LMFDB - Embedded newform 8046.2.a.l.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, 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("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.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:	$64.2476334663$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 21 x^{10} + 116 x^{9} + 106 x^{8} - 774 x^{7} - 63 x^{6} + 2013 x^{5} - 417 x^{4} + \cdots - 375 $ x^12 - 5x^11 - 21x^10 + 116x^9 + 106x^8 - 774x^7 - 63x^6 + 2013x^5 - 417x^4 - 2249x^3 + 761x^2 + 910*x - 375
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.448758$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +0.448758 q^{5} -1.52942 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +0.448758 q^{5} -1.52942 q^{7} -1.00000 q^{8} -0.448758 q^{10} -0.412196 q^{11} +3.36442 q^{13} +1.52942 q^{14} +1.00000 q^{16} +3.96634 q^{17} -5.13026 q^{19} +0.448758 q^{20} +0.412196 q^{22} -2.63703 q^{23} -4.79862 q^{25} -3.36442 q^{26} -1.52942 q^{28} -2.15254 q^{29} -5.88764 q^{31} -1.00000 q^{32} -3.96634 q^{34} -0.686340 q^{35} +10.4747 q^{37} +5.13026 q^{38} -0.448758 q^{40} -5.89535 q^{41} +1.10493 q^{43} -0.412196 q^{44} +2.63703 q^{46} +5.78830 q^{47} -4.66087 q^{49} +4.79862 q^{50} +3.36442 q^{52} -1.69686 q^{53} -0.184976 q^{55} +1.52942 q^{56} +2.15254 q^{58} +11.9240 q^{59} +12.2236 q^{61} +5.88764 q^{62} +1.00000 q^{64} +1.50981 q^{65} -15.8063 q^{67} +3.96634 q^{68} +0.686340 q^{70} +12.3113 q^{71} +5.42784 q^{73} -10.4747 q^{74} -5.13026 q^{76} +0.630422 q^{77} +2.21050 q^{79} +0.448758 q^{80} +5.89535 q^{82} +11.7231 q^{83} +1.77993 q^{85} -1.10493 q^{86} +0.412196 q^{88} +0.410694 q^{89} -5.14562 q^{91} -2.63703 q^{92} -5.78830 q^{94} -2.30224 q^{95} +4.08503 q^{97} +4.66087 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{2} + 12 q^{4} + 5 q^{5} - 6 q^{7} - 12 q^{8}+O(q^{10}) $ 12 * q - 12 * q^2 + 12 * q^4 + 5 * q^5 - 6 * q^7 - 12 * q^8	$ 12 q - 12 q^{2} + 12 q^{4} + 5 q^{5} - 6 q^{7} - 12 q^{8} - 5 q^{10} + 10 q^{11} - q^{13} + 6 q^{14} + 12 q^{16} + 6 q^{17} - 10 q^{19} + 5 q^{20} - 10 q^{22} + 15 q^{23} + 7 q^{25} + q^{26} - 6 q^{28} + 33 q^{29} - 6 q^{31} - 12 q^{32} - 6 q^{34} + 16 q^{35} - 13 q^{37} + 10 q^{38} - 5 q^{40} + 20 q^{41} - 11 q^{43} + 10 q^{44} - 15 q^{46} + 15 q^{47} + 2 q^{49} - 7 q^{50} - q^{52} + 4 q^{53} - 17 q^{55} + 6 q^{56} - 33 q^{58} + 10 q^{59} - 12 q^{61} + 6 q^{62} + 12 q^{64} + 40 q^{65} - 19 q^{67} + 6 q^{68} - 16 q^{70} + 47 q^{71} - 2 q^{73} + 13 q^{74} - 10 q^{76} - 6 q^{77} - 15 q^{79} + 5 q^{80} - 20 q^{82} + 18 q^{83} - 25 q^{85} + 11 q^{86} - 10 q^{88} + 24 q^{89} - 3 q^{91} + 15 q^{92} - 15 q^{94} - 3 q^{95} - 25 q^{97} - 2 q^{98}+O(q^{100}) $ 12 * q - 12 * q^2 + 12 * q^4 + 5 * q^5 - 6 * q^7 - 12 * q^8 - 5 * q^10 + 10 * q^11 - q^13 + 6 * q^14 + 12 * q^16 + 6 * q^17 - 10 * q^19 + 5 * q^20 - 10 * q^22 + 15 * q^23 + 7 * q^25 + q^26 - 6 * q^28 + 33 * q^29 - 6 * q^31 - 12 * q^32 - 6 * q^34 + 16 * q^35 - 13 * q^37 + 10 * q^38 - 5 * q^40 + 20 * q^41 - 11 * q^43 + 10 * q^44 - 15 * q^46 + 15 * q^47 + 2 * q^49 - 7 * q^50 - q^52 + 4 * q^53 - 17 * q^55 + 6 * q^56 - 33 * q^58 + 10 * q^59 - 12 * q^61 + 6 * q^62 + 12 * q^64 + 40 * q^65 - 19 * q^67 + 6 * q^68 - 16 * q^70 + 47 * q^71 - 2 * q^73 + 13 * q^74 - 10 * q^76 - 6 * q^77 - 15 * q^79 + 5 * q^80 - 20 * q^82 + 18 * q^83 - 25 * q^85 + 11 * q^86 - 10 * q^88 + 24 * q^89 - 3 * q^91 + 15 * q^92 - 15 * q^94 - 3 * q^95 - 25 * q^97 - 2 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0.448758	0.200691	0.100345	−	0.994953i	$-0.468005\pi$
$5$	0.448758	0.200691	0.100345	+	0.994953i	$0.468005\pi$
$6$	0	0
$7$	−1.52942	−0.578067	−0.289033	−	0.957319i	$-0.593334\pi$
$7$	−1.52942	−0.578067	−0.289033	+	0.957319i	$0.593334\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−0.448758	−0.141910
$11$	−0.412196	−0.124282	−0.0621409	−	0.998067i	$-0.519793\pi$
$11$	−0.412196	−0.124282	−0.0621409	+	0.998067i	$0.519793\pi$
$12$	0	0
$13$	3.36442	0.933123	0.466561	−	0.884489i	$-0.345493\pi$
$13$	3.36442	0.933123	0.466561	+	0.884489i	$0.345493\pi$
$14$	1.52942	0.408755
$15$	0	0
$16$	1.00000	0.250000
$17$	3.96634	0.961979	0.480989	−	0.876726i	$-0.340278\pi$
$17$	3.96634	0.961979	0.480989	+	0.876726i	$0.340278\pi$
$18$	0	0
$19$	−5.13026	−1.17696	−0.588481	−	0.808511i	$-0.700274\pi$
$19$	−5.13026	−1.17696	−0.588481	+	0.808511i	$0.700274\pi$
$20$	0.448758	0.100345
$21$	0	0
$22$	0.412196	0.0878805
$23$	−2.63703	−0.549858	−0.274929	−	0.961465i	$-0.588654\pi$
$23$	−2.63703	−0.549858	−0.274929	+	0.961465i	$0.588654\pi$
$24$	0	0
$25$	−4.79862	−0.959723
$26$	−3.36442	−0.659817
$27$	0	0
$28$	−1.52942	−0.289033
$29$	−2.15254	−0.399717	−0.199858	−	0.979825i	$-0.564048\pi$
$29$	−2.15254	−0.399717	−0.199858	+	0.979825i	$0.564048\pi$
$30$	0	0
$31$	−5.88764	−1.05745	−0.528726	−	0.848793i	$-0.677330\pi$
$31$	−5.88764	−1.05745	−0.528726	+	0.848793i	$0.677330\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−3.96634	−0.680222
$35$	−0.686340	−0.116013
$36$	0	0
$37$	10.4747	1.72203	0.861013	−	0.508583i	$-0.169830\pi$
$37$	10.4747	1.72203	0.861013	+	0.508583i	$0.169830\pi$
$38$	5.13026	0.832238
$39$	0	0
$40$	−0.448758	−0.0709549
$41$	−5.89535	−0.920700	−0.460350	−	0.887738i	$-0.652276\pi$
$41$	−5.89535	−0.920700	−0.460350	+	0.887738i	$0.652276\pi$
$42$	0	0
$43$	1.10493	0.168500	0.0842498	−	0.996445i	$-0.473151\pi$
$43$	1.10493	0.168500	0.0842498	+	0.996445i	$0.473151\pi$
$44$	−0.412196	−0.0621409
$45$	0	0
$46$	2.63703	0.388808
$47$	5.78830	0.844310	0.422155	−	0.906524i	$-0.361274\pi$
$47$	5.78830	0.844310	0.422155	+	0.906524i	$0.361274\pi$
$48$	0	0
$49$	−4.66087	−0.665839
$50$	4.79862	0.678627
$51$	0	0
$52$	3.36442	0.466561
$53$	−1.69686	−0.233082	−0.116541	−	0.993186i	$-0.537181\pi$
$53$	−1.69686	−0.233082	−0.116541	+	0.993186i	$0.537181\pi$
$54$	0	0
$55$	−0.184976	−0.0249422
$56$	1.52942	0.204378
$57$	0	0
$58$	2.15254	0.282642
$59$	11.9240	1.55237	0.776187	−	0.630503i	$-0.217151\pi$
$59$	11.9240	1.55237	0.776187	+	0.630503i	$0.217151\pi$
$60$	0	0
$61$	12.2236	1.56507	0.782533	−	0.622609i	$-0.213927\pi$
$61$	12.2236	1.56507	0.782533	+	0.622609i	$0.213927\pi$
$62$	5.88764	0.747731
$63$	0	0
$64$	1.00000	0.125000
$65$	1.50981	0.187269
$66$	0	0
$67$	−15.8063	−1.93105	−0.965526	−	0.260307i	$-0.916176\pi$
$67$	−15.8063	−1.93105	−0.965526	+	0.260307i	$0.916176\pi$
$68$	3.96634	0.480989
$69$	0	0
$70$	0.686340	0.0820333
$71$	12.3113	1.46108	0.730541	−	0.682869i	$-0.239268\pi$
$71$	12.3113	1.46108	0.730541	+	0.682869i	$0.239268\pi$
$72$	0	0
$73$	5.42784	0.635280	0.317640	−	0.948211i	$-0.397110\pi$
$73$	5.42784	0.635280	0.317640	+	0.948211i	$0.397110\pi$
$74$	−10.4747	−1.21766
$75$	0	0
$76$	−5.13026	−0.588481
$77$	0.630422	0.0718432
$78$	0	0
$79$	2.21050	0.248701	0.124350	−	0.992238i	$-0.460315\pi$
$79$	2.21050	0.248701	0.124350	+	0.992238i	$0.460315\pi$
$80$	0.448758	0.0501727
$81$	0	0
$82$	5.89535	0.651033
$83$	11.7231	1.28678	0.643390	−	0.765538i	$-0.277527\pi$
$83$	11.7231	1.28678	0.643390	+	0.765538i	$0.277527\pi$
$84$	0	0
$85$	1.77993	0.193060
$86$	−1.10493	−0.119147
$87$	0	0
$88$	0.412196	0.0439403
$89$	0.410694	0.0435335	0.0217667	−	0.999763i	$-0.493071\pi$
$89$	0.410694	0.0435335	0.0217667	+	0.999763i	$0.493071\pi$
$90$	0	0
$91$	−5.14562	−0.539407
$92$	−2.63703	−0.274929
$93$	0	0
$94$	−5.78830	−0.597018
$95$	−2.30224	−0.236205
$96$	0	0
$97$	4.08503	0.414772	0.207386	−	0.978259i	$-0.433504\pi$
$97$	4.08503	0.414772	0.207386	+	0.978259i	$0.433504\pi$
$98$	4.66087	0.470819
$99$	0	0
$100$	−4.79862	−0.479862
$101$	2.71317	0.269971	0.134985	−	0.990848i	$-0.456901\pi$
$101$	2.71317	0.269971	0.134985	+	0.990848i	$0.456901\pi$
$102$	0	0
$103$	−8.11049	−0.799150	−0.399575	−	0.916700i	$-0.630842\pi$
$103$	−8.11049	−0.799150	−0.399575	+	0.916700i	$0.630842\pi$
$104$	−3.36442	−0.329909
$105$	0	0
$106$	1.69686	0.164814
$107$	2.84336	0.274878	0.137439	−	0.990510i	$-0.456113\pi$
$107$	2.84336	0.274878	0.137439	+	0.990510i	$0.456113\pi$
$108$	0	0
$109$	−7.54113	−0.722309	−0.361155	−	0.932506i	$-0.617617\pi$
$109$	−7.54113	−0.722309	−0.361155	+	0.932506i	$0.617617\pi$
$110$	0.184976	0.0176368
$111$	0	0
$112$	−1.52942	−0.144517
$113$	13.2743	1.24874	0.624369	−	0.781130i	$-0.285356\pi$
$113$	13.2743	1.24874	0.624369	+	0.781130i	$0.285356\pi$
$114$	0	0
$115$	−1.18339	−0.110351
$116$	−2.15254	−0.199858
$117$	0	0
$118$	−11.9240	−1.09769
$119$	−6.06621	−0.556088
$120$	0	0
$121$	−10.8301	−0.984554
$122$	−12.2236	−1.10667
$123$	0	0
$124$	−5.88764	−0.528726
$125$	−4.39721	−0.393298
$126$	0	0
$127$	2.16121	0.191776	0.0958882	−	0.995392i	$-0.469431\pi$
$127$	2.16121	0.191776	0.0958882	+	0.995392i	$0.469431\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−1.50981	−0.132419
$131$	−8.82663	−0.771187	−0.385593	−	0.922669i	$-0.626003\pi$
$131$	−8.82663	−0.771187	−0.385593	+	0.922669i	$0.626003\pi$
$132$	0	0
$133$	7.84633	0.680363
$134$	15.8063	1.36546
$135$	0	0
$136$	−3.96634	−0.340111
$137$	16.8659	1.44095	0.720477	−	0.693478i	$-0.243923\pi$
$137$	16.8659	1.44095	0.720477	+	0.693478i	$0.243923\pi$
$138$	0	0
$139$	−6.15934	−0.522428	−0.261214	−	0.965281i	$-0.584123\pi$
$139$	−6.15934	−0.522428	−0.261214	+	0.965281i	$0.584123\pi$
$140$	−0.686340	−0.0580063
$141$	0	0
$142$	−12.3113	−1.03314
$143$	−1.38680	−0.115970
$144$	0	0
$145$	−0.965970	−0.0802194
$146$	−5.42784	−0.449211
$147$	0	0
$148$	10.4747	0.861013
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	−3.51636	−0.286157	−0.143079	−	0.989711i	$-0.545700\pi$
$151$	−3.51636	−0.286157	−0.143079	+	0.989711i	$0.545700\pi$
$152$	5.13026	0.416119
$153$	0	0
$154$	−0.630422	−0.0508008
$155$	−2.64213	−0.212221
$156$	0	0
$157$	8.71888	0.695843	0.347921	−	0.937524i	$-0.386888\pi$
$157$	8.71888	0.695843	0.347921	+	0.937524i	$0.386888\pi$
$158$	−2.21050	−0.175858
$159$	0	0
$160$	−0.448758	−0.0354774
$161$	4.03312	0.317855
$162$	0	0
$163$	−5.58812	−0.437696	−0.218848	−	0.975759i	$-0.570230\pi$
$163$	−5.58812	−0.437696	−0.218848	+	0.975759i	$0.570230\pi$
$164$	−5.89535	−0.460350
$165$	0	0
$166$	−11.7231	−0.909891
$167$	−22.8936	−1.77156	−0.885778	−	0.464108i	$-0.846375\pi$
$167$	−22.8936	−1.77156	−0.885778	+	0.464108i	$0.846375\pi$
$168$	0	0
$169$	−1.68067	−0.129282
$170$	−1.77993	−0.136514
$171$	0	0
$172$	1.10493	0.0842498
$173$	19.4389	1.47791	0.738955	−	0.673754i	$-0.235319\pi$
$173$	19.4389	1.47791	0.738955	+	0.673754i	$0.235319\pi$
$174$	0	0
$175$	7.33911	0.554784
$176$	−0.412196	−0.0310705
$177$	0	0
$178$	−0.410694	−0.0307828
$179$	16.6171	1.24202	0.621011	−	0.783802i	$-0.286722\pi$
$179$	16.6171	1.24202	0.621011	+	0.783802i	$0.286722\pi$
$180$	0	0
$181$	0.435647	0.0323814	0.0161907	−	0.999869i	$-0.494846\pi$
$181$	0.435647	0.0323814	0.0161907	+	0.999869i	$0.494846\pi$
$182$	5.14562	0.381419
$183$	0	0
$184$	2.63703	0.194404
$185$	4.70059	0.345595
$186$	0	0
$187$	−1.63491	−0.119557
$188$	5.78830	0.422155
$189$	0	0
$190$	2.30224	0.167022
$191$	−16.5232	−1.19557	−0.597787	−	0.801655i	$-0.703953\pi$
$191$	−16.5232	−1.19557	−0.597787	+	0.801655i	$0.703953\pi$
$192$	0	0
$193$	−11.9297	−0.858716	−0.429358	−	0.903134i	$-0.641260\pi$
$193$	−11.9297	−0.858716	−0.429358	+	0.903134i	$0.641260\pi$
$194$	−4.08503	−0.293288
$195$	0	0
$196$	−4.66087	−0.332919
$197$	−17.4448	−1.24289	−0.621447	−	0.783456i	$-0.713455\pi$
$197$	−17.4448	−1.24289	−0.621447	+	0.783456i	$0.713455\pi$
$198$	0	0
$199$	21.2904	1.50923	0.754617	−	0.656166i	$-0.227823\pi$
$199$	21.2904	1.50923	0.754617	+	0.656166i	$0.227823\pi$
$200$	4.79862	0.339313
$201$	0	0
$202$	−2.71317	−0.190898
$203$	3.29214	0.231063
$204$	0	0
$205$	−2.64559	−0.184776
$206$	8.11049	0.565085
$207$	0	0
$208$	3.36442	0.233281
$209$	2.11467	0.146275
$210$	0	0
$211$	15.0226	1.03420	0.517098	−	0.855926i	$-0.327012\pi$
$211$	15.0226	1.03420	0.517098	+	0.855926i	$0.327012\pi$
$212$	−1.69686	−0.116541
$213$	0	0
$214$	−2.84336	−0.194368
$215$	0.495844	0.0338163
$216$	0	0
$217$	9.00469	0.611278
$218$	7.54113	0.510750
$219$	0	0
$220$	−0.184976	−0.0124711
$221$	13.3444	0.897644
$222$	0	0
$223$	−14.0252	−0.939196	−0.469598	−	0.882880i	$-0.655601\pi$
$223$	−14.0252	−0.939196	−0.469598	+	0.882880i	$0.655601\pi$
$224$	1.52942	0.102189
$225$	0	0
$226$	−13.2743	−0.882991
$227$	−0.382438	−0.0253833	−0.0126917	−	0.999919i	$-0.504040\pi$
$227$	−0.382438	−0.0253833	−0.0126917	+	0.999919i	$0.504040\pi$
$228$	0	0
$229$	20.5931	1.36083	0.680416	−	0.732826i	$-0.261799\pi$
$229$	20.5931	1.36083	0.680416	+	0.732826i	$0.261799\pi$
$230$	1.18339	0.0780302
$231$	0	0
$232$	2.15254	0.141321
$233$	10.0982	0.661553	0.330776	−	0.943709i	$-0.392689\pi$
$233$	10.0982	0.661553	0.330776	+	0.943709i	$0.392689\pi$
$234$	0	0
$235$	2.59755	0.169445
$236$	11.9240	0.776187
$237$	0	0
$238$	6.06621	0.393214
$239$	−11.1356	−0.720300	−0.360150	−	0.932894i	$-0.617275\pi$
$239$	−11.1356	−0.720300	−0.360150	+	0.932894i	$0.617275\pi$
$240$	0	0
$241$	5.22345	0.336472	0.168236	−	0.985747i	$-0.446193\pi$
$241$	5.22345	0.336472	0.168236	+	0.985747i	$0.446193\pi$
$242$	10.8301	0.696185
$243$	0	0
$244$	12.2236	0.782533
$245$	−2.09160	−0.133628
$246$	0	0
$247$	−17.2604	−1.09825
$248$	5.88764	0.373866
$249$	0	0
$250$	4.39721	0.278104
$251$	−13.7996	−0.871026	−0.435513	−	0.900183i	$-0.643433\pi$
$251$	−13.7996	−0.871026	−0.435513	+	0.900183i	$0.643433\pi$
$252$	0	0
$253$	1.08697	0.0683374
$254$	−2.16121	−0.135606
$255$	0	0
$256$	1.00000	0.0625000
$257$	15.4652	0.964690	0.482345	−	0.875981i	$-0.339785\pi$
$257$	15.4652	0.964690	0.482345	+	0.875981i	$0.339785\pi$
$258$	0	0
$259$	−16.0202	−0.995446
$260$	1.50981	0.0936345
$261$	0	0
$262$	8.82663	0.545311
$263$	25.6673	1.58271	0.791356	−	0.611355i	$-0.209375\pi$
$263$	25.6673	1.58271	0.791356	+	0.611355i	$0.209375\pi$
$264$	0	0
$265$	−0.761480	−0.0467773
$266$	−7.84633	−0.481089
$267$	0	0
$268$	−15.8063	−0.965526
$269$	17.3104	1.05544	0.527718	−	0.849420i	$-0.323048\pi$
$269$	17.3104	1.05544	0.527718	+	0.849420i	$0.323048\pi$
$270$	0	0
$271$	−1.02623	−0.0623391	−0.0311696	−	0.999514i	$-0.509923\pi$
$271$	−1.02623	−0.0623391	−0.0311696	+	0.999514i	$0.509923\pi$
$272$	3.96634	0.240495
$273$	0	0
$274$	−16.8659	−1.01891
$275$	1.97797	0.119276
$276$	0	0
$277$	21.6609	1.30148	0.650739	−	0.759302i	$-0.274459\pi$
$277$	21.6609	1.30148	0.650739	+	0.759302i	$0.274459\pi$
$278$	6.15934	0.369412
$279$	0	0
$280$	0.686340	0.0410167
$281$	23.3056	1.39029	0.695147	−	0.718867i	$-0.255339\pi$
$281$	23.3056	1.39029	0.695147	+	0.718867i	$0.255339\pi$
$282$	0	0
$283$	15.4938	0.921014	0.460507	−	0.887656i	$-0.347668\pi$
$283$	15.4938	0.921014	0.460507	+	0.887656i	$0.347668\pi$
$284$	12.3113	0.730541
$285$	0	0
$286$	1.38680	0.0820033
$287$	9.01648	0.532226
$288$	0	0
$289$	−1.26814	−0.0745964
$290$	0.965970	0.0567237
$291$	0	0
$292$	5.42784	0.317640
$293$	18.4933	1.08039	0.540196	−	0.841539i	$-0.318350\pi$
$293$	18.4933	1.08039	0.540196	+	0.841539i	$0.318350\pi$
$294$	0	0
$295$	5.35099	0.311547
$296$	−10.4747	−0.608828
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	−8.87207	−0.513085
$300$	0	0
$301$	−1.68990	−0.0974041
$302$	3.51636	0.202344
$303$	0	0
$304$	−5.13026	−0.294240
$305$	5.48542	0.314094
$306$	0	0
$307$	20.5286	1.17163	0.585814	−	0.810445i	$-0.300775\pi$
$307$	20.5286	1.17163	0.585814	+	0.810445i	$0.300775\pi$
$308$	0.630422	0.0359216
$309$	0	0
$310$	2.64213	0.150063
$311$	15.1347	0.858212	0.429106	−	0.903254i	$-0.358829\pi$
$311$	15.1347	0.858212	0.429106	+	0.903254i	$0.358829\pi$
$312$	0	0
$313$	−27.6172	−1.56102	−0.780509	−	0.625145i	$-0.785040\pi$
$313$	−27.6172	−1.56102	−0.780509	+	0.625145i	$0.785040\pi$
$314$	−8.71888	−0.492035
$315$	0	0
$316$	2.21050	0.124350
$317$	23.3262	1.31013	0.655064	−	0.755573i	$-0.272642\pi$
$317$	23.3262	1.31013	0.655064	+	0.755573i	$0.272642\pi$
$318$	0	0
$319$	0.887269	0.0496775
$320$	0.448758	0.0250863
$321$	0	0
$322$	−4.03312	−0.224757
$323$	−20.3484	−1.13221
$324$	0	0
$325$	−16.1446	−0.895539
$326$	5.58812	0.309498
$327$	0	0
$328$	5.89535	0.325516
$329$	−8.85275	−0.488068
$330$	0	0
$331$	−31.3199	−1.72149	−0.860747	−	0.509032i	$-0.830003\pi$
$331$	−31.3199	−1.72149	−0.860747	+	0.509032i	$0.830003\pi$
$332$	11.7231	0.643390
$333$	0	0
$334$	22.8936	1.25268
$335$	−7.09322	−0.387544
$336$	0	0
$337$	−3.36124	−0.183099	−0.0915493	−	0.995801i	$-0.529182\pi$
$337$	−3.36124	−0.183099	−0.0915493	+	0.995801i	$0.529182\pi$
$338$	1.68067	0.0914163
$339$	0	0
$340$	1.77993	0.0965301
$341$	2.42686	0.131422
$342$	0	0
$343$	17.8344	0.962966
$344$	−1.10493	−0.0595736
$345$	0	0
$346$	−19.4389	−1.04504
$347$	−11.2162	−0.602116	−0.301058	−	0.953606i	$-0.597340\pi$
$347$	−11.2162	−0.602116	−0.301058	+	0.953606i	$0.597340\pi$
$348$	0	0
$349$	13.4404	0.719448	0.359724	−	0.933059i	$-0.382871\pi$
$349$	13.4404	0.719448	0.359724	+	0.933059i	$0.382871\pi$
$350$	−7.33911	−0.392292
$351$	0	0
$352$	0.412196	0.0219701
$353$	−8.27345	−0.440351	−0.220176	−	0.975460i	$-0.570663\pi$
$353$	−8.27345	−0.440351	−0.220176	+	0.975460i	$0.570663\pi$
$354$	0	0
$355$	5.52479	0.293226
$356$	0.410694	0.0217667
$357$	0	0
$358$	−16.6171	−0.878242
$359$	31.9960	1.68869	0.844343	−	0.535802i	$-0.179991\pi$
$359$	31.9960	1.68869	0.844343	+	0.535802i	$0.179991\pi$
$360$	0	0
$361$	7.31955	0.385239
$362$	−0.435647	−0.0228971
$363$	0	0
$364$	−5.14562	−0.269704
$365$	2.43578	0.127495
$366$	0	0
$367$	−17.2289	−0.899342	−0.449671	−	0.893194i	$-0.648459\pi$
$367$	−17.2289	−0.899342	−0.449671	+	0.893194i	$0.648459\pi$
$368$	−2.63703	−0.137465
$369$	0	0
$370$	−4.70059	−0.244372
$371$	2.59522	0.134737
$372$	0	0
$373$	−4.22649	−0.218839	−0.109420	−	0.993996i	$-0.534899\pi$
$373$	−4.22649	−0.218839	−0.109420	+	0.993996i	$0.534899\pi$
$374$	1.63491	0.0845392
$375$	0	0
$376$	−5.78830	−0.298509
$377$	−7.24205	−0.372985
$378$	0	0
$379$	−4.84751	−0.249000	−0.124500	−	0.992220i	$-0.539733\pi$
$379$	−4.84751	−0.249000	−0.124500	+	0.992220i	$0.539733\pi$
$380$	−2.30224	−0.118103
$381$	0	0
$382$	16.5232	0.845399
$383$	26.5621	1.35726	0.678629	−	0.734481i	$-0.262574\pi$
$383$	26.5621	1.35726	0.678629	+	0.734481i	$0.262574\pi$
$384$	0	0
$385$	0.282907	0.0144183
$386$	11.9297	0.607204
$387$	0	0
$388$	4.08503	0.207386
$389$	31.1841	1.58110	0.790548	−	0.612400i	$-0.209796\pi$
$389$	31.1841	1.58110	0.790548	+	0.612400i	$0.209796\pi$
$390$	0	0
$391$	−10.4593	−0.528952
$392$	4.66087	0.235410
$393$	0	0
$394$	17.4448	0.878858
$395$	0.991979	0.0499119
$396$	0	0
$397$	38.9060	1.95264	0.976319	−	0.216335i	$-0.0694103\pi$
$397$	38.9060	1.95264	0.976319	+	0.216335i	$0.0694103\pi$
$398$	−21.2904	−1.06719
$399$	0	0
$400$	−4.79862	−0.239931
$401$	−27.4187	−1.36923	−0.684613	−	0.728907i	$-0.740029\pi$
$401$	−27.4187	−1.36923	−0.684613	+	0.728907i	$0.740029\pi$
$402$	0	0
$403$	−19.8085	−0.986732
$404$	2.71317	0.134985
$405$	0	0
$406$	−3.29214	−0.163386
$407$	−4.31762	−0.214017
$408$	0	0
$409$	13.5525	0.670129	0.335065	−	0.942195i	$-0.391242\pi$
$409$	13.5525	0.670129	0.335065	+	0.942195i	$0.391242\pi$
$410$	2.64559	0.130656
$411$	0	0
$412$	−8.11049	−0.399575
$413$	−18.2368	−0.897376
$414$	0	0
$415$	5.26085	0.258245
$416$	−3.36442	−0.164954
$417$	0	0
$418$	−2.11467	−0.103432
$419$	−5.66966	−0.276981	−0.138490	−	0.990364i	$-0.544225\pi$
$419$	−5.66966	−0.276981	−0.138490	+	0.990364i	$0.544225\pi$
$420$	0	0
$421$	18.4788	0.900603	0.450301	−	0.892877i	$-0.351317\pi$
$421$	18.4788	0.900603	0.450301	+	0.892877i	$0.351317\pi$
$422$	−15.0226	−0.731287
$423$	0	0
$424$	1.69686	0.0824069
$425$	−19.0329	−0.923234
$426$	0	0
$427$	−18.6950	−0.904713
$428$	2.84336	0.137439
$429$	0	0
$430$	−0.495844	−0.0239117
$431$	13.9772	0.673260	0.336630	−	0.941637i	$-0.390713\pi$
$431$	13.9772	0.673260	0.336630	+	0.941637i	$0.390713\pi$
$432$	0	0
$433$	−23.6058	−1.13442	−0.567211	−	0.823572i	$-0.691978\pi$
$433$	−23.6058	−1.13442	−0.567211	+	0.823572i	$0.691978\pi$
$434$	−9.00469	−0.432239
$435$	0	0
$436$	−7.54113	−0.361155
$437$	13.5286	0.647162
$438$	0	0
$439$	−36.6939	−1.75130	−0.875651	−	0.482944i	$-0.839567\pi$
$439$	−36.6939	−1.75130	−0.875651	+	0.482944i	$0.839567\pi$
$440$	0.184976	0.00881840
$441$	0	0
$442$	−13.3444	−0.634730
$443$	26.2648	1.24788	0.623939	−	0.781473i	$-0.285531\pi$
$443$	26.2648	1.24788	0.623939	+	0.781473i	$0.285531\pi$
$444$	0	0
$445$	0.184302	0.00873676
$446$	14.0252	0.664112
$447$	0	0
$448$	−1.52942	−0.0722584
$449$	15.4702	0.730084	0.365042	−	0.930991i	$-0.381055\pi$
$449$	15.4702	0.730084	0.365042	+	0.930991i	$0.381055\pi$
$450$	0	0
$451$	2.43004	0.114426
$452$	13.2743	0.624369
$453$	0	0
$454$	0.382438	0.0179487
$455$	−2.30914	−0.108254
$456$	0	0
$457$	−34.4833	−1.61306	−0.806531	−	0.591192i	$-0.798657\pi$
$457$	−34.4833	−1.61306	−0.806531	+	0.591192i	$0.798657\pi$
$458$	−20.5931	−0.962253
$459$	0	0
$460$	−1.18339	−0.0551757
$461$	−22.4975	−1.04781	−0.523906	−	0.851776i	$-0.675526\pi$
$461$	−22.4975	−1.04781	−0.523906	+	0.851776i	$0.675526\pi$
$462$	0	0
$463$	20.7183	0.962859	0.481430	−	0.876485i	$-0.340118\pi$
$463$	20.7183	0.962859	0.481430	+	0.876485i	$0.340118\pi$
$464$	−2.15254	−0.0999292
$465$	0	0
$466$	−10.0982	−0.467789
$467$	21.3417	0.987575	0.493787	−	0.869583i	$-0.335612\pi$
$467$	21.3417	0.987575	0.493787	+	0.869583i	$0.335612\pi$
$468$	0	0
$469$	24.1746	1.11628
$470$	−2.59755	−0.119816
$471$	0	0
$472$	−11.9240	−0.548847
$473$	−0.455447	−0.0209415
$474$	0	0
$475$	24.6181	1.12956
$476$	−6.06621	−0.278044
$477$	0	0
$478$	11.1356	0.509329
$479$	−21.6025	−0.987043	−0.493521	−	0.869734i	$-0.664291\pi$
$479$	−21.6025	−0.987043	−0.493521	+	0.869734i	$0.664291\pi$
$480$	0	0
$481$	35.2412	1.60686
$482$	−5.22345	−0.237921
$483$	0	0
$484$	−10.8301	−0.492277
$485$	1.83319	0.0832409
$486$	0	0
$487$	−24.2335	−1.09812	−0.549062	−	0.835782i	$-0.685015\pi$
$487$	−24.2335	−1.09812	−0.549062	+	0.835782i	$0.685015\pi$
$488$	−12.2236	−0.553334
$489$	0	0
$490$	2.09160	0.0944890
$491$	−18.0062	−0.812609	−0.406305	−	0.913738i	$-0.633183\pi$
$491$	−18.0062	−0.812609	−0.406305	+	0.913738i	$0.633183\pi$
$492$	0	0
$493$	−8.53771	−0.384519
$494$	17.2604	0.776580
$495$	0	0
$496$	−5.88764	−0.264363
$497$	−18.8292	−0.844604
$498$	0	0
$499$	−32.4315	−1.45183	−0.725916	−	0.687784i	$-0.758584\pi$
$499$	−32.4315	−1.45183	−0.725916	+	0.687784i	$0.758584\pi$
$500$	−4.39721	−0.196649
$501$	0	0
$502$	13.7996	0.615908
$503$	39.8535	1.77698	0.888489	−	0.458897i	$-0.151755\pi$
$503$	39.8535	1.77698	0.888489	+	0.458897i	$0.151755\pi$
$504$	0	0
$505$	1.21756	0.0541806
$506$	−1.08697	−0.0483218
$507$	0	0
$508$	2.16121	0.0958882
$509$	−25.8808	−1.14715	−0.573573	−	0.819155i	$-0.694443\pi$
$509$	−25.8808	−1.14715	−0.573573	+	0.819155i	$0.694443\pi$
$510$	0	0
$511$	−8.30145	−0.367234
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−15.4652	−0.682139
$515$	−3.63965	−0.160382
$516$	0	0
$517$	−2.38592	−0.104932
$518$	16.0202	0.703887
$519$	0	0
$520$	−1.50981	−0.0662096
$521$	1.80800	0.0792098	0.0396049	−	0.999215i	$-0.487390\pi$
$521$	1.80800	0.0792098	0.0396049	+	0.999215i	$0.487390\pi$
$522$	0	0
$523$	12.7444	0.557274	0.278637	−	0.960396i	$-0.410117\pi$
$523$	12.7444	0.557274	0.278637	+	0.960396i	$0.410117\pi$
$524$	−8.82663	−0.385593
$525$	0	0
$526$	−25.6673	−1.11915
$527$	−23.3524	−1.01725
$528$	0	0
$529$	−16.0461	−0.697656
$530$	0.761480	0.0330766
$531$	0	0
$532$	7.84633	0.340181
$533$	−19.8345	−0.859126
$534$	0	0
$535$	1.27598	0.0551655
$536$	15.8063	0.682730
$537$	0	0
$538$	−17.3104	−0.746306
$539$	1.92119	0.0827517
$540$	0	0
$541$	−1.88791	−0.0811674	−0.0405837	−	0.999176i	$-0.512922\pi$
$541$	−1.88791	−0.0811674	−0.0405837	+	0.999176i	$0.512922\pi$
$542$	1.02623	0.0440804
$543$	0	0
$544$	−3.96634	−0.170055
$545$	−3.38414	−0.144961
$546$	0	0
$547$	32.6811	1.39734	0.698671	−	0.715443i	$-0.253775\pi$
$547$	32.6811	1.39734	0.698671	+	0.715443i	$0.253775\pi$
$548$	16.8659	0.720477
$549$	0	0
$550$	−1.97797	−0.0843410
$551$	11.0431	0.470451
$552$	0	0
$553$	−3.38078	−0.143766
$554$	−21.6609	−0.920283
$555$	0	0
$556$	−6.15934	−0.261214
$557$	−19.7850	−0.838316	−0.419158	−	0.907913i	$-0.637675\pi$
$557$	−19.7850	−0.838316	−0.419158	+	0.907913i	$0.637675\pi$
$558$	0	0
$559$	3.71744	0.157231
$560$	−0.686340	−0.0290032
$561$	0	0
$562$	−23.3056	−0.983087
$563$	0.579433	0.0244202	0.0122101	−	0.999925i	$-0.496113\pi$
$563$	0.579433	0.0244202	0.0122101	+	0.999925i	$0.496113\pi$
$564$	0	0
$565$	5.95693	0.250610
$566$	−15.4938	−0.651255
$567$	0	0
$568$	−12.3113	−0.516571
$569$	37.1560	1.55766	0.778830	−	0.627235i	$-0.215814\pi$
$569$	37.1560	1.55766	0.778830	+	0.627235i	$0.215814\pi$
$570$	0	0
$571$	27.4696	1.14957	0.574784	−	0.818305i	$-0.305086\pi$
$571$	27.4696	1.14957	0.574784	+	0.818305i	$0.305086\pi$
$572$	−1.38680	−0.0579851
$573$	0	0
$574$	−9.01648	−0.376341
$575$	12.6541	0.527712
$576$	0	0
$577$	40.5999	1.69019	0.845097	−	0.534613i	$-0.179543\pi$
$577$	40.5999	1.69019	0.845097	+	0.534613i	$0.179543\pi$
$578$	1.26814	0.0527476
$579$	0	0
$580$	−0.965970	−0.0401097
$581$	−17.9296	−0.743845
$582$	0	0
$583$	0.699440	0.0289678
$584$	−5.42784	−0.224605
$585$	0	0
$586$	−18.4933	−0.763952
$587$	13.9395	0.575344	0.287672	−	0.957729i	$-0.407119\pi$
$587$	13.9395	0.575344	0.287672	+	0.957729i	$0.407119\pi$
$588$	0	0
$589$	30.2051	1.24458
$590$	−5.35099	−0.220297
$591$	0	0
$592$	10.4747	0.430507
$593$	−9.08991	−0.373278	−0.186639	−	0.982429i	$-0.559759\pi$
$593$	−9.08991	−0.373278	−0.186639	+	0.982429i	$0.559759\pi$
$594$	0	0
$595$	−2.72226	−0.111602
$596$	1.00000	0.0409616
$597$	0	0
$598$	8.87207	0.362806
$599$	−26.2211	−1.07136	−0.535682	−	0.844420i	$-0.679946\pi$
$599$	−26.2211	−1.07136	−0.535682	+	0.844420i	$0.679946\pi$
$600$	0	0
$601$	23.6914	0.966393	0.483196	−	0.875512i	$-0.339476\pi$
$601$	23.6914	0.966393	0.483196	+	0.875512i	$0.339476\pi$
$602$	1.68990	0.0688751
$603$	0	0
$604$	−3.51636	−0.143079
$605$	−4.86009	−0.197591
$606$	0	0
$607$	44.2134	1.79457	0.897283	−	0.441457i	$-0.145538\pi$
$607$	44.2134	1.79457	0.897283	+	0.441457i	$0.145538\pi$
$608$	5.13026	0.208059
$609$	0	0
$610$	−5.48542	−0.222098
$611$	19.4743	0.787845
$612$	0	0
$613$	−32.3437	−1.30635	−0.653174	−	0.757208i	$-0.726563\pi$
$613$	−32.3437	−1.30635	−0.653174	+	0.757208i	$0.726563\pi$
$614$	−20.5286	−0.828466
$615$	0	0
$616$	−0.630422	−0.0254004
$617$	−35.5068	−1.42945	−0.714725	−	0.699406i	$-0.753448\pi$
$617$	−35.5068	−1.42945	−0.714725	+	0.699406i	$0.753448\pi$
$618$	0	0
$619$	21.5408	0.865800	0.432900	−	0.901442i	$-0.357490\pi$
$619$	21.5408	0.865800	0.432900	+	0.901442i	$0.357490\pi$
$620$	−2.64213	−0.106110
$621$	0	0
$622$	−15.1347	−0.606848
$623$	−0.628124	−0.0251653
$624$	0	0
$625$	22.0198	0.880792
$626$	27.6172	1.10381
$627$	0	0
$628$	8.71888	0.347921
$629$	41.5461	1.65655
$630$	0	0
$631$	19.9049	0.792400	0.396200	−	0.918164i	$-0.370329\pi$
$631$	19.9049	0.792400	0.396200	+	0.918164i	$0.370329\pi$
$632$	−2.21050	−0.0879289
$633$	0	0
$634$	−23.3262	−0.926401
$635$	0.969861	0.0384877
$636$	0	0
$637$	−15.6811	−0.621309
$638$	−0.887269	−0.0351273
$639$	0	0
$640$	−0.448758	−0.0177387
$641$	47.4766	1.87521	0.937606	−	0.347700i	$-0.113037\pi$
$641$	47.4766	1.87521	0.937606	+	0.347700i	$0.113037\pi$
$642$	0	0
$643$	15.1115	0.595938	0.297969	−	0.954576i	$-0.403691\pi$
$643$	15.1115	0.595938	0.297969	+	0.954576i	$0.403691\pi$
$644$	4.03312	0.158927
$645$	0	0
$646$	20.3484	0.800595
$647$	0.935043	0.0367603	0.0183802	−	0.999831i	$-0.494149\pi$
$647$	0.935043	0.0367603	0.0183802	+	0.999831i	$0.494149\pi$
$648$	0	0
$649$	−4.91503	−0.192932
$650$	16.1446	0.633242
$651$	0	0
$652$	−5.58812	−0.218848
$653$	−6.49482	−0.254162	−0.127081	−	0.991892i	$-0.540561\pi$
$653$	−6.49482	−0.254162	−0.127081	+	0.991892i	$0.540561\pi$
$654$	0	0
$655$	−3.96102	−0.154770
$656$	−5.89535	−0.230175
$657$	0	0
$658$	8.85275	0.345116
$659$	−19.9197	−0.775962	−0.387981	−	0.921667i	$-0.626827\pi$
$659$	−19.9197	−0.775962	−0.387981	+	0.921667i	$0.626827\pi$
$660$	0	0
$661$	51.2734	1.99430	0.997152	−	0.0754194i	$-0.0240296\pi$
$661$	51.2734	1.99430	0.997152	+	0.0754194i	$0.0240296\pi$
$662$	31.3199	1.21728
$663$	0	0
$664$	−11.7231	−0.454946
$665$	3.52110	0.136542
$666$	0	0
$667$	5.67631	0.219787
$668$	−22.8936	−0.885778
$669$	0	0
$670$	7.09322	0.274035
$671$	−5.03850	−0.194509
$672$	0	0
$673$	8.64483	0.333234	0.166617	−	0.986022i	$-0.446716\pi$
$673$	8.64483	0.333234	0.166617	+	0.986022i	$0.446716\pi$
$674$	3.36124	0.129470
$675$	0	0
$676$	−1.68067	−0.0646411
$677$	−20.6047	−0.791904	−0.395952	−	0.918271i	$-0.629585\pi$
$677$	−20.6047	−0.791904	−0.395952	+	0.918271i	$0.629585\pi$
$678$	0	0
$679$	−6.24773	−0.239766
$680$	−1.77993	−0.0682571
$681$	0	0
$682$	−2.42686	−0.0929294
$683$	30.6010	1.17091	0.585457	−	0.810704i	$-0.300915\pi$
$683$	30.6010	1.17091	0.585457	+	0.810704i	$0.300915\pi$
$684$	0	0
$685$	7.56873	0.289186
$686$	−17.8344	−0.680920
$687$	0	0
$688$	1.10493	0.0421249
$689$	−5.70896	−0.217494
$690$	0	0
$691$	13.6881	0.520720	0.260360	−	0.965512i	$-0.416159\pi$
$691$	13.6881	0.520720	0.260360	+	0.965512i	$0.416159\pi$
$692$	19.4389	0.738955
$693$	0	0
$694$	11.2162	0.425760
$695$	−2.76405	−0.104846
$696$	0	0
$697$	−23.3830	−0.885694
$698$	−13.4404	−0.508727
$699$	0	0
$700$	7.33911	0.277392
$701$	−9.14812	−0.345520	−0.172760	−	0.984964i	$-0.555268\pi$
$701$	−9.14812	−0.345520	−0.172760	+	0.984964i	$0.555268\pi$
$702$	0	0
$703$	−53.7378	−2.02676
$704$	−0.412196	−0.0155352
$705$	0	0
$706$	8.27345	0.311375
$707$	−4.14958	−0.156061
$708$	0	0
$709$	48.8099	1.83309	0.916546	−	0.399929i	$-0.130965\pi$
$709$	48.8099	1.83309	0.916546	+	0.399929i	$0.130965\pi$
$710$	−5.52479	−0.207342
$711$	0	0
$712$	−0.410694	−0.0153914
$713$	15.5259	0.581448
$714$	0	0
$715$	−0.622338	−0.0232741
$716$	16.6171	0.621011
$717$	0	0
$718$	−31.9960	−1.19408
$719$	−37.9337	−1.41469	−0.707344	−	0.706870i	$-0.750107\pi$
$719$	−37.9337	−1.41469	−0.707344	+	0.706870i	$0.750107\pi$
$720$	0	0
$721$	12.4044	0.461962
$722$	−7.31955	−0.272405
$723$	0	0
$724$	0.435647	0.0161907
$725$	10.3292	0.383617
$726$	0	0
$727$	30.6546	1.13692	0.568458	−	0.822712i	$-0.307540\pi$
$727$	30.6546	1.13692	0.568458	+	0.822712i	$0.307540\pi$
$728$	5.14562	0.190709
$729$	0	0
$730$	−2.43578	−0.0901524
$731$	4.38251	0.162093
$732$	0	0
$733$	7.53505	0.278313	0.139157	−	0.990270i	$-0.455561\pi$
$733$	7.53505	0.278313	0.139157	+	0.990270i	$0.455561\pi$
$734$	17.2289	0.635931
$735$	0	0
$736$	2.63703	0.0972021
$737$	6.51531	0.239995
$738$	0	0
$739$	−0.680768	−0.0250424	−0.0125212	−	0.999922i	$-0.503986\pi$
$739$	−0.680768	−0.0250424	−0.0125212	+	0.999922i	$0.503986\pi$
$740$	4.70059	0.172797
$741$	0	0
$742$	−2.59522	−0.0952734
$743$	−8.32757	−0.305509	−0.152754	−	0.988264i	$-0.548814\pi$
$743$	−8.32757	−0.305509	−0.152754	+	0.988264i	$0.548814\pi$
$744$	0	0
$745$	0.448758	0.0164412
$746$	4.22649	0.154743
$747$	0	0
$748$	−1.63491	−0.0597783
$749$	−4.34870	−0.158898
$750$	0	0
$751$	−31.8640	−1.16273	−0.581366	−	0.813642i	$-0.697482\pi$
$751$	−31.8640	−1.16273	−0.581366	+	0.813642i	$0.697482\pi$
$752$	5.78830	0.211078
$753$	0	0
$754$	7.24205	0.263740
$755$	−1.57799	−0.0574291
$756$	0	0
$757$	35.9885	1.30802	0.654012	−	0.756484i	$-0.273085\pi$
$757$	35.9885	1.30802	0.654012	+	0.756484i	$0.273085\pi$
$758$	4.84751	0.176069
$759$	0	0
$760$	2.30224	0.0835112
$761$	26.6709	0.966821	0.483410	−	0.875394i	$-0.339398\pi$
$761$	26.6709	0.966821	0.483410	+	0.875394i	$0.339398\pi$
$762$	0	0
$763$	11.5336	0.417543
$764$	−16.5232	−0.597787
$765$	0	0
$766$	−26.5621	−0.959727
$767$	40.1174	1.44856
$768$	0	0
$769$	30.4187	1.09692	0.548462	−	0.836175i	$-0.315213\pi$
$769$	30.4187	1.09692	0.548462	+	0.836175i	$0.315213\pi$
$770$	−0.282907	−0.0101953
$771$	0	0
$772$	−11.9297	−0.429358
$773$	10.8028	0.388550	0.194275	−	0.980947i	$-0.437765\pi$
$773$	10.8028	0.388550	0.194275	+	0.980947i	$0.437765\pi$
$774$	0	0
$775$	28.2525	1.01486
$776$	−4.08503	−0.146644
$777$	0	0
$778$	−31.1841	−1.11800
$779$	30.2447	1.08363
$780$	0	0
$781$	−5.07467	−0.181586
$782$	10.4593	0.374025
$783$	0	0
$784$	−4.66087	−0.166460
$785$	3.91267	0.139649
$786$	0	0
$787$	8.92923	0.318293	0.159146	−	0.987255i	$-0.449126\pi$
$787$	8.92923	0.318293	0.159146	+	0.987255i	$0.449126\pi$
$788$	−17.4448	−0.621447
$789$	0	0
$790$	−0.991979	−0.0352930
$791$	−20.3019	−0.721854
$792$	0	0
$793$	41.1252	1.46040
$794$	−38.9060	−1.38072
$795$	0	0
$796$	21.2904	0.754617
$797$	−23.7646	−0.841784	−0.420892	−	0.907111i	$-0.638283\pi$
$797$	−23.7646	−0.841784	−0.420892	+	0.907111i	$0.638283\pi$
$798$	0	0
$799$	22.9584	0.812209
$800$	4.79862	0.169657
$801$	0	0
$802$	27.4187	0.968189
$803$	−2.23733	−0.0789538
$804$	0	0
$805$	1.80990	0.0637905
$806$	19.8085	0.697725
$807$	0	0
$808$	−2.71317	−0.0954490
$809$	−29.4039	−1.03379	−0.516893	−	0.856050i	$-0.672912\pi$
$809$	−29.4039	−1.03379	−0.516893	+	0.856050i	$0.672912\pi$
$810$	0	0
$811$	35.0491	1.23074	0.615371	−	0.788238i	$-0.289006\pi$
$811$	35.0491	1.23074	0.615371	+	0.788238i	$0.289006\pi$
$812$	3.29214	0.115532
$813$	0	0
$814$	4.31762	0.151333
$815$	−2.50771	−0.0878414
$816$	0	0
$817$	−5.66856	−0.198318
$818$	−13.5525	−0.473853
$819$	0	0
$820$	−2.64559	−0.0923879
$821$	−17.4579	−0.609287	−0.304643	−	0.952467i	$-0.598537\pi$
$821$	−17.4579	−0.609287	−0.304643	+	0.952467i	$0.598537\pi$
$822$	0	0
$823$	16.2840	0.567624	0.283812	−	0.958880i	$-0.408401\pi$
$823$	16.2840	0.567624	0.283812	+	0.958880i	$0.408401\pi$
$824$	8.11049	0.282542
$825$	0	0
$826$	18.2368	0.634541
$827$	31.3272	1.08935	0.544676	−	0.838646i	$-0.316652\pi$
$827$	31.3272	1.08935	0.544676	+	0.838646i	$0.316652\pi$
$828$	0	0
$829$	−3.39656	−0.117967	−0.0589837	−	0.998259i	$-0.518786\pi$
$829$	−3.39656	−0.117967	−0.0589837	+	0.998259i	$0.518786\pi$
$830$	−5.26085	−0.182607
$831$	0	0
$832$	3.36442	0.116640
$833$	−18.4866	−0.640523
$834$	0	0
$835$	−10.2737	−0.355535
$836$	2.11467	0.0731375
$837$	0	0
$838$	5.66966	0.195855
$839$	−12.2041	−0.421331	−0.210666	−	0.977558i	$-0.567563\pi$
$839$	−12.2041	−0.421331	−0.210666	+	0.977558i	$0.567563\pi$
$840$	0	0
$841$	−24.3666	−0.840227
$842$	−18.4788	−0.636822
$843$	0	0
$844$	15.0226	0.517098
$845$	−0.754213	−0.0259457
$846$	0	0
$847$	16.5638	0.569138
$848$	−1.69686	−0.0582705
$849$	0	0
$850$	19.0329	0.652825
$851$	−27.6220	−0.946870
$852$	0	0
$853$	−0.474093	−0.0162326	−0.00811632	−	0.999967i	$-0.502584\pi$
$853$	−0.474093	−0.0162326	−0.00811632	+	0.999967i	$0.502584\pi$
$854$	18.6950	0.639729
$855$	0	0
$856$	−2.84336	−0.0971841
$857$	−29.7904	−1.01762	−0.508810	−	0.860879i	$-0.669914\pi$
$857$	−29.7904	−1.01762	−0.508810	+	0.860879i	$0.669914\pi$
$858$	0	0
$859$	−4.72469	−0.161204	−0.0806021	−	0.996746i	$-0.525684\pi$
$859$	−4.72469	−0.161204	−0.0806021	+	0.996746i	$0.525684\pi$
$860$	0.495844	0.0169082
$861$	0	0
$862$	−13.9772	−0.476067
$863$	−33.2971	−1.13345	−0.566723	−	0.823908i	$-0.691789\pi$
$863$	−33.2971	−1.13345	−0.566723	+	0.823908i	$0.691789\pi$
$864$	0	0
$865$	8.72335	0.296603
$866$	23.6058	0.802158
$867$	0	0
$868$	9.00469	0.305639
$869$	−0.911160	−0.0309090
$870$	0	0
$871$	−53.1792	−1.80191
$872$	7.54113	0.255375
$873$	0	0
$874$	−13.5286	−0.457613
$875$	6.72518	0.227353
$876$	0	0
$877$	−31.5174	−1.06427	−0.532134	−	0.846660i	$-0.678610\pi$
$877$	−31.5174	−1.06427	−0.532134	+	0.846660i	$0.678610\pi$
$878$	36.6939	1.23836
$879$	0	0
$880$	−0.184976	−0.00623555
$881$	17.5526	0.591363	0.295682	−	0.955287i	$-0.404453\pi$
$881$	17.5526	0.591363	0.295682	+	0.955287i	$0.404453\pi$
$882$	0	0
$883$	−24.7111	−0.831594	−0.415797	−	0.909457i	$-0.636497\pi$
$883$	−24.7111	−0.831594	−0.415797	+	0.909457i	$0.636497\pi$
$884$	13.3444	0.448822
$885$	0	0
$886$	−26.2648	−0.882384
$887$	7.55626	0.253715	0.126857	−	0.991921i	$-0.459511\pi$
$887$	7.55626	0.253715	0.126857	+	0.991921i	$0.459511\pi$
$888$	0	0
$889$	−3.30540	−0.110860
$890$	−0.184302	−0.00617782
$891$	0	0
$892$	−14.0252	−0.469598
$893$	−29.6955	−0.993721
$894$	0	0
$895$	7.45706	0.249262
$896$	1.52942	0.0510944
$897$	0	0
$898$	−15.4702	−0.516247
$899$	12.6734	0.422681
$900$	0	0
$901$	−6.73033	−0.224220
$902$	−2.43004	−0.0809116
$903$	0	0
$904$	−13.2743	−0.441496
$905$	0.195500	0.00649865
$906$	0	0
$907$	−58.9226	−1.95649	−0.978246	−	0.207447i	$-0.933485\pi$
$907$	−58.9226	−1.95649	−0.978246	+	0.207447i	$0.933485\pi$
$908$	−0.382438	−0.0126917
$909$	0	0
$910$	2.30914	0.0765471
$911$	41.7691	1.38387	0.691937	−	0.721958i	$-0.256758\pi$
$911$	41.7691	1.38387	0.691937	+	0.721958i	$0.256758\pi$
$912$	0	0
$913$	−4.83223	−0.159923
$914$	34.4833	1.14061
$915$	0	0
$916$	20.5931	0.680416
$917$	13.4996	0.445797
$918$	0	0
$919$	49.9848	1.64885	0.824423	−	0.565974i	$-0.191500\pi$
$919$	49.9848	1.64885	0.824423	+	0.565974i	$0.191500\pi$
$920$	1.18339	0.0390151
$921$	0	0
$922$	22.4975	0.740916
$923$	41.4204	1.36337
$924$	0	0
$925$	−50.2639	−1.65267
$926$	−20.7183	−0.680844
$927$	0	0
$928$	2.15254	0.0706606
$929$	22.3752	0.734106	0.367053	−	0.930200i	$-0.380367\pi$
$929$	22.3752	0.734106	0.367053	+	0.930200i	$0.380367\pi$
$930$	0	0
$931$	23.9115	0.783667
$932$	10.0982	0.330776
$933$	0	0
$934$	−21.3417	−0.698321
$935$	−0.733679	−0.0239939
$936$	0	0
$937$	31.4257	1.02663	0.513316	−	0.858199i	$-0.328417\pi$
$937$	31.4257	1.02663	0.513316	+	0.858199i	$0.328417\pi$
$938$	−24.1746	−0.789327
$939$	0	0
$940$	2.59755	0.0847226
$941$	7.47412	0.243649	0.121825	−	0.992552i	$-0.461125\pi$
$941$	7.47412	0.243649	0.121825	+	0.992552i	$0.461125\pi$
$942$	0	0
$943$	15.5462	0.506254
$944$	11.9240	0.388094
$945$	0	0
$946$	0.455447	0.0148078
$947$	−48.7738	−1.58494	−0.792468	−	0.609914i	$-0.791204\pi$
$947$	−48.7738	−1.58494	−0.792468	+	0.609914i	$0.791204\pi$
$948$	0	0
$949$	18.2615	0.592794
$950$	−24.6181	−0.798718
$951$	0	0
$952$	6.06621	0.196607
$953$	54.8068	1.77537	0.887683	−	0.460455i	$-0.152314\pi$
$953$	54.8068	1.77537	0.887683	+	0.460455i	$0.152314\pi$
$954$	0	0
$955$	−7.41490	−0.239941
$956$	−11.1356	−0.360150
$957$	0	0
$958$	21.6025	0.697945
$959$	−25.7951	−0.832968
$960$	0	0
$961$	3.66434	0.118205
$962$	−35.2412	−1.13622
$963$	0	0
$964$	5.22345	0.168236
$965$	−5.35353	−0.172336
$966$	0	0
$967$	9.63016	0.309685	0.154843	−	0.987939i	$-0.450513\pi$
$967$	9.63016	0.309685	0.154843	+	0.987939i	$0.450513\pi$
$968$	10.8301	0.348092
$969$	0	0
$970$	−1.83319	−0.0588602
$971$	16.8080	0.539395	0.269698	−	0.962945i	$-0.413076\pi$
$971$	16.8080	0.539395	0.269698	+	0.962945i	$0.413076\pi$
$972$	0	0
$973$	9.42022	0.301998
$974$	24.2335	0.776491
$975$	0	0
$976$	12.2236	0.391266
$977$	−4.01585	−0.128478	−0.0642392	−	0.997935i	$-0.520462\pi$
$977$	−4.01585	−0.128478	−0.0642392	+	0.997935i	$0.520462\pi$
$978$	0	0
$979$	−0.169287	−0.00541042
$980$	−2.09160	−0.0668138
$981$	0	0
$982$	18.0062	0.574601
$983$	−4.31416	−0.137600	−0.0688001	−	0.997630i	$-0.521917\pi$
$983$	−4.31416	−0.137600	−0.0688001	+	0.997630i	$0.521917\pi$
$984$	0	0
$985$	−7.82851	−0.249437
$986$	8.53771	0.271896
$987$	0	0
$988$	−17.2604	−0.549125
$989$	−2.91372	−0.0926509
$990$	0	0
$991$	35.7058	1.13423	0.567116	−	0.823638i	$-0.308059\pi$
$991$	35.7058	1.13423	0.567116	+	0.823638i	$0.308059\pi$
$992$	5.88764	0.186933
$993$	0	0
$994$	18.8292	0.597225
$995$	9.55422	0.302889
$996$	0	0
$997$	−32.4431	−1.02748	−0.513742	−	0.857944i	$-0.671741\pi$
$997$	−32.4431	−1.02748	−0.513742	+	0.857944i	$0.671741\pi$
$998$	32.4315	1.02660
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.l.1.6	✓	12
3.2	odd	2		8046.2.a.m.1.7	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.l.1.6	✓	12	1.1	even	1	trivial
8046.2.a.m.1.7	yes	12	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +0.448758 q^{5} -1.52942 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +0.448758 q^{5} -1.52942 q^{7} -1.00000 q^{8} -0.448758 q^{10} -0.412196 q^{11} +3.36442 q^{13} +1.52942 q^{14} +1.00000 q^{16} +3.96634 q^{17} -5.13026 q^{19} +0.448758 q^{20} +0.412196 q^{22} -2.63703 q^{23} -4.79862 q^{25} -3.36442 q^{26} -1.52942 q^{28} -2.15254 q^{29} -5.88764 q^{31} -1.00000 q^{32} -3.96634 q^{34} -0.686340 q^{35} +10.4747 q^{37} +5.13026 q^{38} -0.448758 q^{40} -5.89535 q^{41} +1.10493 q^{43} -0.412196 q^{44} +2.63703 q^{46} +5.78830 q^{47} -4.66087 q^{49} +4.79862 q^{50} +3.36442 q^{52} -1.69686 q^{53} -0.184976 q^{55} +1.52942 q^{56} +2.15254 q^{58} +11.9240 q^{59} +12.2236 q^{61} +5.88764 q^{62} +1.00000 q^{64} +1.50981 q^{65} -15.8063 q^{67} +3.96634 q^{68} +0.686340 q^{70} +12.3113 q^{71} +5.42784 q^{73} -10.4747 q^{74} -5.13026 q^{76} +0.630422 q^{77} +2.21050 q^{79} +0.448758 q^{80} +5.89535 q^{82} +11.7231 q^{83} +1.77993 q^{85} -1.10493 q^{86} +0.412196 q^{88} +0.410694 q^{89} -5.14562 q^{91} -2.63703 q^{92} -5.78830 q^{94} -2.30224 q^{95} +4.08503 q^{97} +4.66087 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{2} + 12 q^{4} + 5 q^{5} - 6 q^{7} - 12 q^{8}+O(q^{10}) \) 12 * q - 12 * q^2 + 12 * q^4 + 5 * q^5 - 6 * q^7 - 12 * q^8	\( 12 q - 12 q^{2} + 12 q^{4} + 5 q^{5} - 6 q^{7} - 12 q^{8} - 5 q^{10} + 10 q^{11} - q^{13} + 6 q^{14} + 12 q^{16} + 6 q^{17} - 10 q^{19} + 5 q^{20} - 10 q^{22} + 15 q^{23} + 7 q^{25} + q^{26} - 6 q^{28} + 33 q^{29} - 6 q^{31} - 12 q^{32} - 6 q^{34} + 16 q^{35} - 13 q^{37} + 10 q^{38} - 5 q^{40} + 20 q^{41} - 11 q^{43} + 10 q^{44} - 15 q^{46} + 15 q^{47} + 2 q^{49} - 7 q^{50} - q^{52} + 4 q^{53} - 17 q^{55} + 6 q^{56} - 33 q^{58} + 10 q^{59} - 12 q^{61} + 6 q^{62} + 12 q^{64} + 40 q^{65} - 19 q^{67} + 6 q^{68} - 16 q^{70} + 47 q^{71} - 2 q^{73} + 13 q^{74} - 10 q^{76} - 6 q^{77} - 15 q^{79} + 5 q^{80} - 20 q^{82} + 18 q^{83} - 25 q^{85} + 11 q^{86} - 10 q^{88} + 24 q^{89} - 3 q^{91} + 15 q^{92} - 15 q^{94} - 3 q^{95} - 25 q^{97} - 2 q^{98}+O(q^{100}) \) 12 * q - 12 * q^2 + 12 * q^4 + 5 * q^5 - 6 * q^7 - 12 * q^8 - 5 * q^10 + 10 * q^11 - q^13 + 6 * q^14 + 12 * q^16 + 6 * q^17 - 10 * q^19 + 5 * q^20 - 10 * q^22 + 15 * q^23 + 7 * q^25 + q^26 - 6 * q^28 + 33 * q^29 - 6 * q^31 - 12 * q^32 - 6 * q^34 + 16 * q^35 - 13 * q^37 + 10 * q^38 - 5 * q^40 + 20 * q^41 - 11 * q^43 + 10 * q^44 - 15 * q^46 + 15 * q^47 + 2 * q^49 - 7 * q^50 - q^52 + 4 * q^53 - 17 * q^55 + 6 * q^56 - 33 * q^58 + 10 * q^59 - 12 * q^61 + 6 * q^62 + 12 * q^64 + 40 * q^65 - 19 * q^67 + 6 * q^68 - 16 * q^70 + 47 * q^71 - 2 * q^73 + 13 * q^74 - 10 * q^76 - 6 * q^77 - 15 * q^79 + 5 * q^80 - 20 * q^82 + 18 * q^83 - 25 * q^85 + 11 * q^86 - 10 * q^88 + 24 * q^89 - 3 * q^91 + 15 * q^92 - 15 * q^94 - 3 * q^95 - 25 * q^97 - 2 * q^98

Introduction

L-functions

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