LMFDB - Embedded newform 8048.2.a.v.1.23

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, 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("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.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:	$64.2636035467$
Analytic rank:	$1$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.23
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.98770 q^{3} +0.688613 q^{5} +1.63548 q^{7} +0.950962 q^{9} +O(q^{10})$	$q+1.98770 q^{3} +0.688613 q^{5} +1.63548 q^{7} +0.950962 q^{9} -1.13439 q^{11} -4.69083 q^{13} +1.36876 q^{15} +3.66450 q^{17} +0.478156 q^{19} +3.25084 q^{21} -4.93137 q^{23} -4.52581 q^{25} -4.07288 q^{27} +3.04650 q^{29} -7.78407 q^{31} -2.25484 q^{33} +1.12621 q^{35} -1.05854 q^{37} -9.32397 q^{39} -10.5669 q^{41} +3.06948 q^{43} +0.654845 q^{45} -11.5743 q^{47} -4.32522 q^{49} +7.28393 q^{51} +6.59289 q^{53} -0.781158 q^{55} +0.950432 q^{57} +2.97184 q^{59} -9.43833 q^{61} +1.55527 q^{63} -3.23016 q^{65} +8.15913 q^{67} -9.80210 q^{69} -11.2051 q^{71} -2.52633 q^{73} -8.99597 q^{75} -1.85527 q^{77} +0.930221 q^{79} -10.9486 q^{81} -1.69982 q^{83} +2.52342 q^{85} +6.05554 q^{87} +1.82823 q^{89} -7.67173 q^{91} -15.4724 q^{93} +0.329264 q^{95} +12.4565 q^{97} -1.07876 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.98770	1.14760	0.573800	−	0.818995i	$-0.305469\pi$
$3$	1.98770	1.14760	0.573800	+	0.818995i	$0.305469\pi$
$4$	0	0
$5$	0.688613	0.307957	0.153979	−	0.988074i	$-0.450791\pi$
$5$	0.688613	0.307957	0.153979	+	0.988074i	$0.450791\pi$
$6$	0	0
$7$	1.63548	0.618152	0.309076	−	0.951037i	$-0.399980\pi$
$7$	1.63548	0.618152	0.309076	+	0.951037i	$0.399980\pi$
$8$	0	0
$9$	0.950962	0.316987
$10$	0	0
$11$	−1.13439	−0.342033	−0.171016	−	0.985268i	$-0.554705\pi$
$11$	−1.13439	−0.342033	−0.171016	+	0.985268i	$0.554705\pi$
$12$	0	0
$13$	−4.69083	−1.30100	−0.650501	−	0.759506i	$-0.725441\pi$
$13$	−4.69083	−1.30100	−0.650501	+	0.759506i	$0.725441\pi$
$14$	0	0
$15$	1.36876	0.353412
$16$	0	0
$17$	3.66450	0.888771	0.444385	−	0.895836i	$-0.353422\pi$
$17$	3.66450	0.888771	0.444385	+	0.895836i	$0.353422\pi$
$18$	0	0
$19$	0.478156	0.109697	0.0548483	−	0.998495i	$-0.482532\pi$
$19$	0.478156	0.109697	0.0548483	+	0.998495i	$0.482532\pi$
$20$	0	0
$21$	3.25084	0.709391
$22$	0	0
$23$	−4.93137	−1.02826	−0.514131	−	0.857712i	$-0.671885\pi$
$23$	−4.93137	−1.02826	−0.514131	+	0.857712i	$0.671885\pi$
$24$	0	0
$25$	−4.52581	−0.905162
$26$	0	0
$27$	−4.07288	−0.783826
$28$	0	0
$29$	3.04650	0.565721	0.282860	−	0.959161i	$-0.408717\pi$
$29$	3.04650	0.565721	0.282860	+	0.959161i	$0.408717\pi$
$30$	0	0
$31$	−7.78407	−1.39806	−0.699030	−	0.715092i	$-0.746385\pi$
$31$	−7.78407	−1.39806	−0.699030	+	0.715092i	$0.746385\pi$
$32$	0	0
$33$	−2.25484	−0.392517
$34$	0	0
$35$	1.12621	0.190364
$36$	0	0
$37$	−1.05854	−0.174023	−0.0870113	−	0.996207i	$-0.527732\pi$
$37$	−1.05854	−0.174023	−0.0870113	+	0.996207i	$0.527732\pi$
$38$	0	0
$39$	−9.32397	−1.49303
$40$	0	0
$41$	−10.5669	−1.65027	−0.825134	−	0.564937i	$-0.808901\pi$
$41$	−10.5669	−1.65027	−0.825134	+	0.564937i	$0.808901\pi$
$42$	0	0
$43$	3.06948	0.468092	0.234046	−	0.972226i	$-0.424803\pi$
$43$	3.06948	0.468092	0.234046	+	0.972226i	$0.424803\pi$
$44$	0	0
$45$	0.654845	0.0976185
$46$	0	0
$47$	−11.5743	−1.68829	−0.844145	−	0.536114i	$-0.819892\pi$
$47$	−11.5743	−1.68829	−0.844145	+	0.536114i	$0.819892\pi$
$48$	0	0
$49$	−4.32522	−0.617889
$50$	0	0
$51$	7.28393	1.01995
$52$	0	0
$53$	6.59289	0.905603	0.452801	−	0.891611i	$-0.350425\pi$
$53$	6.59289	0.905603	0.452801	+	0.891611i	$0.350425\pi$
$54$	0	0
$55$	−0.781158	−0.105331
$56$	0	0
$57$	0.950432	0.125888
$58$	0	0
$59$	2.97184	0.386900	0.193450	−	0.981110i	$-0.438032\pi$
$59$	2.97184	0.386900	0.193450	+	0.981110i	$0.438032\pi$
$60$	0	0
$61$	−9.43833	−1.20845	−0.604227	−	0.796812i	$-0.706518\pi$
$61$	−9.43833	−1.20845	−0.604227	+	0.796812i	$0.706518\pi$
$62$	0	0
$63$	1.55527	0.195946
$64$	0	0
$65$	−3.23016	−0.400653
$66$	0	0
$67$	8.15913	0.996796	0.498398	−	0.866948i	$-0.333922\pi$
$67$	8.15913	0.996796	0.498398	+	0.866948i	$0.333922\pi$
$68$	0	0
$69$	−9.80210	−1.18003
$70$	0	0
$71$	−11.2051	−1.32981	−0.664903	−	0.746930i	$-0.731527\pi$
$71$	−11.2051	−1.32981	−0.664903	+	0.746930i	$0.731527\pi$
$72$	0	0
$73$	−2.52633	−0.295685	−0.147842	−	0.989011i	$-0.547233\pi$
$73$	−2.52633	−0.295685	−0.147842	+	0.989011i	$0.547233\pi$
$74$	0	0
$75$	−8.99597	−1.03876
$76$	0	0
$77$	−1.85527	−0.211428
$78$	0	0
$79$	0.930221	0.104658	0.0523290	−	0.998630i	$-0.483336\pi$
$79$	0.930221	0.104658	0.0523290	+	0.998630i	$0.483336\pi$
$80$	0	0
$81$	−10.9486	−1.21651
$82$	0	0
$83$	−1.69982	−0.186579	−0.0932897	−	0.995639i	$-0.529738\pi$
$83$	−1.69982	−0.186579	−0.0932897	+	0.995639i	$0.529738\pi$
$84$	0	0
$85$	2.52342	0.273703
$86$	0	0
$87$	6.05554	0.649222
$88$	0	0
$89$	1.82823	0.193792	0.0968960	−	0.995295i	$-0.469109\pi$
$89$	1.82823	0.193792	0.0968960	+	0.995295i	$0.469109\pi$
$90$	0	0
$91$	−7.67173	−0.804216
$92$	0	0
$93$	−15.4724	−1.60442
$94$	0	0
$95$	0.329264	0.0337818
$96$	0	0
$97$	12.4565	1.26477	0.632385	−	0.774654i	$-0.282076\pi$
$97$	12.4565	1.26477	0.632385	+	0.774654i	$0.282076\pi$
$98$	0	0
$99$	−1.07876	−0.108420
$100$	0	0
$101$	−3.76622	−0.374753	−0.187377	−	0.982288i	$-0.559998\pi$
$101$	−3.76622	−0.374753	−0.187377	+	0.982288i	$0.559998\pi$
$102$	0	0
$103$	17.1753	1.69233	0.846166	−	0.532920i	$-0.178905\pi$
$103$	17.1753	1.69233	0.846166	+	0.532920i	$0.178905\pi$
$104$	0	0
$105$	2.23857	0.218462
$106$	0	0
$107$	6.01484	0.581477	0.290738	−	0.956803i	$-0.406099\pi$
$107$	6.01484	0.581477	0.290738	+	0.956803i	$0.406099\pi$
$108$	0	0
$109$	−2.39418	−0.229320	−0.114660	−	0.993405i	$-0.536578\pi$
$109$	−2.39418	−0.229320	−0.114660	+	0.993405i	$0.536578\pi$
$110$	0	0
$111$	−2.10406	−0.199709
$112$	0	0
$113$	10.4392	0.982042	0.491021	−	0.871148i	$-0.336624\pi$
$113$	10.4392	0.982042	0.491021	+	0.871148i	$0.336624\pi$
$114$	0	0
$115$	−3.39581	−0.316661
$116$	0	0
$117$	−4.46080	−0.412401
$118$	0	0
$119$	5.99319	0.549395
$120$	0	0
$121$	−9.71315	−0.883014
$122$	0	0
$123$	−21.0038	−1.89385
$124$	0	0
$125$	−6.55960	−0.586708
$126$	0	0
$127$	9.11409	0.808745	0.404372	−	0.914594i	$-0.367490\pi$
$127$	9.11409	0.808745	0.404372	+	0.914594i	$0.367490\pi$
$128$	0	0
$129$	6.10122	0.537182
$130$	0	0
$131$	−8.76960	−0.766203	−0.383102	−	0.923706i	$-0.625144\pi$
$131$	−8.76960	−0.766203	−0.383102	+	0.923706i	$0.625144\pi$
$132$	0	0
$133$	0.782012	0.0678091
$134$	0	0
$135$	−2.80464	−0.241385
$136$	0	0
$137$	−10.1477	−0.866976	−0.433488	−	0.901159i	$-0.642717\pi$
$137$	−10.1477	−0.866976	−0.433488	+	0.901159i	$0.642717\pi$
$138$	0	0
$139$	−17.0009	−1.44200	−0.720999	−	0.692936i	$-0.756317\pi$
$139$	−17.0009	−1.44200	−0.720999	+	0.692936i	$0.756317\pi$
$140$	0	0
$141$	−23.0063	−1.93748
$142$	0	0
$143$	5.32124	0.444985
$144$	0	0
$145$	2.09786	0.174218
$146$	0	0
$147$	−8.59725	−0.709089
$148$	0	0
$149$	9.22613	0.755834	0.377917	−	0.925839i	$-0.376640\pi$
$149$	9.22613	0.755834	0.377917	+	0.925839i	$0.376640\pi$
$150$	0	0
$151$	16.7011	1.35912	0.679558	−	0.733622i	$-0.262172\pi$
$151$	16.7011	1.35912	0.679558	+	0.733622i	$0.262172\pi$
$152$	0	0
$153$	3.48479	0.281729
$154$	0	0
$155$	−5.36021	−0.430543
$156$	0	0
$157$	5.27599	0.421070	0.210535	−	0.977586i	$-0.432479\pi$
$157$	5.27599	0.421070	0.210535	+	0.977586i	$0.432479\pi$
$158$	0	0
$159$	13.1047	1.03927
$160$	0	0
$161$	−8.06514	−0.635622
$162$	0	0
$163$	19.8148	1.55202	0.776008	−	0.630722i	$-0.217241\pi$
$163$	19.8148	1.55202	0.776008	+	0.630722i	$0.217241\pi$
$164$	0	0
$165$	−1.55271	−0.120878
$166$	0	0
$167$	4.80319	0.371682	0.185841	−	0.982580i	$-0.440499\pi$
$167$	4.80319	0.371682	0.185841	+	0.982580i	$0.440499\pi$
$168$	0	0
$169$	9.00385	0.692604
$170$	0	0
$171$	0.454708	0.0347724
$172$	0	0
$173$	2.26005	0.171828	0.0859141	−	0.996303i	$-0.472619\pi$
$173$	2.26005	0.171828	0.0859141	+	0.996303i	$0.472619\pi$
$174$	0	0
$175$	−7.40185	−0.559528
$176$	0	0
$177$	5.90713	0.444007
$178$	0	0
$179$	1.15821	0.0865690	0.0432845	−	0.999063i	$-0.486218\pi$
$179$	1.15821	0.0865690	0.0432845	+	0.999063i	$0.486218\pi$
$180$	0	0
$181$	−18.8463	−1.40083	−0.700417	−	0.713734i	$-0.747003\pi$
$181$	−18.8463	−1.40083	−0.700417	+	0.713734i	$0.747003\pi$
$182$	0	0
$183$	−18.7606	−1.38682
$184$	0	0
$185$	−0.728923	−0.0535915
$186$	0	0
$187$	−4.15698	−0.303988
$188$	0	0
$189$	−6.66109	−0.484523
$190$	0	0
$191$	−6.90350	−0.499520	−0.249760	−	0.968308i	$-0.580352\pi$
$191$	−6.90350	−0.499520	−0.249760	+	0.968308i	$0.580352\pi$
$192$	0	0
$193$	7.69003	0.553540	0.276770	−	0.960936i	$-0.410736\pi$
$193$	7.69003	0.553540	0.276770	+	0.960936i	$0.410736\pi$
$194$	0	0
$195$	−6.42061	−0.459789
$196$	0	0
$197$	−15.5059	−1.10475	−0.552373	−	0.833597i	$-0.686278\pi$
$197$	−15.5059	−1.10475	−0.552373	+	0.833597i	$0.686278\pi$
$198$	0	0
$199$	0.786271	0.0557373	0.0278686	−	0.999612i	$-0.491128\pi$
$199$	0.786271	0.0557373	0.0278686	+	0.999612i	$0.491128\pi$
$200$	0	0
$201$	16.2179	1.14392
$202$	0	0
$203$	4.98248	0.349701
$204$	0	0
$205$	−7.27649	−0.508212
$206$	0	0
$207$	−4.68955	−0.325946
$208$	0	0
$209$	−0.542417	−0.0375198
$210$	0	0
$211$	3.63528	0.250263	0.125131	−	0.992140i	$-0.460065\pi$
$211$	3.63528	0.250263	0.125131	+	0.992140i	$0.460065\pi$
$212$	0	0
$213$	−22.2725	−1.52609
$214$	0	0
$215$	2.11369	0.144152
$216$	0	0
$217$	−12.7307	−0.864213
$218$	0	0
$219$	−5.02160	−0.339328
$220$	0	0
$221$	−17.1895	−1.15629
$222$	0	0
$223$	16.5480	1.10814	0.554068	−	0.832471i	$-0.313075\pi$
$223$	16.5480	1.10814	0.554068	+	0.832471i	$0.313075\pi$
$224$	0	0
$225$	−4.30387	−0.286925
$226$	0	0
$227$	2.79658	0.185616	0.0928079	−	0.995684i	$-0.470416\pi$
$227$	2.79658	0.185616	0.0928079	+	0.995684i	$0.470416\pi$
$228$	0	0
$229$	−25.4607	−1.68249	−0.841245	−	0.540655i	$-0.818177\pi$
$229$	−25.4607	−1.68249	−0.841245	+	0.540655i	$0.818177\pi$
$230$	0	0
$231$	−3.68773	−0.242635
$232$	0	0
$233$	−21.8036	−1.42840	−0.714201	−	0.699940i	$-0.753210\pi$
$233$	−21.8036	−1.42840	−0.714201	+	0.699940i	$0.753210\pi$
$234$	0	0
$235$	−7.97024	−0.519921
$236$	0	0
$237$	1.84900	0.120106
$238$	0	0
$239$	−9.61699	−0.622071	−0.311036	−	0.950398i	$-0.600676\pi$
$239$	−9.61699	−0.622071	−0.311036	+	0.950398i	$0.600676\pi$
$240$	0	0
$241$	−16.3250	−1.05158	−0.525792	−	0.850613i	$-0.676231\pi$
$241$	−16.3250	−1.05158	−0.525792	+	0.850613i	$0.676231\pi$
$242$	0	0
$243$	−9.54384	−0.612237
$244$	0	0
$245$	−2.97840	−0.190283
$246$	0	0
$247$	−2.24295	−0.142715
$248$	0	0
$249$	−3.37874	−0.214119
$250$	0	0
$251$	18.7773	1.18521	0.592606	−	0.805493i	$-0.298099\pi$
$251$	18.7773	1.18521	0.592606	+	0.805493i	$0.298099\pi$
$252$	0	0
$253$	5.59412	0.351699
$254$	0	0
$255$	5.01581	0.314102
$256$	0	0
$257$	17.3296	1.08099	0.540495	−	0.841347i	$-0.318237\pi$
$257$	17.3296	1.08099	0.540495	+	0.841347i	$0.318237\pi$
$258$	0	0
$259$	−1.73121	−0.107572
$260$	0	0
$261$	2.89711	0.179326
$262$	0	0
$263$	−12.5842	−0.775974	−0.387987	−	0.921665i	$-0.626830\pi$
$263$	−12.5842	−0.775974	−0.387987	+	0.921665i	$0.626830\pi$
$264$	0	0
$265$	4.53995	0.278887
$266$	0	0
$267$	3.63398	0.222396
$268$	0	0
$269$	−12.8277	−0.782122	−0.391061	−	0.920365i	$-0.627892\pi$
$269$	−12.8277	−0.782122	−0.391061	+	0.920365i	$0.627892\pi$
$270$	0	0
$271$	7.41634	0.450511	0.225255	−	0.974300i	$-0.427678\pi$
$271$	7.41634	0.450511	0.225255	+	0.974300i	$0.427678\pi$
$272$	0	0
$273$	−15.2491	−0.922919
$274$	0	0
$275$	5.13405	0.309595
$276$	0	0
$277$	−17.6508	−1.06053	−0.530266	−	0.847831i	$-0.677908\pi$
$277$	−17.6508	−1.06053	−0.530266	+	0.847831i	$0.677908\pi$
$278$	0	0
$279$	−7.40236	−0.443167
$280$	0	0
$281$	−1.73057	−0.103237	−0.0516187	−	0.998667i	$-0.516438\pi$
$281$	−1.73057	−0.103237	−0.0516187	+	0.998667i	$0.516438\pi$
$282$	0	0
$283$	25.1645	1.49587	0.747936	−	0.663771i	$-0.231045\pi$
$283$	25.1645	1.49587	0.747936	+	0.663771i	$0.231045\pi$
$284$	0	0
$285$	0.654480	0.0387680
$286$	0	0
$287$	−17.2819	−1.02012
$288$	0	0
$289$	−3.57148	−0.210087
$290$	0	0
$291$	24.7599	1.45145
$292$	0	0
$293$	14.5820	0.851890	0.425945	−	0.904749i	$-0.359942\pi$
$293$	14.5820	0.851890	0.425945	+	0.904749i	$0.359942\pi$
$294$	0	0
$295$	2.04645	0.119149
$296$	0	0
$297$	4.62025	0.268094
$298$	0	0
$299$	23.1322	1.33777
$300$	0	0
$301$	5.02006	0.289352
$302$	0	0
$303$	−7.48613	−0.430067
$304$	0	0
$305$	−6.49935	−0.372152
$306$	0	0
$307$	6.19489	0.353561	0.176781	−	0.984250i	$-0.443432\pi$
$307$	6.19489	0.353561	0.176781	+	0.984250i	$0.443432\pi$
$308$	0	0
$309$	34.1394	1.94212
$310$	0	0
$311$	27.2078	1.54281	0.771406	−	0.636344i	$-0.219554\pi$
$311$	27.2078	1.54281	0.771406	+	0.636344i	$0.219554\pi$
$312$	0	0
$313$	25.7176	1.45365	0.726823	−	0.686824i	$-0.240996\pi$
$313$	25.7176	1.45365	0.726823	+	0.686824i	$0.240996\pi$
$314$	0	0
$315$	1.07098	0.0603430
$316$	0	0
$317$	0.151691	0.00851982	0.00425991	−	0.999991i	$-0.498644\pi$
$317$	0.151691	0.00851982	0.00425991	+	0.999991i	$0.498644\pi$
$318$	0	0
$319$	−3.45593	−0.193495
$320$	0	0
$321$	11.9557	0.667303
$322$	0	0
$323$	1.75220	0.0974950
$324$	0	0
$325$	21.2298	1.17762
$326$	0	0
$327$	−4.75891	−0.263168
$328$	0	0
$329$	−18.9295	−1.04362
$330$	0	0
$331$	9.22563	0.507087	0.253543	−	0.967324i	$-0.418404\pi$
$331$	9.22563	0.507087	0.253543	+	0.967324i	$0.418404\pi$
$332$	0	0
$333$	−1.00663	−0.0551630
$334$	0	0
$335$	5.61848	0.306970
$336$	0	0
$337$	−10.3200	−0.562167	−0.281084	−	0.959683i	$-0.590694\pi$
$337$	−10.3200	−0.562167	−0.281084	+	0.959683i	$0.590694\pi$
$338$	0	0
$339$	20.7501	1.12699
$340$	0	0
$341$	8.83020	0.478182
$342$	0	0
$343$	−18.5221	−1.00010
$344$	0	0
$345$	−6.74985	−0.363400
$346$	0	0
$347$	−25.5483	−1.37150	−0.685752	−	0.727836i	$-0.740526\pi$
$347$	−25.5483	−1.37150	−0.685752	+	0.727836i	$0.740526\pi$
$348$	0	0
$349$	−28.3172	−1.51578	−0.757892	−	0.652380i	$-0.773771\pi$
$349$	−28.3172	−1.51578	−0.757892	+	0.652380i	$0.773771\pi$
$350$	0	0
$351$	19.1052	1.01976
$352$	0	0
$353$	−3.73049	−0.198554	−0.0992769	−	0.995060i	$-0.531653\pi$
$353$	−3.73049	−0.198554	−0.0992769	+	0.995060i	$0.531653\pi$
$354$	0	0
$355$	−7.71601	−0.409523
$356$	0	0
$357$	11.9127	0.630486
$358$	0	0
$359$	21.5508	1.13741	0.568704	−	0.822543i	$-0.307445\pi$
$359$	21.5508	1.13741	0.568704	+	0.822543i	$0.307445\pi$
$360$	0	0
$361$	−18.7714	−0.987967
$362$	0	0
$363$	−19.3069	−1.01335
$364$	0	0
$365$	−1.73967	−0.0910582
$366$	0	0
$367$	−11.7230	−0.611934	−0.305967	−	0.952042i	$-0.598980\pi$
$367$	−11.7230	−0.611934	−0.305967	+	0.952042i	$0.598980\pi$
$368$	0	0
$369$	−10.0487	−0.523114
$370$	0	0
$371$	10.7825	0.559800
$372$	0	0
$373$	−11.0782	−0.573608	−0.286804	−	0.957989i	$-0.592593\pi$
$373$	−11.0782	−0.573608	−0.286804	+	0.957989i	$0.592593\pi$
$374$	0	0
$375$	−13.0385	−0.673307
$376$	0	0
$377$	−14.2906	−0.736004
$378$	0	0
$379$	19.5312	1.00325	0.501626	−	0.865085i	$-0.332736\pi$
$379$	19.5312	1.00325	0.501626	+	0.865085i	$0.332736\pi$
$380$	0	0
$381$	18.1161	0.928116
$382$	0	0
$383$	−9.11161	−0.465582	−0.232791	−	0.972527i	$-0.574786\pi$
$383$	−9.11161	−0.465582	−0.232791	+	0.972527i	$0.574786\pi$
$384$	0	0
$385$	−1.27756	−0.0651107
$386$	0	0
$387$	2.91896	0.148379
$388$	0	0
$389$	−27.0768	−1.37285	−0.686424	−	0.727202i	$-0.740820\pi$
$389$	−27.0768	−1.37285	−0.686424	+	0.727202i	$0.740820\pi$
$390$	0	0
$391$	−18.0710	−0.913889
$392$	0	0
$393$	−17.4314	−0.879296
$394$	0	0
$395$	0.640562	0.0322302
$396$	0	0
$397$	23.5084	1.17985	0.589927	−	0.807457i	$-0.299157\pi$
$397$	23.5084	1.17985	0.589927	+	0.807457i	$0.299157\pi$
$398$	0	0
$399$	1.55441	0.0778177
$400$	0	0
$401$	35.8354	1.78954	0.894768	−	0.446531i	$-0.147341\pi$
$401$	35.8354	1.78954	0.894768	+	0.446531i	$0.147341\pi$
$402$	0	0
$403$	36.5137	1.81888
$404$	0	0
$405$	−7.53932	−0.374632
$406$	0	0
$407$	1.20080	0.0595214
$408$	0	0
$409$	29.5592	1.46161	0.730804	−	0.682587i	$-0.239145\pi$
$409$	29.5592	1.46161	0.730804	+	0.682587i	$0.239145\pi$
$410$	0	0
$411$	−20.1706	−0.994943
$412$	0	0
$413$	4.86037	0.239163
$414$	0	0
$415$	−1.17052	−0.0574585
$416$	0	0
$417$	−33.7927	−1.65484
$418$	0	0
$419$	−7.19483	−0.351491	−0.175745	−	0.984436i	$-0.556234\pi$
$419$	−7.19483	−0.351491	−0.175745	+	0.984436i	$0.556234\pi$
$420$	0	0
$421$	−33.0736	−1.61191	−0.805954	−	0.591978i	$-0.798347\pi$
$421$	−33.0736	−1.61191	−0.805954	+	0.591978i	$0.798347\pi$
$422$	0	0
$423$	−11.0068	−0.535167
$424$	0	0
$425$	−16.5848	−0.804482
$426$	0	0
$427$	−15.4361	−0.747008
$428$	0	0
$429$	10.5770	0.510665
$430$	0	0
$431$	19.4849	0.938553	0.469276	−	0.883051i	$-0.344515\pi$
$431$	19.4849	0.938553	0.469276	+	0.883051i	$0.344515\pi$
$432$	0	0
$433$	30.7444	1.47748	0.738740	−	0.673991i	$-0.235421\pi$
$433$	30.7444	1.47748	0.738740	+	0.673991i	$0.235421\pi$
$434$	0	0
$435$	4.16992	0.199932
$436$	0	0
$437$	−2.35797	−0.112797
$438$	0	0
$439$	−19.0872	−0.910983	−0.455491	−	0.890240i	$-0.650536\pi$
$439$	−19.0872	−0.910983	−0.455491	+	0.890240i	$0.650536\pi$
$440$	0	0
$441$	−4.11312	−0.195863
$442$	0	0
$443$	7.77156	0.369238	0.184619	−	0.982810i	$-0.440895\pi$
$443$	7.77156	0.369238	0.184619	+	0.982810i	$0.440895\pi$
$444$	0	0
$445$	1.25894	0.0596796
$446$	0	0
$447$	18.3388	0.867396
$448$	0	0
$449$	29.9281	1.41239	0.706197	−	0.708015i	$-0.250409\pi$
$449$	29.9281	1.41239	0.706197	+	0.708015i	$0.250409\pi$
$450$	0	0
$451$	11.9870	0.564446
$452$	0	0
$453$	33.1968	1.55972
$454$	0	0
$455$	−5.28285	−0.247664
$456$	0	0
$457$	−6.17193	−0.288711	−0.144355	−	0.989526i	$-0.546111\pi$
$457$	−6.17193	−0.288711	−0.144355	+	0.989526i	$0.546111\pi$
$458$	0	0
$459$	−14.9250	−0.696641
$460$	0	0
$461$	−7.54136	−0.351236	−0.175618	−	0.984458i	$-0.556192\pi$
$461$	−7.54136	−0.351236	−0.175618	+	0.984458i	$0.556192\pi$
$462$	0	0
$463$	−27.2246	−1.26523	−0.632616	−	0.774465i	$-0.718019\pi$
$463$	−27.2246	−1.26523	−0.632616	+	0.774465i	$0.718019\pi$
$464$	0	0
$465$	−10.6545	−0.494091
$466$	0	0
$467$	7.15705	0.331189	0.165594	−	0.986194i	$-0.447046\pi$
$467$	7.15705	0.331189	0.165594	+	0.986194i	$0.447046\pi$
$468$	0	0
$469$	13.3441	0.616171
$470$	0	0
$471$	10.4871	0.483220
$472$	0	0
$473$	−3.48200	−0.160103
$474$	0	0
$475$	−2.16404	−0.0992932
$476$	0	0
$477$	6.26958	0.287064
$478$	0	0
$479$	2.80206	0.128029	0.0640147	−	0.997949i	$-0.479610\pi$
$479$	2.80206	0.128029	0.0640147	+	0.997949i	$0.479610\pi$
$480$	0	0
$481$	4.96542	0.226404
$482$	0	0
$483$	−16.0311	−0.729440
$484$	0	0
$485$	8.57774	0.389495
$486$	0	0
$487$	−28.0145	−1.26946	−0.634730	−	0.772734i	$-0.718889\pi$
$487$	−28.0145	−1.26946	−0.634730	+	0.772734i	$0.718889\pi$
$488$	0	0
$489$	39.3860	1.78110
$490$	0	0
$491$	23.9111	1.07909	0.539547	−	0.841955i	$-0.318595\pi$
$491$	23.9111	1.07909	0.539547	+	0.841955i	$0.318595\pi$
$492$	0	0
$493$	11.1639	0.502796
$494$	0	0
$495$	−0.742852	−0.0333887
$496$	0	0
$497$	−18.3257	−0.822022
$498$	0	0
$499$	−25.9335	−1.16094	−0.580472	−	0.814280i	$-0.697132\pi$
$499$	−25.9335	−1.16094	−0.580472	+	0.814280i	$0.697132\pi$
$500$	0	0
$501$	9.54732	0.426543
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−2.59347	−0.115408
$506$	0	0
$507$	17.8970	0.794832
$508$	0	0
$509$	−24.1748	−1.07153	−0.535764	−	0.844368i	$-0.679976\pi$
$509$	−24.1748	−1.07153	−0.535764	+	0.844368i	$0.679976\pi$
$510$	0	0
$511$	−4.13175	−0.182778
$512$	0	0
$513$	−1.94747	−0.0859830
$514$	0	0
$515$	11.8271	0.521166
$516$	0	0
$517$	13.1299	0.577450
$518$	0	0
$519$	4.49230	0.197190
$520$	0	0
$521$	−3.71497	−0.162756	−0.0813779	−	0.996683i	$-0.525932\pi$
$521$	−3.71497	−0.162756	−0.0813779	+	0.996683i	$0.525932\pi$
$522$	0	0
$523$	−2.17027	−0.0948991	−0.0474495	−	0.998874i	$-0.515109\pi$
$523$	−2.17027	−0.0948991	−0.0474495	+	0.998874i	$0.515109\pi$
$524$	0	0
$525$	−14.7127	−0.642114
$526$	0	0
$527$	−28.5247	−1.24256
$528$	0	0
$529$	1.31843	0.0573229
$530$	0	0
$531$	2.82610	0.122642
$532$	0	0
$533$	49.5674	2.14700
$534$	0	0
$535$	4.14190	0.179070
$536$	0	0
$537$	2.30218	0.0993466
$538$	0	0
$539$	4.90650	0.211338
$540$	0	0
$541$	−14.3982	−0.619025	−0.309513	−	0.950895i	$-0.600166\pi$
$541$	−14.3982	−0.619025	−0.309513	+	0.950895i	$0.600166\pi$
$542$	0	0
$543$	−37.4608	−1.60760
$544$	0	0
$545$	−1.64866	−0.0706208
$546$	0	0
$547$	10.3525	0.442643	0.221321	−	0.975201i	$-0.428963\pi$
$547$	10.3525	0.442643	0.221321	+	0.975201i	$0.428963\pi$
$548$	0	0
$549$	−8.97549	−0.383064
$550$	0	0
$551$	1.45670	0.0620576
$552$	0	0
$553$	1.52135	0.0646945
$554$	0	0
$555$	−1.44888	−0.0615017
$556$	0	0
$557$	−28.3766	−1.20235	−0.601177	−	0.799116i	$-0.705301\pi$
$557$	−28.3766	−1.20235	−0.601177	+	0.799116i	$0.705301\pi$
$558$	0	0
$559$	−14.3984	−0.608988
$560$	0	0
$561$	−8.26284	−0.348857
$562$	0	0
$563$	33.5183	1.41263	0.706313	−	0.707900i	$-0.250357\pi$
$563$	33.5183	1.41263	0.706313	+	0.707900i	$0.250357\pi$
$564$	0	0
$565$	7.18860	0.302427
$566$	0	0
$567$	−17.9061	−0.751985
$568$	0	0
$569$	−31.3435	−1.31399	−0.656994	−	0.753896i	$-0.728172\pi$
$569$	−31.3435	−1.31399	−0.656994	+	0.753896i	$0.728172\pi$
$570$	0	0
$571$	−32.7332	−1.36984	−0.684921	−	0.728617i	$-0.740163\pi$
$571$	−32.7332	−1.36984	−0.684921	+	0.728617i	$0.740163\pi$
$572$	0	0
$573$	−13.7221	−0.573249
$574$	0	0
$575$	22.3185	0.930744
$576$	0	0
$577$	34.9784	1.45617	0.728085	−	0.685487i	$-0.240411\pi$
$577$	34.9784	1.45617	0.728085	+	0.685487i	$0.240411\pi$
$578$	0	0
$579$	15.2855	0.635243
$580$	0	0
$581$	−2.78001	−0.115334
$582$	0	0
$583$	−7.47893	−0.309746
$584$	0	0
$585$	−3.07176	−0.127002
$586$	0	0
$587$	−27.2112	−1.12313	−0.561563	−	0.827434i	$-0.689800\pi$
$587$	−27.2112	−1.12313	−0.561563	+	0.827434i	$0.689800\pi$
$588$	0	0
$589$	−3.72200	−0.153362
$590$	0	0
$591$	−30.8210	−1.26781
$592$	0	0
$593$	−8.86207	−0.363921	−0.181961	−	0.983306i	$-0.558244\pi$
$593$	−8.86207	−0.363921	−0.181961	+	0.983306i	$0.558244\pi$
$594$	0	0
$595$	4.12699	0.169190
$596$	0	0
$597$	1.56287	0.0639641
$598$	0	0
$599$	33.0116	1.34882	0.674409	−	0.738358i	$-0.264398\pi$
$599$	33.0116	1.34882	0.674409	+	0.738358i	$0.264398\pi$
$600$	0	0
$601$	−27.2290	−1.11069	−0.555347	−	0.831619i	$-0.687414\pi$
$601$	−27.2290	−1.11069	−0.555347	+	0.831619i	$0.687414\pi$
$602$	0	0
$603$	7.75902	0.315972
$604$	0	0
$605$	−6.68860	−0.271930
$606$	0	0
$607$	18.8180	0.763798	0.381899	−	0.924204i	$-0.375270\pi$
$607$	18.8180	0.763798	0.381899	+	0.924204i	$0.375270\pi$
$608$	0	0
$609$	9.90368	0.401317
$610$	0	0
$611$	54.2932	2.19647
$612$	0	0
$613$	9.12383	0.368508	0.184254	−	0.982879i	$-0.441013\pi$
$613$	9.12383	0.368508	0.184254	+	0.982879i	$0.441013\pi$
$614$	0	0
$615$	−14.4635	−0.583224
$616$	0	0
$617$	30.4208	1.22469	0.612347	−	0.790589i	$-0.290226\pi$
$617$	30.4208	1.22469	0.612347	+	0.790589i	$0.290226\pi$
$618$	0	0
$619$	−8.07959	−0.324746	−0.162373	−	0.986729i	$-0.551915\pi$
$619$	−8.07959	−0.324746	−0.162373	+	0.986729i	$0.551915\pi$
$620$	0	0
$621$	20.0849	0.805978
$622$	0	0
$623$	2.99003	0.119793
$624$	0	0
$625$	18.1120	0.724481
$626$	0	0
$627$	−1.07816	−0.0430577
$628$	0	0
$629$	−3.87901	−0.154666
$630$	0	0
$631$	14.3458	0.571097	0.285549	−	0.958364i	$-0.407824\pi$
$631$	14.3458	0.571097	0.285549	+	0.958364i	$0.407824\pi$
$632$	0	0
$633$	7.22585	0.287202
$634$	0	0
$635$	6.27608	0.249059
$636$	0	0
$637$	20.2889	0.803874
$638$	0	0
$639$	−10.6557	−0.421532
$640$	0	0
$641$	−29.8780	−1.18011	−0.590055	−	0.807363i	$-0.700894\pi$
$641$	−29.8780	−1.18011	−0.590055	+	0.807363i	$0.700894\pi$
$642$	0	0
$643$	−35.0867	−1.38369	−0.691843	−	0.722048i	$-0.743201\pi$
$643$	−35.0867	−1.38369	−0.691843	+	0.722048i	$0.743201\pi$
$644$	0	0
$645$	4.20138	0.165429
$646$	0	0
$647$	16.1457	0.634751	0.317376	−	0.948300i	$-0.397198\pi$
$647$	16.1457	0.634751	0.317376	+	0.948300i	$0.397198\pi$
$648$	0	0
$649$	−3.37123	−0.132333
$650$	0	0
$651$	−25.3048	−0.991772
$652$	0	0
$653$	3.36679	0.131753	0.0658764	−	0.997828i	$-0.479016\pi$
$653$	3.36679	0.131753	0.0658764	+	0.997828i	$0.479016\pi$
$654$	0	0
$655$	−6.03886	−0.235958
$656$	0	0
$657$	−2.40244	−0.0937283
$658$	0	0
$659$	2.72422	0.106120	0.0530602	−	0.998591i	$-0.483102\pi$
$659$	2.72422	0.106120	0.0530602	+	0.998591i	$0.483102\pi$
$660$	0	0
$661$	−31.3598	−1.21975	−0.609877	−	0.792496i	$-0.708781\pi$
$661$	−31.3598	−1.21975	−0.609877	+	0.792496i	$0.708781\pi$
$662$	0	0
$663$	−34.1676	−1.32696
$664$	0	0
$665$	0.538504	0.0208823
$666$	0	0
$667$	−15.0234	−0.581709
$668$	0	0
$669$	32.8925	1.27170
$670$	0	0
$671$	10.7068	0.413331
$672$	0	0
$673$	−38.8250	−1.49659	−0.748296	−	0.663365i	$-0.769128\pi$
$673$	−38.8250	−1.49659	−0.748296	+	0.663365i	$0.769128\pi$
$674$	0	0
$675$	18.4331	0.709490
$676$	0	0
$677$	43.3006	1.66418	0.832089	−	0.554642i	$-0.187145\pi$
$677$	43.3006	1.66418	0.832089	+	0.554642i	$0.187145\pi$
$678$	0	0
$679$	20.3724	0.781819
$680$	0	0
$681$	5.55878	0.213013
$682$	0	0
$683$	−37.1515	−1.42156	−0.710782	−	0.703412i	$-0.751659\pi$
$683$	−37.1515	−1.42156	−0.710782	+	0.703412i	$0.751659\pi$
$684$	0	0
$685$	−6.98784	−0.266992
$686$	0	0
$687$	−50.6083	−1.93083
$688$	0	0
$689$	−30.9261	−1.17819
$690$	0	0
$691$	−47.0789	−1.79097	−0.895483	−	0.445096i	$-0.853169\pi$
$691$	−47.0789	−1.79097	−0.895483	+	0.445096i	$0.853169\pi$
$692$	0	0
$693$	−1.76429	−0.0670200
$694$	0	0
$695$	−11.7070	−0.444073
$696$	0	0
$697$	−38.7223	−1.46671
$698$	0	0
$699$	−43.3391	−1.63924
$700$	0	0
$701$	−18.2836	−0.690564	−0.345282	−	0.938499i	$-0.612217\pi$
$701$	−18.2836	−0.690564	−0.345282	+	0.938499i	$0.612217\pi$
$702$	0	0
$703$	−0.506147	−0.0190897
$704$	0	0
$705$	−15.8425	−0.596662
$706$	0	0
$707$	−6.15956	−0.231654
$708$	0	0
$709$	4.10860	0.154302	0.0771508	−	0.997019i	$-0.475418\pi$
$709$	4.10860	0.154302	0.0771508	+	0.997019i	$0.475418\pi$
$710$	0	0
$711$	0.884604	0.0331752
$712$	0	0
$713$	38.3862	1.43757
$714$	0	0
$715$	3.66428	0.137036
$716$	0	0
$717$	−19.1157	−0.713889
$718$	0	0
$719$	4.33485	0.161663	0.0808314	−	0.996728i	$-0.474242\pi$
$719$	4.33485	0.161663	0.0808314	+	0.996728i	$0.474242\pi$
$720$	0	0
$721$	28.0898	1.04612
$722$	0	0
$723$	−32.4492	−1.20680
$724$	0	0
$725$	−13.7879	−0.512069
$726$	0	0
$727$	21.4778	0.796568	0.398284	−	0.917262i	$-0.369606\pi$
$727$	21.4778	0.796568	0.398284	+	0.917262i	$0.369606\pi$
$728$	0	0
$729$	13.8754	0.513902
$730$	0	0
$731$	11.2481	0.416026
$732$	0	0
$733$	−18.2602	−0.674457	−0.337228	−	0.941423i	$-0.609489\pi$
$733$	−18.2602	−0.674457	−0.337228	+	0.941423i	$0.609489\pi$
$734$	0	0
$735$	−5.92018	−0.218369
$736$	0	0
$737$	−9.25566	−0.340937
$738$	0	0
$739$	−3.24461	−0.119355	−0.0596775	−	0.998218i	$-0.519007\pi$
$739$	−3.24461	−0.119355	−0.0596775	+	0.998218i	$0.519007\pi$
$740$	0	0
$741$	−4.45831	−0.163780
$742$	0	0
$743$	−20.5402	−0.753547	−0.376773	−	0.926305i	$-0.622966\pi$
$743$	−20.5402	−0.753547	−0.376773	+	0.926305i	$0.622966\pi$
$744$	0	0
$745$	6.35323	0.232765
$746$	0	0
$747$	−1.61646	−0.0591433
$748$	0	0
$749$	9.83713	0.359441
$750$	0	0
$751$	−28.0632	−1.02404	−0.512021	−	0.858973i	$-0.671103\pi$
$751$	−28.0632	−1.02404	−0.512021	+	0.858973i	$0.671103\pi$
$752$	0	0
$753$	37.3237	1.36015
$754$	0	0
$755$	11.5006	0.418549
$756$	0	0
$757$	−33.1412	−1.20454	−0.602268	−	0.798294i	$-0.705736\pi$
$757$	−33.1412	−1.20454	−0.602268	+	0.798294i	$0.705736\pi$
$758$	0	0
$759$	11.1194	0.403610
$760$	0	0
$761$	2.25043	0.0815781	0.0407890	−	0.999168i	$-0.487013\pi$
$761$	2.25043	0.0815781	0.0407890	+	0.999168i	$0.487013\pi$
$762$	0	0
$763$	−3.91561	−0.141755
$764$	0	0
$765$	2.39967	0.0867604
$766$	0	0
$767$	−13.9404	−0.503358
$768$	0	0
$769$	49.3766	1.78056	0.890282	−	0.455409i	$-0.150507\pi$
$769$	49.3766	1.78056	0.890282	+	0.455409i	$0.150507\pi$
$770$	0	0
$771$	34.4461	1.24055
$772$	0	0
$773$	8.11729	0.291959	0.145979	−	0.989288i	$-0.453367\pi$
$773$	8.11729	0.291959	0.145979	+	0.989288i	$0.453367\pi$
$774$	0	0
$775$	35.2293	1.26547
$776$	0	0
$777$	−3.44114	−0.123450
$778$	0	0
$779$	−5.05262	−0.181029
$780$	0	0
$781$	12.7110	0.454837
$782$	0	0
$783$	−12.4080	−0.443427
$784$	0	0
$785$	3.63312	0.129672
$786$	0	0
$787$	−23.6961	−0.844676	−0.422338	−	0.906438i	$-0.638790\pi$
$787$	−23.6961	−0.844676	−0.422338	+	0.906438i	$0.638790\pi$
$788$	0	0
$789$	−25.0136	−0.890509
$790$	0	0
$791$	17.0731	0.607051
$792$	0	0
$793$	44.2735	1.57220
$794$	0	0
$795$	9.02406	0.320051
$796$	0	0
$797$	−14.1640	−0.501714	−0.250857	−	0.968024i	$-0.580712\pi$
$797$	−14.1640	−0.501714	−0.250857	+	0.968024i	$0.580712\pi$
$798$	0	0
$799$	−42.4141	−1.50050
$800$	0	0
$801$	1.73858	0.0614296
$802$	0	0
$803$	2.86586	0.101134
$804$	0	0
$805$	−5.55376	−0.195744
$806$	0	0
$807$	−25.4977	−0.897563
$808$	0	0
$809$	52.5347	1.84702	0.923510	−	0.383574i	$-0.125307\pi$
$809$	52.5347	1.84702	0.923510	+	0.383574i	$0.125307\pi$
$810$	0	0
$811$	38.8247	1.36332	0.681660	−	0.731669i	$-0.261259\pi$
$811$	38.8247	1.36332	0.681660	+	0.731669i	$0.261259\pi$
$812$	0	0
$813$	14.7415	0.517007
$814$	0	0
$815$	13.6447	0.477955
$816$	0	0
$817$	1.46769	0.0513480
$818$	0	0
$819$	−7.29552	−0.254926
$820$	0	0
$821$	20.5192	0.716127	0.358063	−	0.933697i	$-0.383437\pi$
$821$	20.5192	0.716127	0.358063	+	0.933697i	$0.383437\pi$
$822$	0	0
$823$	−50.2425	−1.75134	−0.875672	−	0.482907i	$-0.839581\pi$
$823$	−50.2425	−1.75134	−0.875672	+	0.482907i	$0.839581\pi$
$824$	0	0
$825$	10.2050	0.355291
$826$	0	0
$827$	−2.50670	−0.0871664	−0.0435832	−	0.999050i	$-0.513877\pi$
$827$	−2.50670	−0.0871664	−0.0435832	+	0.999050i	$0.513877\pi$
$828$	0	0
$829$	48.4355	1.68223	0.841117	−	0.540852i	$-0.181898\pi$
$829$	48.4355	1.68223	0.841117	+	0.540852i	$0.181898\pi$
$830$	0	0
$831$	−35.0845	−1.21707
$832$	0	0
$833$	−15.8498	−0.549161
$834$	0	0
$835$	3.30754	0.114462
$836$	0	0
$837$	31.7036	1.09584
$838$	0	0
$839$	−20.8801	−0.720861	−0.360430	−	0.932786i	$-0.617370\pi$
$839$	−20.8801	−0.720861	−0.360430	+	0.932786i	$0.617370\pi$
$840$	0	0
$841$	−19.7188	−0.679960
$842$	0	0
$843$	−3.43986	−0.118475
$844$	0	0
$845$	6.20017	0.213292
$846$	0	0
$847$	−15.8856	−0.545836
$848$	0	0
$849$	50.0194	1.71666
$850$	0	0
$851$	5.22005	0.178941
$852$	0	0
$853$	−22.0276	−0.754211	−0.377105	−	0.926170i	$-0.623081\pi$
$853$	−22.0276	−0.754211	−0.377105	+	0.926170i	$0.623081\pi$
$854$	0	0
$855$	0.313118	0.0107084
$856$	0	0
$857$	28.4292	0.971124	0.485562	−	0.874202i	$-0.338615\pi$
$857$	28.4292	0.971124	0.485562	+	0.874202i	$0.338615\pi$
$858$	0	0
$859$	36.9251	1.25987	0.629935	−	0.776648i	$-0.283082\pi$
$859$	36.9251	1.25987	0.629935	+	0.776648i	$0.283082\pi$
$860$	0	0
$861$	−34.3512	−1.17069
$862$	0	0
$863$	49.4592	1.68361	0.841805	−	0.539782i	$-0.181493\pi$
$863$	49.4592	1.68361	0.841805	+	0.539782i	$0.181493\pi$
$864$	0	0
$865$	1.55630	0.0529157
$866$	0	0
$867$	−7.09903	−0.241096
$868$	0	0
$869$	−1.05524	−0.0357964
$870$	0	0
$871$	−38.2731	−1.29683
$872$	0	0
$873$	11.8457	0.400916
$874$	0	0
$875$	−10.7281	−0.362675
$876$	0	0
$877$	−20.1086	−0.679019	−0.339510	−	0.940603i	$-0.610261\pi$
$877$	−20.1086	−0.679019	−0.339510	+	0.940603i	$0.610261\pi$
$878$	0	0
$879$	28.9847	0.977630
$880$	0	0
$881$	41.6742	1.40404	0.702020	−	0.712157i	$-0.252282\pi$
$881$	41.6742	1.40404	0.702020	+	0.712157i	$0.252282\pi$
$882$	0	0
$883$	−21.4623	−0.722265	−0.361132	−	0.932515i	$-0.617610\pi$
$883$	−21.4623	−0.722265	−0.361132	+	0.932515i	$0.617610\pi$
$884$	0	0
$885$	4.06773	0.136735
$886$	0	0
$887$	−51.6761	−1.73512	−0.867558	−	0.497337i	$-0.834311\pi$
$887$	−51.6761	−1.73512	−0.867558	+	0.497337i	$0.834311\pi$
$888$	0	0
$889$	14.9059	0.499927
$890$	0	0
$891$	12.4200	0.416085
$892$	0	0
$893$	−5.53434	−0.185200
$894$	0	0
$895$	0.797561	0.0266595
$896$	0	0
$897$	45.9799	1.53523
$898$	0	0
$899$	−23.7142	−0.790912
$900$	0	0
$901$	24.1596	0.804873
$902$	0	0
$903$	9.97839	0.332060
$904$	0	0
$905$	−12.9778	−0.431397
$906$	0	0
$907$	32.5649	1.08130	0.540650	−	0.841248i	$-0.318178\pi$
$907$	32.5649	1.08130	0.540650	+	0.841248i	$0.318178\pi$
$908$	0	0
$909$	−3.58153	−0.118792
$910$	0	0
$911$	−38.1120	−1.26271	−0.631354	−	0.775495i	$-0.717500\pi$
$911$	−38.1120	−1.26271	−0.631354	+	0.775495i	$0.717500\pi$
$912$	0	0
$913$	1.92826	0.0638163
$914$	0	0
$915$	−12.9188	−0.427082
$916$	0	0
$917$	−14.3425	−0.473630
$918$	0	0
$919$	−30.5828	−1.00883	−0.504417	−	0.863460i	$-0.668293\pi$
$919$	−30.5828	−1.00883	−0.504417	+	0.863460i	$0.668293\pi$
$920$	0	0
$921$	12.3136	0.405747
$922$	0	0
$923$	52.5614	1.73008
$924$	0	0
$925$	4.79075	0.157519
$926$	0	0
$927$	16.3330	0.536448
$928$	0	0
$929$	2.30989	0.0757851	0.0378926	−	0.999282i	$-0.487936\pi$
$929$	2.30989	0.0757851	0.0378926	+	0.999282i	$0.487936\pi$
$930$	0	0
$931$	−2.06813	−0.0677802
$932$	0	0
$933$	54.0810	1.77053
$934$	0	0
$935$	−2.86255	−0.0936154
$936$	0	0
$937$	−22.8337	−0.745945	−0.372973	−	0.927842i	$-0.621662\pi$
$937$	−22.8337	−0.745945	−0.372973	+	0.927842i	$0.621662\pi$
$938$	0	0
$939$	51.1190	1.66821
$940$	0	0
$941$	34.0397	1.10966	0.554831	−	0.831963i	$-0.312783\pi$
$941$	34.0397	1.10966	0.554831	+	0.831963i	$0.312783\pi$
$942$	0	0
$943$	52.1092	1.69691
$944$	0	0
$945$	−4.58691	−0.149212
$946$	0	0
$947$	19.1614	0.622661	0.311331	−	0.950302i	$-0.399225\pi$
$947$	19.1614	0.622661	0.311331	+	0.950302i	$0.399225\pi$
$948$	0	0
$949$	11.8506	0.384686
$950$	0	0
$951$	0.301517	0.00977736
$952$	0	0
$953$	19.3222	0.625908	0.312954	−	0.949768i	$-0.398681\pi$
$953$	19.3222	0.625908	0.312954	+	0.949768i	$0.398681\pi$
$954$	0	0
$955$	−4.75384	−0.153831
$956$	0	0
$957$	−6.86936	−0.222055
$958$	0	0
$959$	−16.5963	−0.535923
$960$	0	0
$961$	29.5918	0.954574
$962$	0	0
$963$	5.71988	0.184321
$964$	0	0
$965$	5.29545	0.170467
$966$	0	0
$967$	59.2613	1.90572	0.952858	−	0.303416i	$-0.0981272\pi$
$967$	59.2613	1.90572	0.952858	+	0.303416i	$0.0981272\pi$
$968$	0	0
$969$	3.48285	0.111885
$970$	0	0
$971$	−39.7605	−1.27598	−0.637988	−	0.770047i	$-0.720233\pi$
$971$	−39.7605	−1.27598	−0.637988	+	0.770047i	$0.720233\pi$
$972$	0	0
$973$	−27.8046	−0.891373
$974$	0	0
$975$	42.1985	1.35143
$976$	0	0
$977$	23.7820	0.760855	0.380427	−	0.924811i	$-0.375777\pi$
$977$	23.7820	0.760855	0.380427	+	0.924811i	$0.375777\pi$
$978$	0	0
$979$	−2.07393	−0.0662832
$980$	0	0
$981$	−2.27677	−0.0726916
$982$	0	0
$983$	28.7374	0.916581	0.458290	−	0.888802i	$-0.348462\pi$
$983$	28.7374	0.916581	0.458290	+	0.888802i	$0.348462\pi$
$984$	0	0
$985$	−10.6775	−0.340214
$986$	0	0
$987$	−37.6263	−1.19766
$988$	0	0
$989$	−15.1368	−0.481321
$990$	0	0
$991$	−24.1020	−0.765626	−0.382813	−	0.923826i	$-0.625045\pi$
$991$	−24.1020	−0.765626	−0.382813	+	0.923826i	$0.625045\pi$
$992$	0	0
$993$	18.3378	0.581933
$994$	0	0
$995$	0.541436	0.0171647
$996$	0	0
$997$	−15.6418	−0.495381	−0.247690	−	0.968839i	$-0.579672\pi$
$997$	−15.6418	−0.495381	−0.247690	+	0.968839i	$0.579672\pi$
$998$	0	0
$999$	4.31130	0.136403

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.v.1.23		28
4.3	odd	2		4024.2.a.d.1.6	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.d.1.6	✓	28	4.3	odd	2
8048.2.a.v.1.23		28	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:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q+1.98770 q^{3} +0.688613 q^{5} +1.63548 q^{7} +0.950962 q^{9} +O(q^{10})\)	\(q+1.98770 q^{3} +0.688613 q^{5} +1.63548 q^{7} +0.950962 q^{9} -1.13439 q^{11} -4.69083 q^{13} +1.36876 q^{15} +3.66450 q^{17} +0.478156 q^{19} +3.25084 q^{21} -4.93137 q^{23} -4.52581 q^{25} -4.07288 q^{27} +3.04650 q^{29} -7.78407 q^{31} -2.25484 q^{33} +1.12621 q^{35} -1.05854 q^{37} -9.32397 q^{39} -10.5669 q^{41} +3.06948 q^{43} +0.654845 q^{45} -11.5743 q^{47} -4.32522 q^{49} +7.28393 q^{51} +6.59289 q^{53} -0.781158 q^{55} +0.950432 q^{57} +2.97184 q^{59} -9.43833 q^{61} +1.55527 q^{63} -3.23016 q^{65} +8.15913 q^{67} -9.80210 q^{69} -11.2051 q^{71} -2.52633 q^{73} -8.99597 q^{75} -1.85527 q^{77} +0.930221 q^{79} -10.9486 q^{81} -1.69982 q^{83} +2.52342 q^{85} +6.05554 q^{87} +1.82823 q^{89} -7.67173 q^{91} -15.4724 q^{93} +0.329264 q^{95} +12.4565 q^{97} -1.07876 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * q^99

Introduction

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