LMFDB - Embedded newform 8048.2.a.r.1.10

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:	$0$
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} - 4 x^{11} - 22 x^{10} + 67 x^{9} + 180 x^{8} - 383 x^{7} - 674 x^{6} + 875 x^{5} + 1077 x^{4} + \cdots + 2 $ x^12 - 4x^11 - 22x^10 + 67x^9 + 180x^8 - 383x^7 - 674x^6 + 875x^5 + 1077x^4 - 704x^3 - 467x^2 + 210*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1006)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-2.01674$ of defining polynomial
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)

$f(q)$	$=$	$q+3.12415 q^{3} -0.847598 q^{5} +2.65658 q^{7} +6.76030 q^{9} +O(q^{10})$	$q+3.12415 q^{3} -0.847598 q^{5} +2.65658 q^{7} +6.76030 q^{9} +4.27748 q^{11} -1.76366 q^{13} -2.64802 q^{15} -3.42856 q^{17} +3.93089 q^{19} +8.29954 q^{21} -2.31220 q^{23} -4.28158 q^{25} +11.7477 q^{27} -3.19851 q^{29} +5.56438 q^{31} +13.3635 q^{33} -2.25171 q^{35} -4.86688 q^{37} -5.50993 q^{39} +9.50205 q^{41} +12.5008 q^{43} -5.73002 q^{45} -2.79204 q^{47} +0.0573998 q^{49} -10.7113 q^{51} +5.30659 q^{53} -3.62558 q^{55} +12.2807 q^{57} +0.616748 q^{59} +12.6073 q^{61} +17.9593 q^{63} +1.49487 q^{65} -4.58361 q^{67} -7.22365 q^{69} -11.6262 q^{71} +5.49806 q^{73} -13.3763 q^{75} +11.3635 q^{77} -8.21866 q^{79} +16.4207 q^{81} +8.46967 q^{83} +2.90604 q^{85} -9.99261 q^{87} +10.9799 q^{89} -4.68529 q^{91} +17.3840 q^{93} -3.33181 q^{95} -10.8315 q^{97} +28.9170 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 3 q^{3} + 7 q^{5} + 2 q^{7} + 27 q^{9}+O(q^{10}) $ 12 * q + 3 * q^3 + 7 * q^5 + 2 * q^7 + 27 * q^9	$ 12 q + 3 q^{3} + 7 q^{5} + 2 q^{7} + 27 q^{9} - 18 q^{11} - 4 q^{13} + 2 q^{15} + 12 q^{17} + 7 q^{21} + 9 q^{23} + 25 q^{25} + 18 q^{27} + 34 q^{29} + 11 q^{31} + 4 q^{33} - 21 q^{35} - 22 q^{37} - 13 q^{39} + 32 q^{41} + 8 q^{43} + 13 q^{45} - 24 q^{47} + 36 q^{49} - 16 q^{51} - 2 q^{53} + 12 q^{55} + 26 q^{57} - 26 q^{59} + 12 q^{61} - 5 q^{63} + 66 q^{65} + 21 q^{67} + 20 q^{69} - 50 q^{71} + 17 q^{73} + 14 q^{75} + 25 q^{77} + 9 q^{79} + 48 q^{81} - 25 q^{83} + 24 q^{85} + 10 q^{87} + 21 q^{89} + 9 q^{91} + 31 q^{93} - 22 q^{95} + 18 q^{97} - 102 q^{99}+O(q^{100}) $ 12 * q + 3 * q^3 + 7 * q^5 + 2 * q^7 + 27 * q^9 - 18 * q^11 - 4 * q^13 + 2 * q^15 + 12 * q^17 + 7 * q^21 + 9 * q^23 + 25 * q^25 + 18 * q^27 + 34 * q^29 + 11 * q^31 + 4 * q^33 - 21 * q^35 - 22 * q^37 - 13 * q^39 + 32 * q^41 + 8 * q^43 + 13 * q^45 - 24 * q^47 + 36 * q^49 - 16 * q^51 - 2 * q^53 + 12 * q^55 + 26 * q^57 - 26 * q^59 + 12 * q^61 - 5 * q^63 + 66 * q^65 + 21 * q^67 + 20 * q^69 - 50 * q^71 + 17 * q^73 + 14 * q^75 + 25 * q^77 + 9 * q^79 + 48 * q^81 - 25 * q^83 + 24 * q^85 + 10 * q^87 + 21 * q^89 + 9 * q^91 + 31 * q^93 - 22 * q^95 + 18 * q^97 - 102 * 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
$3$	3.12415	1.80373	0.901864	−	0.432021i	$-0.142199\pi$
$3$	3.12415	1.80373	0.901864	+	0.432021i	$0.142199\pi$
$4$	0	0
$5$	−0.847598	−0.379057	−0.189529	−	0.981875i	$-0.560696\pi$
$5$	−0.847598	−0.379057	−0.189529	+	0.981875i	$0.560696\pi$
$6$	0	0
$7$	2.65658	1.00409	0.502046	−	0.864841i	$-0.332581\pi$
$7$	2.65658	1.00409	0.502046	+	0.864841i	$0.332581\pi$
$8$	0	0
$9$	6.76030	2.25343
$10$	0	0
$11$	4.27748	1.28971	0.644854	−	0.764306i	$-0.276918\pi$
$11$	4.27748	1.28971	0.644854	+	0.764306i	$0.276918\pi$
$12$	0	0
$13$	−1.76366	−0.489151	−0.244575	−	0.969630i	$-0.578649\pi$
$13$	−1.76366	−0.489151	−0.244575	+	0.969630i	$0.578649\pi$
$14$	0	0
$15$	−2.64802	−0.683716
$16$	0	0
$17$	−3.42856	−0.831547	−0.415774	−	0.909468i	$-0.636489\pi$
$17$	−3.42856	−0.831547	−0.415774	+	0.909468i	$0.636489\pi$
$18$	0	0
$19$	3.93089	0.901807	0.450904	−	0.892573i	$-0.351102\pi$
$19$	3.93089	0.901807	0.450904	+	0.892573i	$0.351102\pi$
$20$	0	0
$21$	8.29954	1.81111
$22$	0	0
$23$	−2.31220	−0.482127	−0.241063	−	0.970509i	$-0.577496\pi$
$23$	−2.31220	−0.482127	−0.241063	+	0.970509i	$0.577496\pi$
$24$	0	0
$25$	−4.28158	−0.856315
$26$	0	0
$27$	11.7477	2.26085
$28$	0	0
$29$	−3.19851	−0.593948	−0.296974	−	0.954886i	$-0.595977\pi$
$29$	−3.19851	−0.593948	−0.296974	+	0.954886i	$0.595977\pi$
$30$	0	0
$31$	5.56438	0.999393	0.499696	−	0.866201i	$-0.333445\pi$
$31$	5.56438	0.999393	0.499696	+	0.866201i	$0.333445\pi$
$32$	0	0
$33$	13.3635	2.32628
$34$	0	0
$35$	−2.25171	−0.380608
$36$	0	0
$37$	−4.86688	−0.800111	−0.400055	−	0.916491i	$-0.631009\pi$
$37$	−4.86688	−0.800111	−0.400055	+	0.916491i	$0.631009\pi$
$38$	0	0
$39$	−5.50993	−0.882294
$40$	0	0
$41$	9.50205	1.48397	0.741985	−	0.670416i	$-0.233884\pi$
$41$	9.50205	1.48397	0.741985	+	0.670416i	$0.233884\pi$
$42$	0	0
$43$	12.5008	1.90636	0.953180	−	0.302404i	$-0.0977893\pi$
$43$	12.5008	1.90636	0.953180	+	0.302404i	$0.0977893\pi$
$44$	0	0
$45$	−5.73002	−0.854180
$46$	0	0
$47$	−2.79204	−0.407260	−0.203630	−	0.979048i	$-0.565274\pi$
$47$	−2.79204	−0.407260	−0.203630	+	0.979048i	$0.565274\pi$
$48$	0	0
$49$	0.0573998	0.00819998
$50$	0	0
$51$	−10.7113	−1.49988
$52$	0	0
$53$	5.30659	0.728916	0.364458	−	0.931220i	$-0.381254\pi$
$53$	5.30659	0.728916	0.364458	+	0.931220i	$0.381254\pi$
$54$	0	0
$55$	−3.62558	−0.488874
$56$	0	0
$57$	12.2807	1.62661
$58$	0	0
$59$	0.616748	0.0802938	0.0401469	−	0.999194i	$-0.487217\pi$
$59$	0.616748	0.0802938	0.0401469	+	0.999194i	$0.487217\pi$
$60$	0	0
$61$	12.6073	1.61420	0.807102	−	0.590412i	$-0.201035\pi$
$61$	12.6073	1.61420	0.807102	+	0.590412i	$0.201035\pi$
$62$	0	0
$63$	17.9593	2.26265
$64$	0	0
$65$	1.49487	0.185416
$66$	0	0
$67$	−4.58361	−0.559977	−0.279989	−	0.960003i	$-0.590331\pi$
$67$	−4.58361	−0.559977	−0.279989	+	0.960003i	$0.590331\pi$
$68$	0	0
$69$	−7.22365	−0.869625
$70$	0	0
$71$	−11.6262	−1.37978	−0.689891	−	0.723914i	$-0.742341\pi$
$71$	−11.6262	−1.37978	−0.689891	+	0.723914i	$0.742341\pi$
$72$	0	0
$73$	5.49806	0.643500	0.321750	−	0.946825i	$-0.395729\pi$
$73$	5.49806	0.643500	0.321750	+	0.946825i	$0.395729\pi$
$74$	0	0
$75$	−13.3763	−1.54456
$76$	0	0
$77$	11.3635	1.29499
$78$	0	0
$79$	−8.21866	−0.924671	−0.462336	−	0.886705i	$-0.652988\pi$
$79$	−8.21866	−0.924671	−0.462336	+	0.886705i	$0.652988\pi$
$80$	0	0
$81$	16.4207	1.82453
$82$	0	0
$83$	8.46967	0.929667	0.464834	−	0.885398i	$-0.346114\pi$
$83$	8.46967	0.929667	0.464834	+	0.885398i	$0.346114\pi$
$84$	0	0
$85$	2.90604	0.315204
$86$	0	0
$87$	−9.99261	−1.07132
$88$	0	0
$89$	10.9799	1.16387	0.581933	−	0.813237i	$-0.302297\pi$
$89$	10.9799	1.16387	0.581933	+	0.813237i	$0.302297\pi$
$90$	0	0
$91$	−4.68529	−0.491152
$92$	0	0
$93$	17.3840	1.80263
$94$	0	0
$95$	−3.33181	−0.341837
$96$	0	0
$97$	−10.8315	−1.09977	−0.549884	−	0.835241i	$-0.685328\pi$
$97$	−10.8315	−1.09977	−0.549884	+	0.835241i	$0.685328\pi$
$98$	0	0
$99$	28.9170	2.90627
$100$	0	0
$101$	−16.3894	−1.63081	−0.815403	−	0.578894i	$-0.803485\pi$
$101$	−16.3894	−1.63081	−0.815403	+	0.578894i	$0.803485\pi$
$102$	0	0
$103$	15.2688	1.50448	0.752239	−	0.658891i	$-0.228974\pi$
$103$	15.2688	1.50448	0.752239	+	0.658891i	$0.228974\pi$
$104$	0	0
$105$	−7.03467	−0.686514
$106$	0	0
$107$	−4.35389	−0.420906	−0.210453	−	0.977604i	$-0.567494\pi$
$107$	−4.35389	−0.420906	−0.210453	+	0.977604i	$0.567494\pi$
$108$	0	0
$109$	18.5084	1.77279	0.886393	−	0.462934i	$-0.153203\pi$
$109$	18.5084	1.77279	0.886393	+	0.462934i	$0.153203\pi$
$110$	0	0
$111$	−15.2049	−1.44318
$112$	0	0
$113$	−5.18026	−0.487317	−0.243659	−	0.969861i	$-0.578348\pi$
$113$	−5.18026	−0.487317	−0.243659	+	0.969861i	$0.578348\pi$
$114$	0	0
$115$	1.95981	0.182754
$116$	0	0
$117$	−11.9229	−1.10227
$118$	0	0
$119$	−9.10822	−0.834949
$120$	0	0
$121$	7.29683	0.663348
$122$	0	0
$123$	29.6858	2.67668
$124$	0	0
$125$	7.86705	0.703650
$126$	0	0
$127$	−12.6779	−1.12498	−0.562491	−	0.826804i	$-0.690157\pi$
$127$	−12.6779	−1.12498	−0.562491	+	0.826804i	$0.690157\pi$
$128$	0	0
$129$	39.0545	3.43855
$130$	0	0
$131$	1.71647	0.149968	0.0749842	−	0.997185i	$-0.476109\pi$
$131$	1.71647	0.149968	0.0749842	+	0.997185i	$0.476109\pi$
$132$	0	0
$133$	10.4427	0.905497
$134$	0	0
$135$	−9.95735	−0.856992
$136$	0	0
$137$	−1.34232	−0.114682	−0.0573410	−	0.998355i	$-0.518262\pi$
$137$	−1.34232	−0.114682	−0.0573410	+	0.998355i	$0.518262\pi$
$138$	0	0
$139$	−3.48268	−0.295397	−0.147699	−	0.989032i	$-0.547187\pi$
$139$	−3.48268	−0.295397	−0.147699	+	0.989032i	$0.547187\pi$
$140$	0	0
$141$	−8.72273	−0.734587
$142$	0	0
$143$	−7.54401	−0.630862
$144$	0	0
$145$	2.71105	0.225140
$146$	0	0
$147$	0.179326	0.0147905
$148$	0	0
$149$	14.5453	1.19160	0.595798	−	0.803134i	$-0.296836\pi$
$149$	14.5453	1.19160	0.595798	+	0.803134i	$0.296836\pi$
$150$	0	0
$151$	10.7029	0.870986	0.435493	−	0.900192i	$-0.356574\pi$
$151$	10.7029	0.870986	0.435493	+	0.900192i	$0.356574\pi$
$152$	0	0
$153$	−23.1781	−1.87384
$154$	0	0
$155$	−4.71636	−0.378827
$156$	0	0
$157$	−5.37045	−0.428608	−0.214304	−	0.976767i	$-0.568748\pi$
$157$	−5.37045	−0.428608	−0.214304	+	0.976767i	$0.568748\pi$
$158$	0	0
$159$	16.5786	1.31477
$160$	0	0
$161$	−6.14253	−0.484099
$162$	0	0
$163$	−8.08291	−0.633102	−0.316551	−	0.948575i	$-0.602525\pi$
$163$	−8.08291	−0.633102	−0.316551	+	0.948575i	$0.602525\pi$
$164$	0	0
$165$	−11.3269	−0.881795
$166$	0	0
$167$	7.73781	0.598770	0.299385	−	0.954132i	$-0.403219\pi$
$167$	7.73781	0.598770	0.299385	+	0.954132i	$0.403219\pi$
$168$	0	0
$169$	−9.88951	−0.760732
$170$	0	0
$171$	26.5740	2.03216
$172$	0	0
$173$	−12.2730	−0.933099	−0.466550	−	0.884495i	$-0.654503\pi$
$173$	−12.2730	−0.933099	−0.466550	+	0.884495i	$0.654503\pi$
$174$	0	0
$175$	−11.3743	−0.859819
$176$	0	0
$177$	1.92681	0.144828
$178$	0	0
$179$	2.14197	0.160098	0.0800491	−	0.996791i	$-0.474492\pi$
$179$	2.14197	0.160098	0.0800491	+	0.996791i	$0.474492\pi$
$180$	0	0
$181$	−4.63927	−0.344835	−0.172417	−	0.985024i	$-0.555158\pi$
$181$	−4.63927	−0.344835	−0.172417	+	0.985024i	$0.555158\pi$
$182$	0	0
$183$	39.3872	2.91158
$184$	0	0
$185$	4.12516	0.303288
$186$	0	0
$187$	−14.6656	−1.07245
$188$	0	0
$189$	31.2087	2.27010
$190$	0	0
$191$	−13.4792	−0.975324	−0.487662	−	0.873032i	$-0.662150\pi$
$191$	−13.4792	−0.975324	−0.487662	+	0.873032i	$0.662150\pi$
$192$	0	0
$193$	−0.0606435	−0.00436521	−0.00218261	−	0.999998i	$-0.500695\pi$
$193$	−0.0606435	−0.00436521	−0.00218261	+	0.999998i	$0.500695\pi$
$194$	0	0
$195$	4.67020	0.334440
$196$	0	0
$197$	−1.56291	−0.111353	−0.0556764	−	0.998449i	$-0.517732\pi$
$197$	−1.56291	−0.111353	−0.0556764	+	0.998449i	$0.517732\pi$
$198$	0	0
$199$	8.85487	0.627705	0.313852	−	0.949472i	$-0.398380\pi$
$199$	8.85487	0.627705	0.313852	+	0.949472i	$0.398380\pi$
$200$	0	0
$201$	−14.3199	−1.01005
$202$	0	0
$203$	−8.49708	−0.596378
$204$	0	0
$205$	−8.05392	−0.562510
$206$	0	0
$207$	−15.6311	−1.08644
$208$	0	0
$209$	16.8143	1.16307
$210$	0	0
$211$	20.4703	1.40923	0.704616	−	0.709588i	$-0.251119\pi$
$211$	20.4703	1.40923	0.704616	+	0.709588i	$0.251119\pi$
$212$	0	0
$213$	−36.3221	−2.48875
$214$	0	0
$215$	−10.5957	−0.722620
$216$	0	0
$217$	14.7822	1.00348
$218$	0	0
$219$	17.1768	1.16070
$220$	0	0
$221$	6.04680	0.406752
$222$	0	0
$223$	8.33947	0.558452	0.279226	−	0.960225i	$-0.409922\pi$
$223$	8.33947	0.558452	0.279226	+	0.960225i	$0.409922\pi$
$224$	0	0
$225$	−28.9447	−1.92965
$226$	0	0
$227$	5.83203	0.387086	0.193543	−	0.981092i	$-0.438002\pi$
$227$	5.83203	0.387086	0.193543	+	0.981092i	$0.438002\pi$
$228$	0	0
$229$	−27.3179	−1.80522	−0.902609	−	0.430461i	$-0.858351\pi$
$229$	−27.3179	−1.80522	−0.902609	+	0.430461i	$0.858351\pi$
$230$	0	0
$231$	35.5011	2.33580
$232$	0	0
$233$	20.5951	1.34923	0.674614	−	0.738171i	$-0.264310\pi$
$233$	20.5951	1.34923	0.674614	+	0.738171i	$0.264310\pi$
$234$	0	0
$235$	2.36652	0.154375
$236$	0	0
$237$	−25.6763	−1.66786
$238$	0	0
$239$	−21.6130	−1.39803	−0.699015	−	0.715107i	$-0.746378\pi$
$239$	−21.6130	−1.39803	−0.699015	+	0.715107i	$0.746378\pi$
$240$	0	0
$241$	−15.6823	−1.01019	−0.505093	−	0.863065i	$-0.668542\pi$
$241$	−15.6823	−1.01019	−0.505093	+	0.863065i	$0.668542\pi$
$242$	0	0
$243$	16.0576	1.03010
$244$	0	0
$245$	−0.0486520	−0.00310826
$246$	0	0
$247$	−6.93274	−0.441120
$248$	0	0
$249$	26.4605	1.67687
$250$	0	0
$251$	−30.2215	−1.90757	−0.953784	−	0.300494i	$-0.902848\pi$
$251$	−30.2215	−1.90757	−0.953784	+	0.300494i	$0.902848\pi$
$252$	0	0
$253$	−9.89038	−0.621803
$254$	0	0
$255$	9.07889	0.568542
$256$	0	0
$257$	−26.4008	−1.64684	−0.823420	−	0.567433i	$-0.807937\pi$
$257$	−26.4008	−1.64684	−0.823420	+	0.567433i	$0.807937\pi$
$258$	0	0
$259$	−12.9292	−0.803384
$260$	0	0
$261$	−21.6229	−1.33842
$262$	0	0
$263$	−5.65316	−0.348588	−0.174294	−	0.984694i	$-0.555764\pi$
$263$	−5.65316	−0.348588	−0.174294	+	0.984694i	$0.555764\pi$
$264$	0	0
$265$	−4.49786	−0.276301
$266$	0	0
$267$	34.3028	2.09930
$268$	0	0
$269$	−8.57325	−0.522720	−0.261360	−	0.965241i	$-0.584171\pi$
$269$	−8.57325	−0.522720	−0.261360	+	0.965241i	$0.584171\pi$
$270$	0	0
$271$	21.9711	1.33465	0.667323	−	0.744768i	$-0.267440\pi$
$271$	21.9711	1.33465	0.667323	+	0.744768i	$0.267440\pi$
$272$	0	0
$273$	−14.6375	−0.885904
$274$	0	0
$275$	−18.3144	−1.10440
$276$	0	0
$277$	5.39064	0.323892	0.161946	−	0.986800i	$-0.448223\pi$
$277$	5.39064	0.323892	0.161946	+	0.986800i	$0.448223\pi$
$278$	0	0
$279$	37.6169	2.25206
$280$	0	0
$281$	14.8976	0.888718	0.444359	−	0.895849i	$-0.353431\pi$
$281$	14.8976	0.888718	0.444359	+	0.895849i	$0.353431\pi$
$282$	0	0
$283$	−27.5431	−1.63727	−0.818635	−	0.574314i	$-0.805269\pi$
$283$	−27.5431	−1.63727	−0.818635	+	0.574314i	$0.805269\pi$
$284$	0	0
$285$	−10.4091	−0.616580
$286$	0	0
$287$	25.2429	1.49004
$288$	0	0
$289$	−5.24500	−0.308529
$290$	0	0
$291$	−33.8391	−1.98368
$292$	0	0
$293$	−17.7574	−1.03740	−0.518698	−	0.854958i	$-0.673583\pi$
$293$	−17.7574	−1.03740	−0.518698	+	0.854958i	$0.673583\pi$
$294$	0	0
$295$	−0.522755	−0.0304360
$296$	0	0
$297$	50.2507	2.91584
$298$	0	0
$299$	4.07793	0.235833
$300$	0	0
$301$	33.2094	1.91416
$302$	0	0
$303$	−51.2029	−2.94153
$304$	0	0
$305$	−10.6860	−0.611876
$306$	0	0
$307$	14.9753	0.854688	0.427344	−	0.904089i	$-0.359449\pi$
$307$	14.9753	0.854688	0.427344	+	0.904089i	$0.359449\pi$
$308$	0	0
$309$	47.7019	2.71367
$310$	0	0
$311$	1.38379	0.0784676	0.0392338	−	0.999230i	$-0.487508\pi$
$311$	1.38379	0.0784676	0.0392338	+	0.999230i	$0.487508\pi$
$312$	0	0
$313$	9.33893	0.527868	0.263934	−	0.964541i	$-0.414980\pi$
$313$	9.33893	0.527868	0.263934	+	0.964541i	$0.414980\pi$
$314$	0	0
$315$	−15.2222	−0.857675
$316$	0	0
$317$	−0.0693793	−0.00389673	−0.00194837	−	0.999998i	$-0.500620\pi$
$317$	−0.0693793	−0.00389673	−0.00194837	+	0.999998i	$0.500620\pi$
$318$	0	0
$319$	−13.6815	−0.766020
$320$	0	0
$321$	−13.6022	−0.759200
$322$	0	0
$323$	−13.4773	−0.749895
$324$	0	0
$325$	7.55124	0.418867
$326$	0	0
$327$	57.8231	3.19762
$328$	0	0
$329$	−7.41726	−0.408927
$330$	0	0
$331$	−7.18873	−0.395128	−0.197564	−	0.980290i	$-0.563303\pi$
$331$	−7.18873	−0.395128	−0.197564	+	0.980290i	$0.563303\pi$
$332$	0	0
$333$	−32.9016	−1.80300
$334$	0	0
$335$	3.88506	0.212264
$336$	0	0
$337$	26.1380	1.42383	0.711913	−	0.702268i	$-0.247829\pi$
$337$	26.1380	1.42383	0.711913	+	0.702268i	$0.247829\pi$
$338$	0	0
$339$	−16.1839	−0.878988
$340$	0	0
$341$	23.8015	1.28893
$342$	0	0
$343$	−18.4435	−0.995858
$344$	0	0
$345$	6.12275	0.329638
$346$	0	0
$347$	−31.8216	−1.70827	−0.854137	−	0.520048i	$-0.825914\pi$
$347$	−31.8216	−1.70827	−0.854137	+	0.520048i	$0.825914\pi$
$348$	0	0
$349$	−22.8810	−1.22479	−0.612397	−	0.790551i	$-0.709794\pi$
$349$	−22.8810	−1.22479	−0.612397	+	0.790551i	$0.709794\pi$
$350$	0	0
$351$	−20.7190	−1.10590
$352$	0	0
$353$	29.0731	1.54740	0.773702	−	0.633550i	$-0.218403\pi$
$353$	29.0731	1.54740	0.773702	+	0.633550i	$0.218403\pi$
$354$	0	0
$355$	9.85438	0.523016
$356$	0	0
$357$	−28.4554	−1.50602
$358$	0	0
$359$	−17.5920	−0.928471	−0.464235	−	0.885712i	$-0.653671\pi$
$359$	−17.5920	−0.928471	−0.464235	+	0.885712i	$0.653671\pi$
$360$	0	0
$361$	−3.54813	−0.186743
$362$	0	0
$363$	22.7964	1.19650
$364$	0	0
$365$	−4.66015	−0.243923
$366$	0	0
$367$	13.4614	0.702679	0.351339	−	0.936248i	$-0.385726\pi$
$367$	13.4614	0.702679	0.351339	+	0.936248i	$0.385726\pi$
$368$	0	0
$369$	64.2367	3.34403
$370$	0	0
$371$	14.0974	0.731899
$372$	0	0
$373$	19.8601	1.02832	0.514158	−	0.857695i	$-0.328104\pi$
$373$	19.8601	1.02832	0.514158	+	0.857695i	$0.328104\pi$
$374$	0	0
$375$	24.5778	1.26919
$376$	0	0
$377$	5.64107	0.290530
$378$	0	0
$379$	21.8391	1.12180	0.560901	−	0.827883i	$-0.310455\pi$
$379$	21.8391	1.12180	0.560901	+	0.827883i	$0.310455\pi$
$380$	0	0
$381$	−39.6076	−2.02916
$382$	0	0
$383$	−32.2292	−1.64683	−0.823416	−	0.567438i	$-0.807935\pi$
$383$	−32.2292	−1.64683	−0.823416	+	0.567438i	$0.807935\pi$
$384$	0	0
$385$	−9.63164	−0.490874
$386$	0	0
$387$	84.5094	4.29585
$388$	0	0
$389$	−18.4804	−0.936994	−0.468497	−	0.883465i	$-0.655204\pi$
$389$	−18.4804	−0.936994	−0.468497	+	0.883465i	$0.655204\pi$
$390$	0	0
$391$	7.92750	0.400911
$392$	0	0
$393$	5.36249	0.270502
$394$	0	0
$395$	6.96612	0.350504
$396$	0	0
$397$	2.88486	0.144787	0.0723936	−	0.997376i	$-0.476936\pi$
$397$	2.88486	0.144787	0.0723936	+	0.997376i	$0.476936\pi$
$398$	0	0
$399$	32.6245	1.63327
$400$	0	0
$401$	1.42681	0.0712513	0.0356256	−	0.999365i	$-0.488658\pi$
$401$	1.42681	0.0712513	0.0356256	+	0.999365i	$0.488658\pi$
$402$	0	0
$403$	−9.81367	−0.488854
$404$	0	0
$405$	−13.9182	−0.691600
$406$	0	0
$407$	−20.8180	−1.03191
$408$	0	0
$409$	6.81823	0.337140	0.168570	−	0.985690i	$-0.446085\pi$
$409$	6.81823	0.337140	0.168570	+	0.985690i	$0.446085\pi$
$410$	0	0
$411$	−4.19360	−0.206855
$412$	0	0
$413$	1.63844	0.0806223
$414$	0	0
$415$	−7.17888	−0.352397
$416$	0	0
$417$	−10.8804	−0.532816
$418$	0	0
$419$	25.9565	1.26806	0.634030	−	0.773309i	$-0.281400\pi$
$419$	25.9565	1.26806	0.634030	+	0.773309i	$0.281400\pi$
$420$	0	0
$421$	−10.1183	−0.493135	−0.246568	−	0.969126i	$-0.579303\pi$
$421$	−10.1183	−0.493135	−0.246568	+	0.969126i	$0.579303\pi$
$422$	0	0
$423$	−18.8750	−0.917734
$424$	0	0
$425$	14.6796	0.712067
$426$	0	0
$427$	33.4923	1.62081
$428$	0	0
$429$	−23.5686	−1.13790
$430$	0	0
$431$	13.2111	0.636355	0.318177	−	0.948031i	$-0.396929\pi$
$431$	13.2111	0.636355	0.318177	+	0.948031i	$0.396929\pi$
$432$	0	0
$433$	21.1411	1.01598	0.507988	−	0.861364i	$-0.330390\pi$
$433$	21.1411	1.01598	0.507988	+	0.861364i	$0.330390\pi$
$434$	0	0
$435$	8.46972	0.406092
$436$	0	0
$437$	−9.08899	−0.434785
$438$	0	0
$439$	−37.9917	−1.81325	−0.906623	−	0.421942i	$-0.861349\pi$
$439$	−37.9917	−1.81325	−0.906623	+	0.421942i	$0.861349\pi$
$440$	0	0
$441$	0.388040	0.0184781
$442$	0	0
$443$	36.9899	1.75744	0.878722	−	0.477334i	$-0.158397\pi$
$443$	36.9899	1.75744	0.878722	+	0.477334i	$0.158397\pi$
$444$	0	0
$445$	−9.30653	−0.441172
$446$	0	0
$447$	45.4416	2.14932
$448$	0	0
$449$	7.96930	0.376095	0.188047	−	0.982160i	$-0.439784\pi$
$449$	7.96930	0.376095	0.188047	+	0.982160i	$0.439784\pi$
$450$	0	0
$451$	40.6448	1.91389
$452$	0	0
$453$	33.4373	1.57102
$454$	0	0
$455$	3.97124	0.186175
$456$	0	0
$457$	25.5154	1.19356	0.596780	−	0.802405i	$-0.296446\pi$
$457$	25.5154	1.19356	0.596780	+	0.802405i	$0.296446\pi$
$458$	0	0
$459$	−40.2777	−1.88000
$460$	0	0
$461$	7.39934	0.344622	0.172311	−	0.985043i	$-0.444877\pi$
$461$	7.39934	0.344622	0.172311	+	0.985043i	$0.444877\pi$
$462$	0	0
$463$	−6.20299	−0.288277	−0.144139	−	0.989557i	$-0.546041\pi$
$463$	−6.20299	−0.288277	−0.144139	+	0.989557i	$0.546041\pi$
$464$	0	0
$465$	−14.7346	−0.683301
$466$	0	0
$467$	42.4362	1.96371	0.981857	−	0.189622i	$-0.0607261\pi$
$467$	42.4362	1.96371	0.981857	+	0.189622i	$0.0607261\pi$
$468$	0	0
$469$	−12.1767	−0.562269
$470$	0	0
$471$	−16.7781	−0.773092
$472$	0	0
$473$	53.4721	2.45865
$474$	0	0
$475$	−16.8304	−0.772232
$476$	0	0
$477$	35.8741	1.64256
$478$	0	0
$479$	24.7049	1.12880	0.564398	−	0.825503i	$-0.309108\pi$
$479$	24.7049	1.12880	0.564398	+	0.825503i	$0.309108\pi$
$480$	0	0
$481$	8.58352	0.391375
$482$	0	0
$483$	−19.1902	−0.873183
$484$	0	0
$485$	9.18073	0.416876
$486$	0	0
$487$	2.67070	0.121021	0.0605105	−	0.998168i	$-0.480727\pi$
$487$	2.67070	0.121021	0.0605105	+	0.998168i	$0.480727\pi$
$488$	0	0
$489$	−25.2522	−1.14194
$490$	0	0
$491$	−15.0860	−0.680820	−0.340410	−	0.940277i	$-0.610566\pi$
$491$	−15.0860	−0.680820	−0.340410	+	0.940277i	$0.610566\pi$
$492$	0	0
$493$	10.9663	0.493896
$494$	0	0
$495$	−24.5100	−1.10164
$496$	0	0
$497$	−30.8860	−1.38543
$498$	0	0
$499$	4.86763	0.217905	0.108953	−	0.994047i	$-0.465250\pi$
$499$	4.86763	0.217905	0.108953	+	0.994047i	$0.465250\pi$
$500$	0	0
$501$	24.1741	1.08002
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	13.8916	0.618169
$506$	0	0
$507$	−30.8963	−1.37215
$508$	0	0
$509$	3.70485	0.164215	0.0821073	−	0.996623i	$-0.473835\pi$
$509$	3.70485	0.164215	0.0821073	+	0.996623i	$0.473835\pi$
$510$	0	0
$511$	14.6060	0.646133
$512$	0	0
$513$	46.1790	2.03885
$514$	0	0
$515$	−12.9418	−0.570283
$516$	0	0
$517$	−11.9429	−0.525247
$518$	0	0
$519$	−38.3427	−1.68306
$520$	0	0
$521$	−36.3349	−1.59186	−0.795930	−	0.605388i	$-0.793018\pi$
$521$	−36.3349	−1.59186	−0.795930	+	0.605388i	$0.793018\pi$
$522$	0	0
$523$	33.0412	1.44479	0.722395	−	0.691481i	$-0.243041\pi$
$523$	33.0412	1.44479	0.722395	+	0.691481i	$0.243041\pi$
$524$	0	0
$525$	−35.5351	−1.55088
$526$	0	0
$527$	−19.0778	−0.831042
$528$	0	0
$529$	−17.6537	−0.767554
$530$	0	0
$531$	4.16940	0.180937
$532$	0	0
$533$	−16.7584	−0.725885
$534$	0	0
$535$	3.69035	0.159548
$536$	0	0
$537$	6.69182	0.288773
$538$	0	0
$539$	0.245527	0.0105756
$540$	0	0
$541$	−38.7565	−1.66627	−0.833135	−	0.553069i	$-0.813456\pi$
$541$	−38.7565	−1.66627	−0.833135	+	0.553069i	$0.813456\pi$
$542$	0	0
$543$	−14.4938	−0.621988
$544$	0	0
$545$	−15.6877	−0.671988
$546$	0	0
$547$	−38.6037	−1.65058	−0.825288	−	0.564712i	$-0.808987\pi$
$547$	−38.6037	−1.65058	−0.825288	+	0.564712i	$0.808987\pi$
$548$	0	0
$549$	85.2293	3.63750
$550$	0	0
$551$	−12.5730	−0.535627
$552$	0	0
$553$	−21.8335	−0.928455
$554$	0	0
$555$	12.8876	0.547049
$556$	0	0
$557$	3.55613	0.150678	0.0753390	−	0.997158i	$-0.475996\pi$
$557$	3.55613	0.150678	0.0753390	+	0.997158i	$0.475996\pi$
$558$	0	0
$559$	−22.0472	−0.932497
$560$	0	0
$561$	−45.8174	−1.93441
$562$	0	0
$563$	−17.8449	−0.752071	−0.376035	−	0.926605i	$-0.622713\pi$
$563$	−17.8449	−0.752071	−0.376035	+	0.926605i	$0.622713\pi$
$564$	0	0
$565$	4.39077	0.184721
$566$	0	0
$567$	43.6229	1.83199
$568$	0	0
$569$	−20.0168	−0.839148	−0.419574	−	0.907721i	$-0.637820\pi$
$569$	−20.0168	−0.839148	−0.419574	+	0.907721i	$0.637820\pi$
$570$	0	0
$571$	−21.7640	−0.910797	−0.455398	−	0.890288i	$-0.650503\pi$
$571$	−21.7640	−0.910797	−0.455398	+	0.890288i	$0.650503\pi$
$572$	0	0
$573$	−42.1112	−1.75922
$574$	0	0
$575$	9.89986	0.412853
$576$	0	0
$577$	20.7634	0.864394	0.432197	−	0.901779i	$-0.357739\pi$
$577$	20.7634	0.864394	0.432197	+	0.901779i	$0.357739\pi$
$578$	0	0
$579$	−0.189459	−0.00787366
$580$	0	0
$581$	22.5003	0.933471
$582$	0	0
$583$	22.6988	0.940089
$584$	0	0
$585$	10.1058	0.417823
$586$	0	0
$587$	27.3031	1.12692	0.563460	−	0.826144i	$-0.309470\pi$
$587$	27.3031	1.12692	0.563460	+	0.826144i	$0.309470\pi$
$588$	0	0
$589$	21.8730	0.901260
$590$	0	0
$591$	−4.88277	−0.200850
$592$	0	0
$593$	35.0036	1.43743	0.718714	−	0.695306i	$-0.244731\pi$
$593$	35.0036	1.43743	0.718714	+	0.695306i	$0.244731\pi$
$594$	0	0
$595$	7.72011	0.316494
$596$	0	0
$597$	27.6639	1.13221
$598$	0	0
$599$	22.7487	0.929486	0.464743	−	0.885446i	$-0.346147\pi$
$599$	22.7487	0.929486	0.464743	+	0.885446i	$0.346147\pi$
$600$	0	0
$601$	−28.0292	−1.14334	−0.571668	−	0.820485i	$-0.693704\pi$
$601$	−28.0292	−1.14334	−0.571668	+	0.820485i	$0.693704\pi$
$602$	0	0
$603$	−30.9866	−1.26187
$604$	0	0
$605$	−6.18478	−0.251447
$606$	0	0
$607$	−4.42084	−0.179436	−0.0897182	−	0.995967i	$-0.528597\pi$
$607$	−4.42084	−0.179436	−0.0897182	+	0.995967i	$0.528597\pi$
$608$	0	0
$609$	−26.5461	−1.07570
$610$	0	0
$611$	4.92420	0.199212
$612$	0	0
$613$	20.4937	0.827733	0.413866	−	0.910338i	$-0.364178\pi$
$613$	20.4937	0.827733	0.413866	+	0.910338i	$0.364178\pi$
$614$	0	0
$615$	−25.1616	−1.01461
$616$	0	0
$617$	−1.43340	−0.0577066	−0.0288533	−	0.999584i	$-0.509186\pi$
$617$	−1.43340	−0.0577066	−0.0288533	+	0.999584i	$0.509186\pi$
$618$	0	0
$619$	−36.4678	−1.46577	−0.732883	−	0.680355i	$-0.761826\pi$
$619$	−36.4678	−1.46577	−0.732883	+	0.680355i	$0.761826\pi$
$620$	0	0
$621$	−27.1631	−1.09002
$622$	0	0
$623$	29.1689	1.16863
$624$	0	0
$625$	14.7398	0.589592
$626$	0	0
$627$	52.5303	2.09786
$628$	0	0
$629$	16.6864	0.665330
$630$	0	0
$631$	36.8662	1.46762	0.733810	−	0.679355i	$-0.237740\pi$
$631$	36.8662	1.46762	0.733810	+	0.679355i	$0.237740\pi$
$632$	0	0
$633$	63.9522	2.54187
$634$	0	0
$635$	10.7458	0.426433
$636$	0	0
$637$	−0.101234	−0.00401102
$638$	0	0
$639$	−78.5969	−3.10924
$640$	0	0
$641$	−10.4368	−0.412231	−0.206115	−	0.978528i	$-0.566082\pi$
$641$	−10.4368	−0.412231	−0.206115	+	0.978528i	$0.566082\pi$
$642$	0	0
$643$	−10.4820	−0.413369	−0.206685	−	0.978408i	$-0.566267\pi$
$643$	−10.4820	−0.413369	−0.206685	+	0.978408i	$0.566267\pi$
$644$	0	0
$645$	−33.1025	−1.30341
$646$	0	0
$647$	−21.0218	−0.826451	−0.413225	−	0.910629i	$-0.635598\pi$
$647$	−21.0218	−0.826451	−0.413225	+	0.910629i	$0.635598\pi$
$648$	0	0
$649$	2.63813	0.103556
$650$	0	0
$651$	46.1818	1.81001
$652$	0	0
$653$	−28.3163	−1.10810	−0.554052	−	0.832482i	$-0.686919\pi$
$653$	−28.3163	−1.10810	−0.554052	+	0.832482i	$0.686919\pi$
$654$	0	0
$655$	−1.45487	−0.0568466
$656$	0	0
$657$	37.1686	1.45008
$658$	0	0
$659$	−21.8087	−0.849546	−0.424773	−	0.905300i	$-0.639646\pi$
$659$	−21.8087	−0.849546	−0.424773	+	0.905300i	$0.639646\pi$
$660$	0	0
$661$	4.08815	0.159011	0.0795054	−	0.996834i	$-0.474666\pi$
$661$	4.08815	0.159011	0.0795054	+	0.996834i	$0.474666\pi$
$662$	0	0
$663$	18.8911	0.733669
$664$	0	0
$665$	−8.85122	−0.343235
$666$	0	0
$667$	7.39558	0.286358
$668$	0	0
$669$	26.0537	1.00730
$670$	0	0
$671$	53.9276	2.08185
$672$	0	0
$673$	−11.9627	−0.461130	−0.230565	−	0.973057i	$-0.574057\pi$
$673$	−11.9627	−0.461130	−0.230565	+	0.973057i	$0.574057\pi$
$674$	0	0
$675$	−50.2988	−1.93600
$676$	0	0
$677$	41.9343	1.61167	0.805834	−	0.592142i	$-0.201717\pi$
$677$	41.9343	1.61167	0.805834	+	0.592142i	$0.201717\pi$
$678$	0	0
$679$	−28.7746	−1.10427
$680$	0	0
$681$	18.2201	0.698197
$682$	0	0
$683$	8.40231	0.321505	0.160753	−	0.986995i	$-0.448608\pi$
$683$	8.40231	0.321505	0.160753	+	0.986995i	$0.448608\pi$
$684$	0	0
$685$	1.13775	0.0434711
$686$	0	0
$687$	−85.3452	−3.25612
$688$	0	0
$689$	−9.35901	−0.356550
$690$	0	0
$691$	13.5228	0.514433	0.257216	−	0.966354i	$-0.417195\pi$
$691$	13.5228	0.514433	0.257216	+	0.966354i	$0.417195\pi$
$692$	0	0
$693$	76.8203	2.91816
$694$	0	0
$695$	2.95192	0.111973
$696$	0	0
$697$	−32.5783	−1.23399
$698$	0	0
$699$	64.3421	2.43364
$700$	0	0
$701$	−28.6758	−1.08307	−0.541535	−	0.840678i	$-0.682157\pi$
$701$	−28.6758	−1.08307	−0.541535	+	0.840678i	$0.682157\pi$
$702$	0	0
$703$	−19.1312	−0.721546
$704$	0	0
$705$	7.39337	0.278451
$706$	0	0
$707$	−43.5397	−1.63748
$708$	0	0
$709$	17.8535	0.670504	0.335252	−	0.942129i	$-0.391179\pi$
$709$	17.8535	0.670504	0.335252	+	0.942129i	$0.391179\pi$
$710$	0	0
$711$	−55.5606	−2.08368
$712$	0	0
$713$	−12.8660	−0.481834
$714$	0	0
$715$	6.39429	0.239133
$716$	0	0
$717$	−67.5222	−2.52166
$718$	0	0
$719$	−24.1926	−0.902233	−0.451117	−	0.892465i	$-0.648974\pi$
$719$	−24.1926	−0.902233	−0.451117	+	0.892465i	$0.648974\pi$
$720$	0	0
$721$	40.5627	1.51063
$722$	0	0
$723$	−48.9938	−1.82210
$724$	0	0
$725$	13.6947	0.508607
$726$	0	0
$727$	44.5908	1.65378	0.826890	−	0.562364i	$-0.190108\pi$
$727$	44.5908	1.65378	0.826890	+	0.562364i	$0.190108\pi$
$728$	0	0
$729$	0.904180	0.0334881
$730$	0	0
$731$	−42.8598	−1.58523
$732$	0	0
$733$	48.0199	1.77366	0.886828	−	0.462099i	$-0.152903\pi$
$733$	48.0199	1.77366	0.886828	+	0.462099i	$0.152903\pi$
$734$	0	0
$735$	−0.151996	−0.00560646
$736$	0	0
$737$	−19.6063	−0.722207
$738$	0	0
$739$	−42.1877	−1.55190	−0.775950	−	0.630795i	$-0.782729\pi$
$739$	−42.1877	−1.55190	−0.775950	+	0.630795i	$0.782729\pi$
$740$	0	0
$741$	−21.6589	−0.795660
$742$	0	0
$743$	8.78644	0.322343	0.161172	−	0.986926i	$-0.448473\pi$
$743$	8.78644	0.322343	0.161172	+	0.986926i	$0.448473\pi$
$744$	0	0
$745$	−12.3286	−0.451683
$746$	0	0
$747$	57.2575	2.09494
$748$	0	0
$749$	−11.5664	−0.422628
$750$	0	0
$751$	22.7867	0.831497	0.415748	−	0.909480i	$-0.363520\pi$
$751$	22.7867	0.831497	0.415748	+	0.909480i	$0.363520\pi$
$752$	0	0
$753$	−94.4166	−3.44073
$754$	0	0
$755$	−9.07172	−0.330154
$756$	0	0
$757$	−31.9717	−1.16203	−0.581015	−	0.813893i	$-0.697344\pi$
$757$	−31.9717	−1.16203	−0.581015	+	0.813893i	$0.697344\pi$
$758$	0	0
$759$	−30.8990	−1.12156
$760$	0	0
$761$	−48.1520	−1.74551	−0.872754	−	0.488160i	$-0.837668\pi$
$761$	−48.1520	−1.74551	−0.872754	+	0.488160i	$0.837668\pi$
$762$	0	0
$763$	49.1691	1.78004
$764$	0	0
$765$	19.6457	0.710291
$766$	0	0
$767$	−1.08773	−0.0392758
$768$	0	0
$769$	−0.686945	−0.0247719	−0.0123859	−	0.999923i	$-0.503943\pi$
$769$	−0.686945	−0.0247719	−0.0123859	+	0.999923i	$0.503943\pi$
$770$	0	0
$771$	−82.4801	−2.97045
$772$	0	0
$773$	16.7685	0.603120	0.301560	−	0.953447i	$-0.402493\pi$
$773$	16.7685	0.603120	0.301560	+	0.953447i	$0.402493\pi$
$774$	0	0
$775$	−23.8243	−0.855795
$776$	0	0
$777$	−40.3929	−1.44909
$778$	0	0
$779$	37.3515	1.33826
$780$	0	0
$781$	−49.7310	−1.77952
$782$	0	0
$783$	−37.5752	−1.34283
$784$	0	0
$785$	4.55198	0.162467
$786$	0	0
$787$	−9.38091	−0.334393	−0.167197	−	0.985924i	$-0.553471\pi$
$787$	−9.38091	−0.334393	−0.167197	+	0.985924i	$0.553471\pi$
$788$	0	0
$789$	−17.6613	−0.628759
$790$	0	0
$791$	−13.7617	−0.489311
$792$	0	0
$793$	−22.2350	−0.789589
$794$	0	0
$795$	−14.0520	−0.498372
$796$	0	0
$797$	−16.1010	−0.570326	−0.285163	−	0.958479i	$-0.592048\pi$
$797$	−16.1010	−0.570326	−0.285163	+	0.958479i	$0.592048\pi$
$798$	0	0
$799$	9.57265	0.338656
$800$	0	0
$801$	74.2273	2.62269
$802$	0	0
$803$	23.5179	0.829927
$804$	0	0
$805$	5.20640	0.183501
$806$	0	0
$807$	−26.7841	−0.942845
$808$	0	0
$809$	−31.6285	−1.11200	−0.556000	−	0.831182i	$-0.687664\pi$
$809$	−31.6285	−1.11200	−0.556000	+	0.831182i	$0.687664\pi$
$810$	0	0
$811$	−20.6295	−0.724398	−0.362199	−	0.932101i	$-0.617974\pi$
$811$	−20.6295	−0.724398	−0.362199	+	0.932101i	$0.617974\pi$
$812$	0	0
$813$	68.6408	2.40734
$814$	0	0
$815$	6.85106	0.239982
$816$	0	0
$817$	49.1394	1.71917
$818$	0	0
$819$	−31.6740	−1.10678
$820$	0	0
$821$	−13.5312	−0.472243	−0.236121	−	0.971724i	$-0.575876\pi$
$821$	−13.5312	−0.472243	−0.236121	+	0.971724i	$0.575876\pi$
$822$	0	0
$823$	−41.4859	−1.44611	−0.723054	−	0.690792i	$-0.757262\pi$
$823$	−41.4859	−1.44611	−0.723054	+	0.690792i	$0.757262\pi$
$824$	0	0
$825$	−57.2168	−1.99203
$826$	0	0
$827$	13.0455	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$827$	13.0455	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$828$	0	0
$829$	−25.9202	−0.900245	−0.450123	−	0.892967i	$-0.648620\pi$
$829$	−25.9202	−0.900245	−0.450123	+	0.892967i	$0.648620\pi$
$830$	0	0
$831$	16.8411	0.584213
$832$	0	0
$833$	−0.196799	−0.00681867
$834$	0	0
$835$	−6.55856	−0.226968
$836$	0	0
$837$	65.3688	2.25948
$838$	0	0
$839$	−33.3056	−1.14984	−0.574919	−	0.818210i	$-0.694966\pi$
$839$	−33.3056	−1.14984	−0.574919	+	0.818210i	$0.694966\pi$
$840$	0	0
$841$	−18.7696	−0.647226
$842$	0	0
$843$	46.5424	1.60301
$844$	0	0
$845$	8.38233	0.288361
$846$	0	0
$847$	19.3846	0.666062
$848$	0	0
$849$	−86.0489	−2.95319
$850$	0	0
$851$	11.2532	0.385755
$852$	0	0
$853$	34.8594	1.19356	0.596782	−	0.802404i	$-0.296446\pi$
$853$	34.8594	1.19356	0.596782	+	0.802404i	$0.296446\pi$
$854$	0	0
$855$	−22.5240	−0.770306
$856$	0	0
$857$	−49.2032	−1.68075	−0.840374	−	0.542007i	$-0.817665\pi$
$857$	−49.2032	−1.68075	−0.840374	+	0.542007i	$0.817665\pi$
$858$	0	0
$859$	−42.6362	−1.45473	−0.727364	−	0.686252i	$-0.759255\pi$
$859$	−42.6362	−1.45473	−0.727364	+	0.686252i	$0.759255\pi$
$860$	0	0
$861$	78.8626	2.68763
$862$	0	0
$863$	−7.39158	−0.251612	−0.125806	−	0.992055i	$-0.540152\pi$
$863$	−7.39158	−0.251612	−0.125806	+	0.992055i	$0.540152\pi$
$864$	0	0
$865$	10.4026	0.353698
$866$	0	0
$867$	−16.3862	−0.556503
$868$	0	0
$869$	−35.1551	−1.19256
$870$	0	0
$871$	8.08392	0.273913
$872$	0	0
$873$	−73.2239	−2.47825
$874$	0	0
$875$	20.8994	0.706529
$876$	0	0
$877$	−38.9550	−1.31542	−0.657709	−	0.753272i	$-0.728474\pi$
$877$	−38.9550	−1.31542	−0.657709	+	0.753272i	$0.728474\pi$
$878$	0	0
$879$	−55.4766	−1.87118
$880$	0	0
$881$	−40.7977	−1.37451	−0.687255	−	0.726417i	$-0.741184\pi$
$881$	−40.7977	−1.37451	−0.687255	+	0.726417i	$0.741184\pi$
$882$	0	0
$883$	−48.4479	−1.63040	−0.815201	−	0.579178i	$-0.803374\pi$
$883$	−48.4479	−1.63040	−0.815201	+	0.579178i	$0.803374\pi$
$884$	0	0
$885$	−1.63316	−0.0548982
$886$	0	0
$887$	−33.2105	−1.11510	−0.557549	−	0.830144i	$-0.688258\pi$
$887$	−33.2105	−1.11510	−0.557549	+	0.830144i	$0.688258\pi$
$888$	0	0
$889$	−33.6798	−1.12958
$890$	0	0
$891$	70.2394	2.35311
$892$	0	0
$893$	−10.9752	−0.367270
$894$	0	0
$895$	−1.81553	−0.0606864
$896$	0	0
$897$	12.7400	0.425378
$898$	0	0
$899$	−17.7977	−0.593587
$900$	0	0
$901$	−18.1939	−0.606128
$902$	0	0
$903$	103.751	3.45262
$904$	0	0
$905$	3.93224	0.130712
$906$	0	0
$907$	40.4111	1.34183	0.670915	−	0.741535i	$-0.265902\pi$
$907$	40.4111	1.34183	0.670915	+	0.741535i	$0.265902\pi$
$908$	0	0
$909$	−110.797	−3.67491
$910$	0	0
$911$	−45.7379	−1.51536	−0.757682	−	0.652624i	$-0.773668\pi$
$911$	−45.7379	−1.51536	−0.757682	+	0.652624i	$0.773668\pi$
$912$	0	0
$913$	36.2288	1.19900
$914$	0	0
$915$	−33.3845	−1.10366
$916$	0	0
$917$	4.55992	0.150582
$918$	0	0
$919$	48.7558	1.60831	0.804153	−	0.594423i	$-0.202619\pi$
$919$	48.7558	1.60831	0.804153	+	0.594423i	$0.202619\pi$
$920$	0	0
$921$	46.7852	1.54162
$922$	0	0
$923$	20.5047	0.674921
$924$	0	0
$925$	20.8379	0.685147
$926$	0	0
$927$	103.221	3.39024
$928$	0	0
$929$	28.2302	0.926202	0.463101	−	0.886305i	$-0.346737\pi$
$929$	28.2302	0.926202	0.463101	+	0.886305i	$0.346737\pi$
$930$	0	0
$931$	0.225632	0.00739480
$932$	0	0
$933$	4.32317	0.141534
$934$	0	0
$935$	12.4305	0.406521
$936$	0	0
$937$	16.8083	0.549102	0.274551	−	0.961573i	$-0.411471\pi$
$937$	16.8083	0.549102	0.274551	+	0.961573i	$0.411471\pi$
$938$	0	0
$939$	29.1762	0.952130
$940$	0	0
$941$	−5.47458	−0.178466	−0.0892330	−	0.996011i	$-0.528442\pi$
$941$	−5.47458	−0.178466	−0.0892330	+	0.996011i	$0.528442\pi$
$942$	0	0
$943$	−21.9706	−0.715462
$944$	0	0
$945$	−26.4525	−0.860499
$946$	0	0
$947$	4.12540	0.134058	0.0670288	−	0.997751i	$-0.478648\pi$
$947$	4.12540	0.134058	0.0670288	+	0.997751i	$0.478648\pi$
$948$	0	0
$949$	−9.69670	−0.314768
$950$	0	0
$951$	−0.216751	−0.00702864
$952$	0	0
$953$	−58.3063	−1.88873	−0.944363	−	0.328904i	$-0.893321\pi$
$953$	−58.3063	−1.88873	−0.944363	+	0.328904i	$0.893321\pi$
$954$	0	0
$955$	11.4250	0.369704
$956$	0	0
$957$	−42.7432	−1.38169
$958$	0	0
$959$	−3.56597	−0.115151
$960$	0	0
$961$	−0.0376471	−0.00121442
$962$	0	0
$963$	−29.4336	−0.948483
$964$	0	0
$965$	0.0514013	0.00165467
$966$	0	0
$967$	3.05680	0.0983001	0.0491500	−	0.998791i	$-0.484349\pi$
$967$	3.05680	0.0983001	0.0491500	+	0.998791i	$0.484349\pi$
$968$	0	0
$969$	−42.1050	−1.35261
$970$	0	0
$971$	21.8221	0.700306	0.350153	−	0.936693i	$-0.386130\pi$
$971$	21.8221	0.700306	0.350153	+	0.936693i	$0.386130\pi$
$972$	0	0
$973$	−9.25201	−0.296606
$974$	0	0
$975$	23.5912	0.755522
$976$	0	0
$977$	7.73277	0.247393	0.123697	−	0.992320i	$-0.460525\pi$
$977$	7.73277	0.247393	0.123697	+	0.992320i	$0.460525\pi$
$978$	0	0
$979$	46.9662	1.50105
$980$	0	0
$981$	125.122	3.99485
$982$	0	0
$983$	47.3156	1.50913	0.754567	−	0.656223i	$-0.227847\pi$
$983$	47.3156	1.50913	0.754567	+	0.656223i	$0.227847\pi$
$984$	0	0
$985$	1.32472	0.0422091
$986$	0	0
$987$	−23.1726	−0.737592
$988$	0	0
$989$	−28.9044	−0.919107
$990$	0	0
$991$	−18.0763	−0.574211	−0.287106	−	0.957899i	$-0.592693\pi$
$991$	−18.0763	−0.574211	−0.287106	+	0.957899i	$0.592693\pi$
$992$	0	0
$993$	−22.4586	−0.712703
$994$	0	0
$995$	−7.50537	−0.237936
$996$	0	0
$997$	43.5329	1.37870	0.689351	−	0.724427i	$-0.257896\pi$
$997$	43.5329	1.37870	0.689351	+	0.724427i	$0.257896\pi$
$998$	0	0
$999$	−57.1748	−1.80893

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.r.1.10		12
4.3	odd	2		1006.2.a.i.1.3	✓	12
12.11	even	2		9054.2.a.bj.1.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1006.2.a.i.1.3	✓	12	4.3	odd	2
8048.2.a.r.1.10		12	1.1	even	1	trivial
9054.2.a.bj.1.7		12	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:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q+3.12415 q^{3} -0.847598 q^{5} +2.65658 q^{7} +6.76030 q^{9} +O(q^{10})\)	\(q+3.12415 q^{3} -0.847598 q^{5} +2.65658 q^{7} +6.76030 q^{9} +4.27748 q^{11} -1.76366 q^{13} -2.64802 q^{15} -3.42856 q^{17} +3.93089 q^{19} +8.29954 q^{21} -2.31220 q^{23} -4.28158 q^{25} +11.7477 q^{27} -3.19851 q^{29} +5.56438 q^{31} +13.3635 q^{33} -2.25171 q^{35} -4.86688 q^{37} -5.50993 q^{39} +9.50205 q^{41} +12.5008 q^{43} -5.73002 q^{45} -2.79204 q^{47} +0.0573998 q^{49} -10.7113 q^{51} +5.30659 q^{53} -3.62558 q^{55} +12.2807 q^{57} +0.616748 q^{59} +12.6073 q^{61} +17.9593 q^{63} +1.49487 q^{65} -4.58361 q^{67} -7.22365 q^{69} -11.6262 q^{71} +5.49806 q^{73} -13.3763 q^{75} +11.3635 q^{77} -8.21866 q^{79} +16.4207 q^{81} +8.46967 q^{83} +2.90604 q^{85} -9.99261 q^{87} +10.9799 q^{89} -4.68529 q^{91} +17.3840 q^{93} -3.33181 q^{95} -10.8315 q^{97} +28.9170 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{3} + 7 q^{5} + 2 q^{7} + 27 q^{9}+O(q^{10}) \) 12 * q + 3 * q^3 + 7 * q^5 + 2 * q^7 + 27 * q^9	\( 12 q + 3 q^{3} + 7 q^{5} + 2 q^{7} + 27 q^{9} - 18 q^{11} - 4 q^{13} + 2 q^{15} + 12 q^{17} + 7 q^{21} + 9 q^{23} + 25 q^{25} + 18 q^{27} + 34 q^{29} + 11 q^{31} + 4 q^{33} - 21 q^{35} - 22 q^{37} - 13 q^{39} + 32 q^{41} + 8 q^{43} + 13 q^{45} - 24 q^{47} + 36 q^{49} - 16 q^{51} - 2 q^{53} + 12 q^{55} + 26 q^{57} - 26 q^{59} + 12 q^{61} - 5 q^{63} + 66 q^{65} + 21 q^{67} + 20 q^{69} - 50 q^{71} + 17 q^{73} + 14 q^{75} + 25 q^{77} + 9 q^{79} + 48 q^{81} - 25 q^{83} + 24 q^{85} + 10 q^{87} + 21 q^{89} + 9 q^{91} + 31 q^{93} - 22 q^{95} + 18 q^{97} - 102 q^{99}+O(q^{100}) \) 12 * q + 3 * q^3 + 7 * q^5 + 2 * q^7 + 27 * q^9 - 18 * q^11 - 4 * q^13 + 2 * q^15 + 12 * q^17 + 7 * q^21 + 9 * q^23 + 25 * q^25 + 18 * q^27 + 34 * q^29 + 11 * q^31 + 4 * q^33 - 21 * q^35 - 22 * q^37 - 13 * q^39 + 32 * q^41 + 8 * q^43 + 13 * q^45 - 24 * q^47 + 36 * q^49 - 16 * q^51 - 2 * q^53 + 12 * q^55 + 26 * q^57 - 26 * q^59 + 12 * q^61 - 5 * q^63 + 66 * q^65 + 21 * q^67 + 20 * q^69 - 50 * q^71 + 17 * q^73 + 14 * q^75 + 25 * q^77 + 9 * q^79 + 48 * q^81 - 25 * q^83 + 24 * q^85 + 10 * q^87 + 21 * q^89 + 9 * q^91 + 31 * q^93 - 22 * q^95 + 18 * q^97 - 102 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more