.841889\pi$
$67$	−14.3923	−1.75830	−0.879150	+	0.476545i	$0.841889\pi$
$68$	0	0
$69$	0	0
$70$	−1.26795	−0.151549
$71$	−9.46410	−1.12318	−0.561591	−	0.827415i	$-0.689811\pi$
$71$	−9.46410	−1.12318	−0.561591	+	0.827415i	$0.689811\pi$
$72$	0	0
$73$	−4.92820	−0.576803	−0.288401	−	0.957510i	$-0.593124\pi$
$73$	−4.92820	−0.576803	−0.288401	+	0.957510i	$0.593124\pi$
$74$	−0.928203	−0.107901
$75$	0	0
$76$	−2.73205	−0.313388
$77$	0	0
$78$	0	0
$79$	−8.39230	−0.944208	−0.472104	−	0.881543i	$-0.656505\pi$
$79$	−8.39230	−0.944208	−0.472104	+	0.881543i	$0.656505\pi$
$80$	5.00000	0.559017
$81$	0	0
$82$	−6.00000	−0.662589
$83$	−2.19615	−0.241059	−0.120530	−	0.992710i	$-0.538459\pi$
$83$	−2.19615	−0.241059	−0.120530	+	0.992710i	$0.538459\pi$
$84$	0	0
$85$	0	0
$86$	22.0526	2.37799
$87$	0	0
$88$	0	0
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	1.46410	0.153480
$92$	−4.73205	−0.493350
$93$	0	0
$94$	−22.3923	−2.30959
$95$	2.73205	0.280302
$96$	0	0
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	−11.1962	−1.13098
$99$	0	0
$100$	1.00000	0.100000
$101$	3.46410	0.344691	0.172345	−	0.985037i	$-0.444865\pi$
$101$	3.46410	0.344691	0.172345	+	0.985037i	$0.444865\pi$
$102$	0	0
$103$	10.1962	1.00466	0.502328	−	0.864677i	$-0.332477\pi$
$103$	10.1962	1.00466	0.502328	+	0.864677i	$0.332477\pi$
$104$	−3.46410	−0.339683
$105$	0	0
$106$	−12.0000	−1.16554
$107$	0.928203	0.0897328	0.0448664	−	0.998993i	$-0.485714\pi$
$107$	0.928203	0.0897328	0.0448664	+	0.998993i	$0.485714\pi$
$108$	0	0
$109$	10.5359	1.00916	0.504578	−	0.863366i	$-0.331648\pi$
$109$	10.5359	1.00916	0.504578	+	0.863366i	$0.331648\pi$
$110$	0	0
$111$	0	0
$112$	−3.66025	−0.345861
$113$	0.928203	0.0873180	0.0436590	−	0.999046i	$-0.486098\pi$
$113$	0.928203	0.0873180	0.0436590	+	0.999046i	$0.486098\pi$
$114$	0	0
$115$	4.73205	0.441266
$116$	−3.46410	−0.321634
$117$	0	0
$118$	−6.58846	−0.606517
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	3.46410	0.313625
$123$	0	0
$124$	4.19615	0.376826
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−10.5885	−0.939574	−0.469787	−	0.882780i	$-0.655669\pi$
$127$	−10.5885	−0.939574	−0.469787	+	0.882780i	$0.655669\pi$
$128$	12.1244	1.07165
$129$	0	0
$130$	−3.46410	−0.303822
$131$	9.46410	0.826882	0.413441	−	0.910531i	$-0.364327\pi$
$131$	9.46410	0.826882	0.413441	+	0.910531i	$0.364327\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	−24.9282	−2.15347
$135$	0	0
$136$	0	0
$137$	8.53590	0.729271	0.364636	−	0.931150i	$-0.381194\pi$
$137$	8.53590	0.729271	0.364636	+	0.931150i	$0.381194\pi$
$138$	0	0
$139$	−9.07180	−0.769460	−0.384730	−	0.923029i	$-0.625705\pi$
$139$	−9.07180	−0.769460	−0.384730	+	0.923029i	$0.625705\pi$
$140$	−0.732051	−0.0618696
$141$	0	0
$142$	−16.3923	−1.37561
$143$	0	0
$144$	0	0
$145$	3.46410	0.287678
$146$	−8.53590	−0.706436
$147$	0	0
$148$	−0.535898	−0.0440506
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	18.0526	1.46910	0.734548	−	0.678557i	$-0.237394\pi$
$151$	18.0526	1.46910	0.734548	+	0.678557i	$0.237394\pi$
$152$	4.73205	0.383820
$153$	0	0
$154$	0	0
$155$	−4.19615	−0.337043
$156$	0	0
$157$	−9.07180	−0.724008	−0.362004	−	0.932177i	$-0.617907\pi$
$157$	−9.07180	−0.724008	−0.362004	+	0.932177i	$0.617907\pi$
$158$	−14.5359	−1.15641
$159$	0	0
$160$	5.19615	0.410792
$161$	−3.46410	−0.273009
$162$	0	0
$163$	10.1962	0.798624	0.399312	−	0.916815i	$-0.369249\pi$
$163$	10.1962	0.798624	0.399312	+	0.916815i	$0.369249\pi$
$164$	−3.46410	−0.270501
$165$	0	0
$166$	−3.80385	−0.295236
$167$	−15.4641	−1.19665	−0.598324	−	0.801254i	$-0.704166\pi$
$167$	−15.4641	−1.19665	−0.598324	+	0.801254i	$0.704166\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	12.7321	0.970810
$173$	0.928203	0.0705700	0.0352850	−	0.999377i	$-0.488766\pi$
$173$	0.928203	0.0705700	0.0352850	+	0.999377i	$0.488766\pi$
$174$	0	0
$175$	0.732051	0.0553378
$176$	0	0
$177$	0	0
$178$	1.73205	0.129823
$179$	−18.9282	−1.41476	−0.707380	−	0.706833i	$-0.750123\pi$
$179$	−18.9282	−1.41476	−0.707380	+	0.706833i	$0.750123\pi$
$180$	0	0
$181$	−2.39230	−0.177819	−0.0889093	−	0.996040i	$-0.528338\pi$
$181$	−2.39230	−0.177819	−0.0889093	+	0.996040i	$0.528338\pi$
$182$	2.53590	0.187973
$183$	0	0
$184$	8.19615	0.604228
$185$	0.535898	0.0394000
$186$	0	0
$187$	0	0
$188$	−12.9282	−0.942886
$189$	0	0
$190$	4.73205	0.343299
$191$	−1.26795	−0.0917456	−0.0458728	−	0.998947i	$-0.514607\pi$
$191$	−1.26795	−0.0917456	−0.0458728	+	0.998947i	$0.514607\pi$
$192$	0	0
$193$	−2.39230	−0.172202	−0.0861009	−	0.996286i	$-0.527441\pi$
$193$	−2.39230	−0.172202	−0.0861009	+	0.996286i	$0.527441\pi$
$194$	−6.92820	−0.497416
$195$	0	0
$196$	−6.46410	−0.461722
$197$	−24.2487	−1.72765	−0.863825	−	0.503793i	$-0.831938\pi$
$197$	−24.2487	−1.72765	−0.863825	+	0.503793i	$0.831938\pi$
$198$	0	0
$199$	12.3923	0.878467	0.439234	−	0.898373i	$-0.355250\pi$
$199$	12.3923	0.878467	0.439234	+	0.898373i	$0.355250\pi$
$200$	−1.73205	−0.122474
$201$	0	0
$202$	6.00000	0.422159
$203$	−2.53590	−0.177985
$204$	0	0
$205$	3.46410	0.241943
$206$	17.6603	1.23045
$207$	0	0
$208$	−10.0000	−0.693375
$209$	0	0
$210$	0	0
$211$	−7.80385	−0.537239	−0.268620	−	0.963246i	$-0.586567\pi$
$211$	−7.80385	−0.537239	−0.268620	+	0.963246i	$0.586567\pi$
$212$	−6.92820	−0.475831
$213$	0	0
$214$	1.60770	0.109900
$215$	−12.7321	−0.868319
$216$	0	0
$217$	3.07180	0.208527
$218$	18.2487	1.23596
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−0.535898	−0.0358864	−0.0179432	−	0.999839i	$-0.505712\pi$
$223$	−0.535898	−0.0358864	−0.0179432	+	0.999839i	$0.505712\pi$
$224$	−3.80385	−0.254155
$225$	0	0
$226$	1.60770	0.106942
$227$	6.00000	0.398234	0.199117	−	0.979976i	$-0.436193\pi$
$227$	6.00000	0.398234	0.199117	+	0.979976i	$0.436193\pi$
$228$	0	0
$229$	−0.535898	−0.0354132	−0.0177066	−	0.999843i	$-0.505636\pi$
$229$	−0.535898	−0.0354132	−0.0177066	+	0.999843i	$0.505636\pi$
$230$	8.19615	0.540438
$231$	0	0
$232$	6.00000	0.393919
$233$	7.85641	0.514690	0.257345	−	0.966320i	$-0.417152\pi$
$233$	7.85641	0.514690	0.257345	+	0.966320i	$0.417152\pi$
$234$	0	0
$235$	12.9282	0.843343
$236$	−3.80385	−0.247609
$237$	0	0
$238$	0	0
$239$	−1.26795	−0.0820168	−0.0410084	−	0.999159i	$-0.513057\pi$
$239$	−1.26795	−0.0820168	−0.0410084	+	0.999159i	$0.513057\pi$
$240$	0	0
$241$	−14.3923	−0.927090	−0.463545	−	0.886073i	$-0.653423\pi$
$241$	−14.3923	−0.927090	−0.463545	+	0.886073i	$0.653423\pi$
$242$	−19.0526	−1.22474
$243$	0	0
$244$	2.00000	0.128037
$245$	6.46410	0.412976
$246$	0	0
$247$	−5.46410	−0.347672
$248$	−7.26795	−0.461515
$249$	0	0
$250$	−1.73205	−0.109545
$251$	−5.07180	−0.320129	−0.160064	−	0.987107i	$-0.551170\pi$
$251$	−5.07180	−0.320129	−0.160064	+	0.987107i	$0.551170\pi$
$252$	0	0
$253$	0	0
$254$	−18.3397	−1.15074
$255$	0	0
$256$	19.0000	1.18750
$257$	0.928203	0.0578997	0.0289499	−	0.999581i	$-0.490784\pi$
$257$	0.928203	0.0578997	0.0289499	+	0.999581i	$0.490784\pi$
$258$	0	0
$259$	−0.392305	−0.0243766
$260$	−2.00000	−0.124035
$261$	0	0
$262$	16.3923	1.01272
$263$	3.46410	0.213606	0.106803	−	0.994280i	$-0.465939\pi$
$263$	3.46410	0.213606	0.106803	+	0.994280i	$0.465939\pi$
$264$	0	0
$265$	6.92820	0.425596
$266$	−3.46410	−0.212398
$267$	0	0
$268$	−14.3923	−0.879150
$269$	28.3923	1.73111	0.865555	−	0.500814i	$-0.166966\pi$
$269$	28.3923	1.73111	0.865555	+	0.500814i	$0.166966\pi$
$270$	0	0
$271$	2.92820	0.177876	0.0889378	−	0.996037i	$-0.471653\pi$
$271$	2.92820	0.177876	0.0889378	+	0.996037i	$0.471653\pi$
$272$	0	0
$273$	0	0
$274$	14.7846	0.893171
$275$	0	0
$276$	0	0
$277$	15.8564	0.952719	0.476360	−	0.879251i	$-0.341956\pi$
$277$	15.8564	0.952719	0.476360	+	0.879251i	$0.341956\pi$
$278$	−15.7128	−0.942392
$279$	0	0
$280$	1.26795	0.0757745
$281$	5.32051	0.317395	0.158697	−	0.987327i	$-0.449271\pi$
$281$	5.32051	0.317395	0.158697	+	0.987327i	$0.449271\pi$
$282$	0	0
$283$	4.53590	0.269631	0.134816	−	0.990871i	$-0.456956\pi$
$283$	4.53590	0.269631	0.134816	+	0.990871i	$0.456956\pi$
$284$	−9.46410	−0.561591
$285$	0	0
$286$	0	0
$287$	−2.53590	−0.149689
$288$	0	0
$289$	−17.0000	−1.00000
$290$	6.00000	0.352332
$291$	0	0
$292$	−4.92820	−0.288401
$293$	27.4641	1.60447	0.802235	−	0.597008i	$-0.203644\pi$
$293$	27.4641	1.60447	0.802235	+	0.597008i	$0.203644\pi$
$294$	0	0
$295$	3.80385	0.221469
$296$	0.928203	0.0539507
$297$	0	0
$298$	10.3923	0.602010
$299$	−9.46410	−0.547323
$300$	0	0
$301$	9.32051	0.537225
$302$	31.2679	1.79927
$303$	0	0
$304$	13.6603	0.783469
$305$	−2.00000	−0.114520
$306$	0	0
$307$	0.143594	0.00819532	0.00409766	−	0.999992i	$-0.498696\pi$
$307$	0.143594	0.00819532	0.00409766	+	0.999992i	$0.498696\pi$
$308$	0	0
$309$	0	0
$310$	−7.26795	−0.412792
$311$	13.8564	0.785725	0.392862	−	0.919597i	$-0.371485\pi$
$311$	13.8564	0.785725	0.392862	+	0.919597i	$0.371485\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	−15.7128	−0.886725
$315$	0	0
$316$	−8.39230	−0.472104
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−6.00000	−0.334367
$323$	0	0
$324$	0	0
$325$	2.00000	0.110940
$326$	17.6603	0.978111
$327$	0	0
$328$	6.00000	0.331295
$329$	−9.46410	−0.521773
$330$	0	0
$331$	2.92820	0.160949	0.0804743	−	0.996757i	$-0.474357\pi$
$331$	2.92820	0.160949	0.0804743	+	0.996757i	$0.474357\pi$
$332$	−2.19615	−0.120530
$333$	0	0
$334$	−26.7846	−1.46559
$335$	14.3923	0.786336
$336$	0	0
$337$	30.3923	1.65557	0.827787	−	0.561042i	$-0.189599\pi$
$337$	30.3923	1.65557	0.827787	+	0.561042i	$0.189599\pi$
$338$	−15.5885	−0.847900
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−9.85641	−0.532196
$344$	−22.0526	−1.18899
$345$	0	0
$346$	1.60770	0.0864302
$347$	12.9282	0.694022	0.347011	−	0.937861i	$-0.387197\pi$
$347$	12.9282	0.694022	0.347011	+	0.937861i	$0.387197\pi$
$348$	0	0
$349$	1.32051	0.0706852	0.0353426	−	0.999375i	$-0.488748\pi$
$349$	1.32051	0.0706852	0.0353426	+	0.999375i	$0.488748\pi$
$350$	1.26795	0.0677747
$351$	0	0
$352$	0	0
$353$	29.3205	1.56057	0.780287	−	0.625422i	$-0.215073\pi$
$353$	29.3205	1.56057	0.780287	+	0.625422i	$0.215073\pi$
$354$	0	0
$355$	9.46410	0.502302
$356$	1.00000	0.0529999
$357$	0	0
$358$	−32.7846	−1.73272
$359$	8.87564	0.468439	0.234219	−	0.972184i	$-0.424747\pi$
$359$	8.87564	0.468439	0.234219	+	0.972184i	$0.424747\pi$
$360$	0	0
$361$	−11.5359	−0.607153
$362$	−4.14359	−0.217782
$363$	0	0
$364$	1.46410	0.0767398
$365$	4.92820	0.257954
$366$	0	0
$367$	30.3923	1.58647	0.793233	−	0.608919i	$-0.208396\pi$
$367$	30.3923	1.58647	0.793233	+	0.608919i	$0.208396\pi$
$368$	23.6603	1.23338
$369$	0	0
$370$	0.928203	0.0482550
$371$	−5.07180	−0.263315
$372$	0	0
$373$	−23.8564	−1.23524	−0.617619	−	0.786477i	$-0.711903\pi$
$373$	−23.8564	−1.23524	−0.617619	+	0.786477i	$0.711903\pi$
$374$	0	0
$375$	0	0
$376$	22.3923	1.15479
$377$	−6.92820	−0.356821
$378$	0	0
$379$	−19.8038	−1.01726	−0.508628	−	0.860987i	$-0.669847\pi$
$379$	−19.8038	−1.01726	−0.508628	+	0.860987i	$0.669847\pi$
$380$	2.73205	0.140151
$381$	0	0
$382$	−2.19615	−0.112365
$383$	−9.80385	−0.500953	−0.250477	−	0.968123i	$-0.580587\pi$
$383$	−9.80385	−0.500953	−0.250477	+	0.968123i	$0.580587\pi$
$384$	0	0
$385$	0	0
$386$	−4.14359	−0.210903
$387$	0	0
$388$	−4.00000	−0.203069
$389$	4.14359	0.210089	0.105044	−	0.994468i	$-0.466502\pi$
$389$	4.14359	0.210089	0.105044	+	0.994468i	$0.466502\pi$
$390$	0	0
$391$	0	0
$392$	11.1962	0.565491
$393$	0	0
$394$	−42.0000	−2.11593
$395$	8.39230	0.422263
$396$	0	0
$397$	13.3205	0.668537	0.334269	−	0.942478i	$-0.391511\pi$
$397$	13.3205	0.668537	0.334269	+	0.942478i	$0.391511\pi$
$398$	21.4641	1.07590
$399$	0	0
$400$	−5.00000	−0.250000
$401$	35.3205	1.76382	0.881911	−	0.471416i	$-0.156257\pi$
$401$	35.3205	1.76382	0.881911	+	0.471416i	$0.156257\pi$
$402$	0	0
$403$	8.39230	0.418050
$404$	3.46410	0.172345
$405$	0	0
$406$	−4.39230	−0.217986
$407$	0	0
$408$	0	0
$409$	24.3923	1.20612	0.603061	−	0.797695i	$-0.293948\pi$
$409$	24.3923	1.20612	0.603061	+	0.797695i	$0.293948\pi$
$410$	6.00000	0.296319
$411$	0	0
$412$	10.1962	0.502328
$413$	−2.78461	−0.137022
$414$	0	0
$415$	2.19615	0.107805
$416$	−10.3923	−0.509525
$417$	0	0
$418$	0	0
$419$	−8.87564	−0.433604	−0.216802	−	0.976216i	$-0.569563\pi$
$419$	−8.87564	−0.433604	−0.216802	+	0.976216i	$0.569563\pi$
$420$	0	0
$421$	22.7846	1.11045	0.555227	−	0.831699i	$-0.312631\pi$
$421$	22.7846	1.11045	0.555227	+	0.831699i	$0.312631\pi$
$422$	−13.5167	−0.657981
$423$	0	0
$424$	12.0000	0.582772
$425$	0	0
$426$	0	0
$427$	1.46410	0.0708528
$428$	0.928203	0.0448664
$429$	0	0
$430$	−22.0526	−1.06347
$431$	15.1244	0.728515	0.364257	−	0.931298i	$-0.381323\pi$
$431$	15.1244	0.728515	0.364257	+	0.931298i	$0.381323\pi$
$432$	0	0
$433$	22.7846	1.09496	0.547479	−	0.836819i	$-0.315588\pi$
$433$	22.7846	1.09496	0.547479	+	0.836819i	$0.315588\pi$
$434$	5.32051	0.255393
$435$	0	0
$436$	10.5359	0.504578
$437$	12.9282	0.618440
$438$	0	0
$439$	−38.7321	−1.84858	−0.924290	−	0.381691i	$-0.875342\pi$
$439$	−38.7321	−1.84858	−0.924290	+	0.381691i	$0.875342\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−20.5359	−0.975690	−0.487845	−	0.872930i	$-0.662217\pi$
$443$	−20.5359	−0.975690	−0.487845	+	0.872930i	$0.662217\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	−0.928203	−0.0439517
$447$	0	0
$448$	0.732051	0.0345861
$449$	−19.8564	−0.937082	−0.468541	−	0.883442i	$-0.655220\pi$
$449$	−19.8564	−0.937082	−0.468541	+	0.883442i	$0.655220\pi$
$450$	0	0
$451$	0	0
$452$	0.928203	0.0436590
$453$	0	0
$454$	10.3923	0.487735
$455$	−1.46410	−0.0686381
$456$	0	0
$457$	14.0000	0.654892	0.327446	−	0.944870i	$-0.393812\pi$
$457$	14.0000	0.654892	0.327446	+	0.944870i	$0.393812\pi$
$458$	−0.928203	−0.0433721
$459$	0	0
$460$	4.73205	0.220633
$461$	−25.1769	−1.17261	−0.586303	−	0.810092i	$-0.699417\pi$
$461$	−25.1769	−1.17261	−0.586303	+	0.810092i	$0.699417\pi$
$462$	0	0
$463$	−16.9282	−0.786720	−0.393360	−	0.919384i	$-0.628687\pi$
$463$	−16.9282	−0.786720	−0.393360	+	0.919384i	$0.628687\pi$
$464$	17.3205	0.804084
$465$	0	0
$466$	13.6077	0.630364
$467$	−27.4641	−1.27089	−0.635444	−	0.772147i	$-0.719183\pi$
$467$	−27.4641	−1.27089	−0.635444	+	0.772147i	$0.719183\pi$
$468$	0	0
$469$	−10.5359	−0.486503
$470$	22.3923	1.03288
$471$	0	0
$472$	6.58846	0.303258
$473$	0	0
$474$	0	0
$475$	−2.73205	−0.125355
$476$	0	0
$477$	0	0
$478$	−2.19615	−0.100450
$479$	−2.53590	−0.115868	−0.0579341	−	0.998320i	$-0.518451\pi$
$479$	−2.53590	−0.115868	−0.0579341	+	0.998320i	$0.518451\pi$
$480$	0	0
$481$	−1.07180	−0.0488697
$482$	−24.9282	−1.13545
$483$	0	0
$484$	−11.0000	−0.500000
$485$	4.00000	0.181631
$486$	0	0
$487$	−34.0000	−1.54069	−0.770344	−	0.637629i	$-0.779915\pi$
$487$	−34.0000	−1.54069	−0.770344	+	0.637629i	$0.779915\pi$
$488$	−3.46410	−0.156813
$489$	0	0
$490$	11.1962	0.505791
$491$	20.1962	0.911440	0.455720	−	0.890123i	$-0.349382\pi$
$491$	20.1962	0.911440	0.455720	+	0.890123i	$0.349382\pi$
$492$	0	0
$493$	0	0
$494$	−9.46410	−0.425810
$495$	0	0
$496$	−20.9808	−0.942064
$497$	−6.92820	−0.310772
$498$	0	0
$499$	−2.73205	−0.122303	−0.0611517	−	0.998128i	$-0.519477\pi$
$499$	−2.73205	−0.122303	−0.0611517	+	0.998128i	$0.519477\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−8.78461	−0.392076
$503$	−9.80385	−0.437132	−0.218566	−	0.975822i	$-0.570138\pi$
$503$	−9.80385	−0.437132	−0.218566	+	0.975822i	$0.570138\pi$
$504$	0	0
$505$	−3.46410	−0.154150
$506$	0	0
$507$	0	0
$508$	−10.5885	−0.469787
$509$	−16.3923	−0.726576	−0.363288	−	0.931677i	$-0.618346\pi$
$509$	−16.3923	−0.726576	−0.363288	+	0.931677i	$0.618346\pi$
$510$	0	0
$511$	−3.60770	−0.159595
$512$	8.66025	0.382733
$513$	0	0
$514$	1.60770	0.0709124
$515$	−10.1962	−0.449296
$516$	0	0
$517$	0	0
$518$	−0.679492	−0.0298552
$519$	0	0
$520$	3.46410	0.151911
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	32.2487	1.41014	0.705069	−	0.709139i	$-0.250916\pi$
$523$	32.2487	1.41014	0.705069	+	0.709139i	$0.250916\pi$
$524$	9.46410	0.413441
$525$	0	0
$526$	6.00000	0.261612
$527$	0	0
$528$	0	0
$529$	−0.607695	−0.0264215
$530$	12.0000	0.521247
$531$	0	0
$532$	−2.00000	−0.0867110
$533$	−6.92820	−0.300094
$534$	0	0
$535$	−0.928203	−0.0401297
$536$	24.9282	1.07673
$537$	0	0
$538$	49.1769	2.12017
$539$	0	0
$540$	0	0
$541$	42.3923	1.82259	0.911294	−	0.411757i	$-0.135085\pi$
$541$	42.3923	1.82259	0.911294	+	0.411757i	$0.135085\pi$
$542$	5.07180	0.217852
$543$	0	0
$544$	0	0
$545$	−10.5359	−0.451308
$546$	0	0
$547$	38.5885	1.64992	0.824962	−	0.565189i	$-0.191197\pi$
$547$	38.5885	1.64992	0.824962	+	0.565189i	$0.191197\pi$
$548$	8.53590	0.364636
$549$	0	0
$550$	0	0
$551$	9.46410	0.403184
$552$	0	0
$553$	−6.14359	−0.261252
$554$	27.4641	1.16684
$555$	0	0
$556$	−9.07180	−0.384730
$557$	−38.7846	−1.64336	−0.821678	−	0.569952i	$-0.806962\pi$
$557$	−38.7846	−1.64336	−0.821678	+	0.569952i	$0.806962\pi$
$558$	0	0
$559$	25.4641	1.07702
$560$	3.66025	0.154674
$561$	0	0
$562$	9.21539	0.388728
$563$	−7.26795	−0.306308	−0.153154	−	0.988202i	$-0.548943\pi$
$563$	−7.26795	−0.306308	−0.153154	+	0.988202i	$0.548943\pi$
$564$	0	0
$565$	−0.928203	−0.0390498
$566$	7.85641	0.330229
$567$	0	0
$568$	16.3923	0.687806
$569$	−1.60770	−0.0673981	−0.0336990	−	0.999432i	$-0.510729\pi$
$569$	−1.60770	−0.0673981	−0.0336990	+	0.999432i	$0.510729\pi$
$570$	0	0
$571$	31.9090	1.33535	0.667674	−	0.744453i	$-0.267290\pi$
$571$	31.9090	1.33535	0.667674	+	0.744453i	$0.267290\pi$
$572$	0	0
$573$	0	0
$574$	−4.39230	−0.183331
$575$	−4.73205	−0.197340
$576$	0	0
$577$	−27.0718	−1.12701	−0.563507	−	0.826111i	$-0.690548\pi$
$577$	−27.0718	−1.12701	−0.563507	+	0.826111i	$0.690548\pi$
$578$	−29.4449	−1.22474
$579$	0	0
$580$	3.46410	0.143839
$581$	−1.60770	−0.0666984
$582$	0	0
$583$	0	0
$584$	8.53590	0.353218
$585$	0	0
$586$	47.5692	1.96507
$587$	−12.9282	−0.533604	−0.266802	−	0.963751i	$-0.585967\pi$
$587$	−12.9282	−0.533604	−0.266802	+	0.963751i	$0.585967\pi$
$588$	0	0
$589$	−11.4641	−0.472370
$590$	6.58846	0.271242
$591$	0	0
$592$	2.67949	0.110126
$593$	36.2487	1.48856	0.744278	−	0.667870i	$-0.232794\pi$
$593$	36.2487	1.48856	0.744278	+	0.667870i	$0.232794\pi$
$594$	0	0
$595$	0	0
$596$	6.00000	0.245770
$597$	0	0
$598$	−16.3923	−0.670331
$599$	36.5885	1.49496	0.747482	−	0.664282i	$-0.231263\pi$
$599$	36.5885	1.49496	0.747482	+	0.664282i	$0.231263\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	16.1436	0.657964
$603$	0	0
$604$	18.0526	0.734548
$605$	11.0000	0.447214
$606$	0	0
$607$	23.4641	0.952379	0.476189	−	0.879343i	$-0.342018\pi$
$607$	23.4641	0.952379	0.476189	+	0.879343i	$0.342018\pi$
$608$	14.1962	0.575730
$609$	0	0
$610$	−3.46410	−0.140257
$611$	−25.8564	−1.04604
$612$	0	0
$613$	14.9282	0.602944	0.301472	−	0.953475i	$-0.402522\pi$
$613$	14.9282	0.602944	0.301472	+	0.953475i	$0.402522\pi$
$614$	0.248711	0.0100372
$615$	0	0
$616$	0	0
$617$	17.3205	0.697297	0.348649	−	0.937253i	$-0.386641\pi$
$617$	17.3205	0.697297	0.348649	+	0.937253i	$0.386641\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	−4.19615	−0.168522
$621$	0	0
$622$	24.0000	0.962312
$623$	0.732051	0.0293290
$624$	0	0
$625$	1.00000	0.0400000
$626$	−38.1051	−1.52299
$627$	0	0
$628$	−9.07180	−0.362004
$629$	0	0
$630$	0	0
$631$	28.7846	1.14590	0.572949	−	0.819591i	$-0.305799\pi$
$631$	28.7846	1.14590	0.572949	+	0.819591i	$0.305799\pi$
$632$	14.5359	0.578207
$633$	0	0
$634$	10.3923	0.412731
$635$	10.5885	0.420190
$636$	0	0
$637$	−12.9282	−0.512234
$638$	0	0
$639$	0	0
$640$	−12.1244	−0.479257
$641$	38.7846	1.53190	0.765950	−	0.642900i	$-0.222269\pi$
$641$	38.7846	1.53190	0.765950	+	0.642900i	$0.222269\pi$
$642$	0	0
$643$	−39.5692	−1.56046	−0.780229	−	0.625494i	$-0.784897\pi$
$643$	−39.5692	−1.56046	−0.780229	+	0.625494i	$0.784897\pi$
$644$	−3.46410	−0.136505
$645$	0	0
$646$	0	0
$647$	−5.41154	−0.212750	−0.106375	−	0.994326i	$-0.533924\pi$
$647$	−5.41154	−0.212750	−0.106375	+	0.994326i	$0.533924\pi$
$648$	0	0
$649$	0	0
$650$	3.46410	0.135873
$651$	0	0
$652$	10.1962	0.399312
$653$	11.0718	0.433273	0.216636	−	0.976252i	$-0.430491\pi$
$653$	11.0718	0.433273	0.216636	+	0.976252i	$0.430491\pi$
$654$	0	0
$655$	−9.46410	−0.369793
$656$	17.3205	0.676252
$657$	0	0
$658$	−16.3923	−0.639039
$659$	−23.3205	−0.908438	−0.454219	−	0.890890i	$-0.650082\pi$
$659$	−23.3205	−0.908438	−0.454219	+	0.890890i	$0.650082\pi$
$660$	0	0
$661$	−4.92820	−0.191685	−0.0958424	−	0.995397i	$-0.530554\pi$
$661$	−4.92820	−0.191685	−0.0958424	+	0.995397i	$0.530554\pi$
$662$	5.07180	0.197121
$663$	0	0
$664$	3.80385	0.147618
$665$	2.00000	0.0775567
$666$	0	0
$667$	16.3923	0.634713
$668$	−15.4641	−0.598324
$669$	0	0
$670$	24.9282	0.963061
$671$	0	0
$672$	0	0
$673$	−26.6410	−1.02694	−0.513468	−	0.858109i	$-0.671639\pi$
$673$	−26.6410	−1.02694	−0.513468	+	0.858109i	$0.671639\pi$
$674$	52.6410	2.02766
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−4.14359	−0.159251	−0.0796256	−	0.996825i	$-0.525372\pi$
$677$	−4.14359	−0.159251	−0.0796256	+	0.996825i	$0.525372\pi$
$678$	0	0
$679$	−2.92820	−0.112374
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−4.73205	−0.181067	−0.0905334	−	0.995893i	$-0.528857\pi$
$683$	−4.73205	−0.181067	−0.0905334	+	0.995893i	$0.528857\pi$
$684$	0	0
$685$	−8.53590	−0.326140
$686$	−17.0718	−0.651804
$687$	0	0
$688$	−63.6603	−2.42702
$689$	−13.8564	−0.527887
$690$	0	0
$691$	−41.1769	−1.56644	−0.783222	−	0.621742i	$-0.786425\pi$
$691$	−41.1769	−1.56644	−0.783222	+	0.621742i	$0.786425\pi$
$692$	0.928203	0.0352850
$693$	0	0
$694$	22.3923	0.850000
$695$	9.07180	0.344113
$696$	0	0
$697$	0	0
$698$	2.28719	0.0865713
$699$	0	0
$700$	0.732051	0.0276689
$701$	−7.85641	−0.296732	−0.148366	−	0.988932i	$-0.547401\pi$
$701$	−7.85641	−0.296732	−0.148366	+	0.988932i	$0.547401\pi$
$702$	0	0
$703$	1.46410	0.0552196
$704$	0	0
$705$	0	0
$706$	50.7846	1.91130
$707$	2.53590	0.0953723
$708$	0	0
$709$	−28.2487	−1.06090	−0.530451	−	0.847715i	$-0.677978\pi$
$709$	−28.2487	−1.06090	−0.530451	+	0.847715i	$0.677978\pi$
$710$	16.3923	0.615192
$711$	0	0
$712$	−1.73205	−0.0649113
$713$	−19.8564	−0.743628
$714$	0	0
$715$	0	0
$716$	−18.9282	−0.707380
$717$	0	0
$718$	15.3731	0.573718
$719$	−4.98076	−0.185751	−0.0928755	−	0.995678i	$-0.529606\pi$
$719$	−4.98076	−0.185751	−0.0928755	+	0.995678i	$0.529606\pi$
$720$	0	0
$721$	7.46410	0.277978
$722$	−19.9808	−0.743607
$723$	0	0
$724$	−2.39230	−0.0889093
$725$	−3.46410	−0.128654
$726$	0	0
$727$	8.33975	0.309304	0.154652	−	0.987969i	$-0.450574\pi$
$727$	8.33975	0.309304	0.154652	+	0.987969i	$0.450574\pi$
$728$	−2.53590	−0.0939866
$729$	0	0
$730$	8.53590	0.315928
$731$	0	0
$732$	0	0
$733$	52.7846	1.94964	0.974822	−	0.222984i	$-0.0715799\pi$
$733$	52.7846	1.94964	0.974822	+	0.222984i	$0.0715799\pi$
$734$	52.6410	1.94302
$735$	0	0
$736$	24.5885	0.906343
$737$	0	0
$738$	0	0
$739$	−7.80385	−0.287069	−0.143535	−	0.989645i	$-0.545847\pi$
$739$	−7.80385	−0.287069	−0.143535	+	0.989645i	$0.545847\pi$
$740$	0.535898	0.0197000
$741$	0	0
$742$	−8.78461	−0.322493
$743$	41.9090	1.53749	0.768745	−	0.639555i	$-0.220881\pi$
$743$	41.9090	1.53749	0.768745	+	0.639555i	$0.220881\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	−41.3205	−1.51285
$747$	0	0
$748$	0	0
$749$	0.679492	0.0248281
$750$	0	0
$751$	−43.7128	−1.59510	−0.797552	−	0.603251i	$-0.793872\pi$
$751$	−43.7128	−1.59510	−0.797552	+	0.603251i	$0.793872\pi$
$752$	64.6410	2.35722
$753$	0	0
$754$	−12.0000	−0.437014
$755$	−18.0526	−0.657000
$756$	0	0
$757$	−5.85641	−0.212855	−0.106427	−	0.994320i	$-0.533941\pi$
$757$	−5.85641	−0.212855	−0.106427	+	0.994320i	$0.533941\pi$
$758$	−34.3013	−1.24588
$759$	0	0
$760$	−4.73205	−0.171650
$761$	32.1051	1.16381	0.581905	−	0.813257i	$-0.302308\pi$
$761$	32.1051	1.16381	0.581905	+	0.813257i	$0.302308\pi$
$762$	0	0
$763$	7.71281	0.279223
$764$	−1.26795	−0.0458728
$765$	0	0
$766$	−16.9808	−0.613540
$767$	−7.60770	−0.274698
$768$	0	0
$769$	15.6077	0.562828	0.281414	−	0.959586i	$-0.409197\pi$
$769$	15.6077	0.562828	0.281414	+	0.959586i	$0.409197\pi$
$770$	0	0
$771$	0	0
$772$	−2.39230	−0.0861009
$773$	−17.3205	−0.622975	−0.311488	−	0.950250i	$-0.600827\pi$
$773$	−17.3205	−0.622975	−0.311488	+	0.950250i	$0.600827\pi$
$774$	0	0
$775$	4.19615	0.150730
$776$	6.92820	0.248708
$777$	0	0
$778$	7.17691	0.257305
$779$	9.46410	0.339087
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	32.3205	1.15430
$785$	9.07180	0.323786
$786$	0	0
$787$	−21.9090	−0.780970	−0.390485	−	0.920609i	$-0.627693\pi$
$787$	−21.9090	−0.780970	−0.390485	+	0.920609i	$0.627693\pi$
$788$	−24.2487	−0.863825
$789$	0	0
$790$	14.5359	0.517164
$791$	0.679492	0.0241600
$792$	0	0
$793$	4.00000	0.142044
$794$	23.0718	0.818787
$795$	0	0
$796$	12.3923	0.439234
$797$	−31.8564	−1.12841	−0.564206	−	0.825634i	$-0.690818\pi$
$797$	−31.8564	−1.12841	−0.564206	+	0.825634i	$0.690818\pi$
$798$	0	0
$799$	0	0
$800$	−5.19615	−0.183712
$801$	0	0
$802$	61.1769	2.16023
$803$	0	0
$804$	0	0
$805$	3.46410	0.122094
$806$	14.5359	0.512005
$807$	0	0
$808$	−6.00000	−0.211079
$809$	−43.8564	−1.54191	−0.770955	−	0.636890i	$-0.780220\pi$
$809$	−43.8564	−1.54191	−0.770955	+	0.636890i	$0.780220\pi$
$810$	0	0
$811$	24.3923	0.856530	0.428265	−	0.903653i	$-0.359125\pi$
$811$	24.3923	0.856530	0.428265	+	0.903653i	$0.359125\pi$
$812$	−2.53590	−0.0889926
$813$	0	0
$814$	0	0
$815$	−10.1962	−0.357156
$816$	0	0
$817$	−34.7846	−1.21696
$818$	42.2487	1.47719
$819$	0	0
$820$	3.46410	0.120972
$821$	−21.4641	−0.749102	−0.374551	−	0.927206i	$-0.622203\pi$
$821$	−21.4641	−0.749102	−0.374551	+	0.927206i	$0.622203\pi$
$822$	0	0
$823$	9.60770	0.334903	0.167452	−	0.985880i	$-0.446446\pi$
$823$	9.60770	0.334903	0.167452	+	0.985880i	$0.446446\pi$
$824$	−17.6603	−0.615224
$825$	0	0
$826$	−4.82309	−0.167817
$827$	−22.9808	−0.799119	−0.399560	−	0.916707i	$-0.630837\pi$
$827$	−22.9808	−0.799119	−0.399560	+	0.916707i	$0.630837\pi$
$828$	0	0
$829$	−51.5692	−1.79107	−0.895537	−	0.444988i	$-0.853208\pi$
$829$	−51.5692	−1.79107	−0.895537	+	0.444988i	$0.853208\pi$
$830$	3.80385	0.132033
$831$	0	0
$832$	2.00000	0.0693375
$833$	0	0
$834$	0	0
$835$	15.4641	0.535157
$836$	0	0
$837$	0	0
$838$	−15.3731	−0.531054
$839$	19.5167	0.673790	0.336895	−	0.941542i	$-0.390623\pi$
$839$	19.5167	0.673790	0.336895	+	0.941542i	$0.390623\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	39.4641	1.36002
$843$	0	0
$844$	−7.80385	−0.268620
$845$	9.00000	0.309609
$846$	0	0
$847$	−8.05256	−0.276689
$848$	34.6410	1.18958
$849$	0	0
$850$	0	0
$851$	2.53590	0.0869295
$852$	0	0
$853$	−22.0000	−0.753266	−0.376633	−	0.926363i	$-0.622918\pi$
$853$	−22.0000	−0.753266	−0.376633	+	0.926363i	$0.622918\pi$
$854$	2.53590	0.0867767
$855$	0	0
$856$	−1.60770	−0.0549499
$857$	−52.6410	−1.79818	−0.899091	−	0.437761i	$-0.855772\pi$
$857$	−52.6410	−1.79818	−0.899091	+	0.437761i	$0.855772\pi$
$858$	0	0
$859$	38.8372	1.32511	0.662554	−	0.749015i	$-0.269473\pi$
$859$	38.8372	1.32511	0.662554	+	0.749015i	$0.269473\pi$
$860$	−12.7321	−0.434159
$861$	0	0
$862$	26.1962	0.892244
$863$	0.339746	0.0115651	0.00578254	−	0.999983i	$-0.498159\pi$
$863$	0.339746	0.0115651	0.00578254	+	0.999983i	$0.498159\pi$
$864$	0	0
$865$	−0.928203	−0.0315599
$866$	39.4641	1.34104
$867$	0	0
$868$	3.07180	0.104264
$869$	0	0
$870$	0	0
$871$	−28.7846	−0.975329
$872$	−18.2487	−0.617979
$873$	0	0
$874$	22.3923	0.757431
$875$	−0.732051	−0.0247478
$876$	0	0
$877$	−17.6077	−0.594570	−0.297285	−	0.954789i	$-0.596081\pi$
$877$	−17.6077	−0.594570	−0.297285	+	0.954789i	$0.596081\pi$
$878$	−67.0859	−2.26404
$879$	0	0
$880$	0	0
$881$	−39.0333	−1.31507	−0.657533	−	0.753426i	$-0.728400\pi$
$881$	−39.0333	−1.31507	−0.657533	+	0.753426i	$0.728400\pi$
$882$	0	0
$883$	5.12436	0.172448	0.0862241	−	0.996276i	$-0.472520\pi$
$883$	5.12436	0.172448	0.0862241	+	0.996276i	$0.472520\pi$
$884$	0	0
$885$	0	0
$886$	−35.5692	−1.19497
$887$	42.5885	1.42998	0.714990	−	0.699134i	$-0.246431\pi$
$887$	42.5885	1.42998	0.714990	+	0.699134i	$0.246431\pi$
$888$	0	0
$889$	−7.75129	−0.259970
$890$	−1.73205	−0.0580585
$891$	0	0
$892$	−0.535898	−0.0179432
$893$	35.3205	1.18196
$894$	0	0
$895$	18.9282	0.632700
$896$	8.87564	0.296514
$897$	0	0
$898$	−34.3923	−1.14769
$899$	−14.5359	−0.484799
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−1.60770	−0.0534711
$905$	2.39230	0.0795229
$906$	0	0
$907$	−15.0718	−0.500451	−0.250225	−	0.968188i	$-0.580505\pi$
$907$	−15.0718	−0.500451	−0.250225	+	0.968188i	$0.580505\pi$
$908$	6.00000	0.199117
$909$	0	0
$910$	−2.53590	−0.0840642
$911$	42.2487	1.39976	0.699881	−	0.714259i	$-0.253236\pi$
$911$	42.2487	1.39976	0.699881	+	0.714259i	$0.253236\pi$
$912$	0	0
$913$	0	0
$914$	24.2487	0.802076
$915$	0	0
$916$	−0.535898	−0.0177066
$917$	6.92820	0.228789
$918$	0	0
$919$	1.66025	0.0547667	0.0273834	−	0.999625i	$-0.491283\pi$
$919$	1.66025	0.0547667	0.0273834	+	0.999625i	$0.491283\pi$
$920$	−8.19615	−0.270219
$921$	0	0
$922$	−43.6077	−1.43614
$923$	−18.9282	−0.623029
$924$	0	0
$925$	−0.535898	−0.0176202
$926$	−29.3205	−0.963532
$927$	0	0
$928$	18.0000	0.590879
$929$	4.14359	0.135947	0.0679734	−	0.997687i	$-0.478347\pi$
$929$	4.14359	0.135947	0.0679734	+	0.997687i	$0.478347\pi$
$930$	0	0
$931$	17.6603	0.578791
$932$	7.85641	0.257345
$933$	0	0
$934$	−47.5692	−1.55651
$935$	0	0
$936$	0	0
$937$	50.0000	1.63343	0.816714	−	0.577042i	$-0.195793\pi$
$937$	50.0000	1.63343	0.816714	+	0.577042i	$0.195793\pi$
$938$	−18.2487	−0.595842
$939$	0	0
$940$	12.9282	0.421671
$941$	33.0333	1.07686	0.538428	−	0.842672i	$-0.319018\pi$
$941$	33.0333	1.07686	0.538428	+	0.842672i	$0.319018\pi$
$942$	0	0
$943$	16.3923	0.533807
$944$	19.0192	0.619023
$945$	0	0
$946$	0	0
$947$	−10.3923	−0.337705	−0.168852	−	0.985641i	$-0.554006\pi$
$947$	−10.3923	−0.337705	−0.168852	+	0.985641i	$0.554006\pi$
$948$	0	0
$949$	−9.85641	−0.319952
$950$	−4.73205	−0.153528
$951$	0	0
$952$	0	0
$953$	41.3205	1.33850	0.669251	−	0.743036i	$-0.266615\pi$
$953$	41.3205	1.33850	0.669251	+	0.743036i	$0.266615\pi$
$954$	0	0
$955$	1.26795	0.0410299
$956$	−1.26795	−0.0410084
$957$	0	0
$958$	−4.39230	−0.141909
$959$	6.24871	0.201781
$960$	0	0
$961$	−13.3923	−0.432010
$962$	−1.85641	−0.0598529
$963$	0	0
$964$	−14.3923	−0.463545
$965$	2.39230	0.0770110
$966$	0	0
$967$	49.9090	1.60496	0.802482	−	0.596676i	$-0.203512\pi$
$967$	49.9090	1.60496	0.802482	+	0.596676i	$0.203512\pi$
$968$	19.0526	0.612372
$969$	0	0
$970$	6.92820	0.222451
$971$	6.92820	0.222337	0.111168	−	0.993802i	$-0.464541\pi$
$971$	6.92820	0.222337	0.111168	+	0.993802i	$0.464541\pi$
$972$	0	0
$973$	−6.64102	−0.212901
$974$	−58.8897	−1.88695
$975$	0	0
$976$	−10.0000	−0.320092
$977$	47.0718	1.50596	0.752980	−	0.658043i	$-0.228616\pi$
$977$	47.0718	1.50596	0.752980	+	0.658043i	$0.228616\pi$
$978$	0	0
$979$	0	0
$980$	6.46410	0.206488
$981$	0	0
$982$	34.9808	1.11628
$983$	3.46410	0.110488	0.0552438	−	0.998473i	$-0.482406\pi$
$983$	3.46410	0.110488	0.0552438	+	0.998473i	$0.482406\pi$
$984$	0	0
$985$	24.2487	0.772628
$986$	0	0
$987$	0	0
$988$	−5.46410	−0.173836
$989$	−60.2487	−1.91580
$990$	0	0
$991$	−30.4449	−0.967113	−0.483556	−	0.875313i	$-0.660655\pi$
$991$	−30.4449	−0.967113	−0.483556	+	0.875313i	$0.660655\pi$
$992$	−21.8038	−0.692273
$993$	0	0
$994$	−12.0000	−0.380617
$995$	−12.3923	−0.392862
$996$	0	0
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	−4.73205	−0.149790
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.f.1.2		2
3.2	odd	2		445.2.a.b.1.1	✓	2
12.11	even	2		7120.2.a.s.1.2		2
15.14	odd	2		2225.2.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.b.1.1	✓	2	3.2	odd	2
2225.2.a.d.1.2		2	15.14	odd	2
4005.2.a.f.1.2		2	1.1	even	1	trivial
7120.2.a.s.1.2		2	12.11	even	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:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.732051 q^{7} -1.73205 q^{8} +O(q^{10})\)	\(q+1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.732051 q^{7} -1.73205 q^{8} -1.73205 q^{10} +2.00000 q^{13} +1.26795 q^{14} -5.00000 q^{16} -2.73205 q^{19} -1.00000 q^{20} -4.73205 q^{23} +1.00000 q^{25} +3.46410 q^{26} +0.732051 q^{28} -3.46410 q^{29} +4.19615 q^{31} -5.19615 q^{32} -0.732051 q^{35} -0.535898 q^{37} -4.73205 q^{38} +1.73205 q^{40} -3.46410 q^{41} +12.7321 q^{43} -8.19615 q^{46} -12.9282 q^{47} -6.46410 q^{49} +1.73205 q^{50} +2.00000 q^{52} -6.92820 q^{53} -1.26795 q^{56} -6.00000 q^{58} -3.80385 q^{59} +2.00000 q^{61} +7.26795 q^{62} +1.00000 q^{64} -2.00000 q^{65} -14.3923 q^{67} -1.26795 q^{70} -9.46410 q^{71} -4.92820 q^{73} -0.928203 q^{74} -2.73205 q^{76} -8.39230 q^{79} +5.00000 q^{80} -6.00000 q^{82} -2.19615 q^{83} +22.0526 q^{86} +1.00000 q^{89} +1.46410 q^{91} -4.73205 q^{92} -22.3923 q^{94} +2.73205 q^{95} -4.00000 q^{97} -11.1962 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^4 - 2 * q^5 - 2 * q^7	\( 2 q + 2 q^{4} - 2 q^{5} - 2 q^{7} + 4 q^{13} + 6 q^{14} - 10 q^{16} - 2 q^{19} - 2 q^{20} - 6 q^{23} + 2 q^{25} - 2 q^{28} - 2 q^{31} + 2 q^{35} - 8 q^{37} - 6 q^{38} + 22 q^{43} - 6 q^{46} - 12 q^{47} - 6 q^{49} + 4 q^{52} - 6 q^{56} - 12 q^{58} - 18 q^{59} + 4 q^{61} + 18 q^{62} + 2 q^{64} - 4 q^{65} - 8 q^{67} - 6 q^{70} - 12 q^{71} + 4 q^{73} + 12 q^{74} - 2 q^{76} + 4 q^{79} + 10 q^{80} - 12 q^{82} + 6 q^{83} + 6 q^{86} + 2 q^{89} - 4 q^{91} - 6 q^{92} - 24 q^{94} + 2 q^{95} - 8 q^{97} - 12 q^{98}+O(q^{100}) \) 2 * q + 2 * q^4 - 2 * q^5 - 2 * q^7 + 4 * q^13 + 6 * q^14 - 10 * q^16 - 2 * q^19 - 2 * q^20 - 6 * q^23 + 2 * q^25 - 2 * q^28 - 2 * q^31 + 2 * q^35 - 8 * q^37 - 6 * q^38 + 22 * q^43 - 6 * q^46 - 12 * q^47 - 6 * q^49 + 4 * q^52 - 6 * q^56 - 12 * q^58 - 18 * q^59 + 4 * q^61 + 18 * q^62 + 2 * q^64 - 4 * q^65 - 8 * q^67 - 6 * q^70 - 12 * q^71 + 4 * q^73 + 12 * q^74 - 2 * q^76 + 4 * q^79 + 10 * q^80 - 12 * q^82 + 6 * q^83 + 6 * q^86 + 2 * q^89 - 4 * q^91 - 6 * q^92 - 24 * q^94 + 2 * q^95 - 8 * q^97 - 12 * q^98

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