LMFDB - Embedded newform 8048.2.a.w.1.15

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:	$29$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
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)

$f(q)$	$=$	$q+0.502218 q^{3} -3.38571 q^{5} -3.39706 q^{7} -2.74778 q^{9} +O(q^{10})$	$q+0.502218 q^{3} -3.38571 q^{5} -3.39706 q^{7} -2.74778 q^{9} +0.762333 q^{11} +5.78788 q^{13} -1.70036 q^{15} -6.42545 q^{17} -2.41615 q^{19} -1.70606 q^{21} -3.05365 q^{23} +6.46301 q^{25} -2.88664 q^{27} -6.29511 q^{29} +2.62920 q^{31} +0.382857 q^{33} +11.5015 q^{35} -11.9577 q^{37} +2.90678 q^{39} +1.26229 q^{41} -12.2275 q^{43} +9.30317 q^{45} -4.26461 q^{47} +4.54003 q^{49} -3.22698 q^{51} +3.63282 q^{53} -2.58103 q^{55} -1.21343 q^{57} -10.0117 q^{59} -1.14744 q^{61} +9.33437 q^{63} -19.5961 q^{65} +0.489215 q^{67} -1.53360 q^{69} -1.37860 q^{71} +9.41461 q^{73} +3.24584 q^{75} -2.58969 q^{77} +4.52016 q^{79} +6.79361 q^{81} +8.13873 q^{83} +21.7547 q^{85} -3.16152 q^{87} -8.29889 q^{89} -19.6618 q^{91} +1.32043 q^{93} +8.18036 q^{95} -16.6311 q^{97} -2.09472 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9}+O(q^{10}) $ 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9	$ 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9} + 27 q^{11} + 16 q^{13} + 14 q^{15} - 15 q^{17} + 14 q^{19} + q^{21} + 25 q^{23} + 21 q^{25} + 25 q^{27} - 13 q^{29} + 27 q^{31} - 9 q^{33} + 29 q^{35} + 35 q^{37} + 38 q^{39} - 30 q^{41} + 38 q^{43} + q^{45} + 35 q^{47} + 14 q^{49} + 21 q^{51} + 2 q^{53} + 25 q^{55} - 25 q^{57} + 40 q^{59} + 10 q^{61} + 56 q^{63} - 50 q^{65} + 31 q^{67} + 11 q^{69} + 65 q^{71} - 23 q^{73} + 32 q^{75} + 13 q^{77} + 44 q^{79} - 7 q^{81} + 41 q^{83} + 26 q^{85} + 25 q^{87} - 48 q^{89} + 44 q^{91} + 25 q^{93} + 75 q^{95} - 18 q^{97} + 80 q^{99}+O(q^{100}) $ 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9 + 27 * q^11 + 16 * q^13 + 14 * q^15 - 15 * q^17 + 14 * q^19 + q^21 + 25 * q^23 + 21 * q^25 + 25 * q^27 - 13 * q^29 + 27 * q^31 - 9 * q^33 + 29 * q^35 + 35 * q^37 + 38 * q^39 - 30 * q^41 + 38 * q^43 + q^45 + 35 * q^47 + 14 * q^49 + 21 * q^51 + 2 * q^53 + 25 * q^55 - 25 * q^57 + 40 * q^59 + 10 * q^61 + 56 * q^63 - 50 * q^65 + 31 * q^67 + 11 * q^69 + 65 * q^71 - 23 * q^73 + 32 * q^75 + 13 * q^77 + 44 * q^79 - 7 * q^81 + 41 * q^83 + 26 * q^85 + 25 * q^87 - 48 * q^89 + 44 * q^91 + 25 * q^93 + 75 * q^95 - 18 * q^97 + 80 * 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
$3$	0.502218	0.289956	0.144978	−	0.989435i	$-0.453689\pi$
$3$	0.502218	0.289956	0.144978	+	0.989435i	$0.453689\pi$
$4$	0	0
$5$	−3.38571	−1.51413	−0.757067	−	0.653337i	$-0.773368\pi$
$5$	−3.38571	−1.51413	−0.757067	+	0.653337i	$0.773368\pi$
$6$	0	0
$7$	−3.39706	−1.28397	−0.641985	−	0.766718i	$-0.721889\pi$
$7$	−3.39706	−1.28397	−0.641985	+	0.766718i	$0.721889\pi$
$8$	0	0
$9$	−2.74778	−0.915926
$10$	0	0
$11$	0.762333	0.229852	0.114926	−	0.993374i	$-0.463337\pi$
$11$	0.762333	0.229852	0.114926	+	0.993374i	$0.463337\pi$
$12$	0	0
$13$	5.78788	1.60527	0.802635	−	0.596470i	$-0.203431\pi$
$13$	5.78788	1.60527	0.802635	+	0.596470i	$0.203431\pi$
$14$	0	0
$15$	−1.70036	−0.439032
$16$	0	0
$17$	−6.42545	−1.55840	−0.779200	−	0.626775i	$-0.784375\pi$
$17$	−6.42545	−1.55840	−0.779200	+	0.626775i	$0.784375\pi$
$18$	0	0
$19$	−2.41615	−0.554302	−0.277151	−	0.960826i	$-0.589390\pi$
$19$	−2.41615	−0.554302	−0.277151	+	0.960826i	$0.589390\pi$
$20$	0	0
$21$	−1.70606	−0.372294
$22$	0	0
$23$	−3.05365	−0.636731	−0.318365	−	0.947968i	$-0.603134\pi$
$23$	−3.05365	−0.636731	−0.318365	+	0.947968i	$0.603134\pi$
$24$	0	0
$25$	6.46301	1.29260
$26$	0	0
$27$	−2.88664	−0.555533
$28$	0	0
$29$	−6.29511	−1.16897	−0.584487	−	0.811403i	$-0.698704\pi$
$29$	−6.29511	−1.16897	−0.584487	+	0.811403i	$0.698704\pi$
$30$	0	0
$31$	2.62920	0.472218	0.236109	−	0.971727i	$-0.424128\pi$
$31$	2.62920	0.472218	0.236109	+	0.971727i	$0.424128\pi$
$32$	0	0
$33$	0.382857	0.0666468
$34$	0	0
$35$	11.5015	1.94410
$36$	0	0
$37$	−11.9577	−1.96584	−0.982918	−	0.184045i	$-0.941081\pi$
$37$	−11.9577	−1.96584	−0.982918	+	0.184045i	$0.941081\pi$
$38$	0	0
$39$	2.90678	0.465457
$40$	0	0
$41$	1.26229	0.197137	0.0985686	−	0.995130i	$-0.468574\pi$
$41$	1.26229	0.197137	0.0985686	+	0.995130i	$0.468574\pi$
$42$	0	0
$43$	−12.2275	−1.86468	−0.932340	−	0.361584i	$-0.882236\pi$
$43$	−12.2275	−1.86468	−0.932340	+	0.361584i	$0.882236\pi$
$44$	0	0
$45$	9.30317	1.38683
$46$	0	0
$47$	−4.26461	−0.622057	−0.311029	−	0.950401i	$-0.600673\pi$
$47$	−4.26461	−0.622057	−0.311029	+	0.950401i	$0.600673\pi$
$48$	0	0
$49$	4.54003	0.648576
$50$	0	0
$51$	−3.22698	−0.451867
$52$	0	0
$53$	3.63282	0.499006	0.249503	−	0.968374i	$-0.419733\pi$
$53$	3.63282	0.499006	0.249503	+	0.968374i	$0.419733\pi$
$54$	0	0
$55$	−2.58103	−0.348027
$56$	0	0
$57$	−1.21343	−0.160723
$58$	0	0
$59$	−10.0117	−1.30341	−0.651706	−	0.758472i	$-0.725946\pi$
$59$	−10.0117	−1.30341	−0.651706	+	0.758472i	$0.725946\pi$
$60$	0	0
$61$	−1.14744	−0.146915	−0.0734573	−	0.997298i	$-0.523403\pi$
$61$	−1.14744	−0.146915	−0.0734573	+	0.997298i	$0.523403\pi$
$62$	0	0
$63$	9.33437	1.17602
$64$	0	0
$65$	−19.5961	−2.43060
$66$	0	0
$67$	0.489215	0.0597671	0.0298836	−	0.999553i	$-0.490486\pi$
$67$	0.489215	0.0597671	0.0298836	+	0.999553i	$0.490486\pi$
$68$	0	0
$69$	−1.53360	−0.184624
$70$	0	0
$71$	−1.37860	−0.163610	−0.0818049	−	0.996648i	$-0.526068\pi$
$71$	−1.37860	−0.163610	−0.0818049	+	0.996648i	$0.526068\pi$
$72$	0	0
$73$	9.41461	1.10190	0.550948	−	0.834539i	$-0.314266\pi$
$73$	9.41461	1.10190	0.550948	+	0.834539i	$0.314266\pi$
$74$	0	0
$75$	3.24584	0.374797
$76$	0	0
$77$	−2.58969	−0.295123
$78$	0	0
$79$	4.52016	0.508558	0.254279	−	0.967131i	$-0.418162\pi$
$79$	4.52016	0.508558	0.254279	+	0.967131i	$0.418162\pi$
$80$	0	0
$81$	6.79361	0.754846
$82$	0	0
$83$	8.13873	0.893342	0.446671	−	0.894698i	$-0.352609\pi$
$83$	8.13873	0.893342	0.446671	+	0.894698i	$0.352609\pi$
$84$	0	0
$85$	21.7547	2.35963
$86$	0	0
$87$	−3.16152	−0.338950
$88$	0	0
$89$	−8.29889	−0.879681	−0.439840	−	0.898076i	$-0.644965\pi$
$89$	−8.29889	−0.879681	−0.439840	+	0.898076i	$0.644965\pi$
$90$	0	0
$91$	−19.6618	−2.06112
$92$	0	0
$93$	1.32043	0.136922
$94$	0	0
$95$	8.18036	0.839288
$96$	0	0
$97$	−16.6311	−1.68863	−0.844317	−	0.535844i	$-0.819994\pi$
$97$	−16.6311	−1.68863	−0.844317	+	0.535844i	$0.819994\pi$
$98$	0	0
$99$	−2.09472	−0.210527
$100$	0	0
$101$	−1.73811	−0.172948	−0.0864742	−	0.996254i	$-0.527560\pi$
$101$	−1.73811	−0.172948	−0.0864742	+	0.996254i	$0.527560\pi$
$102$	0	0
$103$	15.5421	1.53141	0.765703	−	0.643194i	$-0.222391\pi$
$103$	15.5421	1.53141	0.765703	+	0.643194i	$0.222391\pi$
$104$	0	0
$105$	5.77624	0.563703
$106$	0	0
$107$	−5.37853	−0.519962	−0.259981	−	0.965614i	$-0.583716\pi$
$107$	−5.37853	−0.519962	−0.259981	+	0.965614i	$0.583716\pi$
$108$	0	0
$109$	5.80450	0.555971	0.277985	−	0.960585i	$-0.410333\pi$
$109$	5.80450	0.555971	0.277985	+	0.960585i	$0.410333\pi$
$110$	0	0
$111$	−6.00537	−0.570005
$112$	0	0
$113$	−14.4463	−1.35900	−0.679498	−	0.733677i	$-0.737802\pi$
$113$	−14.4463	−1.35900	−0.679498	+	0.733677i	$0.737802\pi$
$114$	0	0
$115$	10.3388	0.964096
$116$	0	0
$117$	−15.9038	−1.47031
$118$	0	0
$119$	21.8277	2.00094
$120$	0	0
$121$	−10.4188	−0.947168
$122$	0	0
$123$	0.633947	0.0571610
$124$	0	0
$125$	−4.95333	−0.443040
$126$	0	0
$127$	11.2556	0.998772	0.499386	−	0.866380i	$-0.333559\pi$
$127$	11.2556	0.998772	0.499386	+	0.866380i	$0.333559\pi$
$128$	0	0
$129$	−6.14088	−0.540674
$130$	0	0
$131$	6.27420	0.548180	0.274090	−	0.961704i	$-0.411623\pi$
$131$	6.27420	0.548180	0.274090	+	0.961704i	$0.411623\pi$
$132$	0	0
$133$	8.20780	0.711707
$134$	0	0
$135$	9.77330	0.841152
$136$	0	0
$137$	−2.72256	−0.232604	−0.116302	−	0.993214i	$-0.537104\pi$
$137$	−2.72256	−0.232604	−0.116302	+	0.993214i	$0.537104\pi$
$138$	0	0
$139$	19.6661	1.66805	0.834027	−	0.551723i	$-0.186030\pi$
$139$	19.6661	1.66805	0.834027	+	0.551723i	$0.186030\pi$
$140$	0	0
$141$	−2.14176	−0.180369
$142$	0	0
$143$	4.41229	0.368974
$144$	0	0
$145$	21.3134	1.76998
$146$	0	0
$147$	2.28009	0.188058
$148$	0	0
$149$	−23.2371	−1.90366	−0.951828	−	0.306633i	$-0.900798\pi$
$149$	−23.2371	−1.90366	−0.951828	+	0.306633i	$0.900798\pi$
$150$	0	0
$151$	−19.5918	−1.59436	−0.797179	−	0.603743i	$-0.793675\pi$
$151$	−19.5918	−1.59436	−0.797179	+	0.603743i	$0.793675\pi$
$152$	0	0
$153$	17.6557	1.42738
$154$	0	0
$155$	−8.90170	−0.715002
$156$	0	0
$157$	8.18143	0.652949	0.326475	−	0.945206i	$-0.394139\pi$
$157$	8.18143	0.652949	0.326475	+	0.945206i	$0.394139\pi$
$158$	0	0
$159$	1.82447	0.144690
$160$	0	0
$161$	10.3735	0.817543
$162$	0	0
$163$	2.92258	0.228914	0.114457	−	0.993428i	$-0.463487\pi$
$163$	2.92258	0.228914	0.114457	+	0.993428i	$0.463487\pi$
$164$	0	0
$165$	−1.29624	−0.100912
$166$	0	0
$167$	23.3087	1.80368	0.901840	−	0.432069i	$-0.142216\pi$
$167$	23.3087	1.80368	0.901840	+	0.432069i	$0.142216\pi$
$168$	0	0
$169$	20.4996	1.57689
$170$	0	0
$171$	6.63903	0.507700
$172$	0	0
$173$	−9.00678	−0.684773	−0.342386	−	0.939559i	$-0.611235\pi$
$173$	−9.00678	−0.684773	−0.342386	+	0.939559i	$0.611235\pi$
$174$	0	0
$175$	−21.9553	−1.65966
$176$	0	0
$177$	−5.02805	−0.377932
$178$	0	0
$179$	−6.56494	−0.490687	−0.245343	−	0.969436i	$-0.578901\pi$
$179$	−6.56494	−0.490687	−0.245343	+	0.969436i	$0.578901\pi$
$180$	0	0
$181$	−20.6535	−1.53517	−0.767583	−	0.640950i	$-0.778541\pi$
$181$	−20.6535	−1.53517	−0.767583	+	0.640950i	$0.778541\pi$
$182$	0	0
$183$	−0.576265	−0.0425987
$184$	0	0
$185$	40.4853	2.97654
$186$	0	0
$187$	−4.89833	−0.358201
$188$	0	0
$189$	9.80608	0.713287
$190$	0	0
$191$	−1.53898	−0.111357	−0.0556784	−	0.998449i	$-0.517732\pi$
$191$	−1.53898	−0.111357	−0.0556784	+	0.998449i	$0.517732\pi$
$192$	0	0
$193$	−20.1342	−1.44929	−0.724647	−	0.689120i	$-0.757997\pi$
$193$	−20.1342	−1.44929	−0.724647	+	0.689120i	$0.757997\pi$
$194$	0	0
$195$	−9.84150	−0.704764
$196$	0	0
$197$	4.41975	0.314894	0.157447	−	0.987527i	$-0.449674\pi$
$197$	4.41975	0.314894	0.157447	+	0.987527i	$0.449674\pi$
$198$	0	0
$199$	3.26548	0.231484	0.115742	−	0.993279i	$-0.463075\pi$
$199$	3.26548	0.231484	0.115742	+	0.993279i	$0.463075\pi$
$200$	0	0
$201$	0.245692	0.0173298
$202$	0	0
$203$	21.3849	1.50093
$204$	0	0
$205$	−4.27376	−0.298492
$206$	0	0
$207$	8.39076	0.583198
$208$	0	0
$209$	−1.84191	−0.127407
$210$	0	0
$211$	−9.90148	−0.681646	−0.340823	−	0.940128i	$-0.610706\pi$
$211$	−9.90148	−0.681646	−0.340823	+	0.940128i	$0.610706\pi$
$212$	0	0
$213$	−0.692357	−0.0474395
$214$	0	0
$215$	41.3988	2.82337
$216$	0	0
$217$	−8.93155	−0.606313
$218$	0	0
$219$	4.72818	0.319501
$220$	0	0
$221$	−37.1898	−2.50165
$222$	0	0
$223$	−8.91331	−0.596879	−0.298440	−	0.954429i	$-0.596466\pi$
$223$	−8.91331	−0.596879	−0.298440	+	0.954429i	$0.596466\pi$
$224$	0	0
$225$	−17.7589	−1.18393
$226$	0	0
$227$	9.51947	0.631830	0.315915	−	0.948788i	$-0.397689\pi$
$227$	9.51947	0.631830	0.315915	+	0.948788i	$0.397689\pi$
$228$	0	0
$229$	8.75667	0.578657	0.289329	−	0.957230i	$-0.406568\pi$
$229$	8.75667	0.578657	0.289329	+	0.957230i	$0.406568\pi$
$230$	0	0
$231$	−1.30059	−0.0855725
$232$	0	0
$233$	−6.91416	−0.452962	−0.226481	−	0.974016i	$-0.572722\pi$
$233$	−6.91416	−0.452962	−0.226481	+	0.974016i	$0.572722\pi$
$234$	0	0
$235$	14.4387	0.941878
$236$	0	0
$237$	2.27011	0.147459
$238$	0	0
$239$	24.4090	1.57889	0.789444	−	0.613822i	$-0.210369\pi$
$239$	24.4090	1.57889	0.789444	+	0.613822i	$0.210369\pi$
$240$	0	0
$241$	10.9548	0.705662	0.352831	−	0.935687i	$-0.385219\pi$
$241$	10.9548	0.705662	0.352831	+	0.935687i	$0.385219\pi$
$242$	0	0
$243$	12.0718	0.774405
$244$	0	0
$245$	−15.3712	−0.982032
$246$	0	0
$247$	−13.9844	−0.889805
$248$	0	0
$249$	4.08742	0.259029
$250$	0	0
$251$	−17.3150	−1.09291	−0.546456	−	0.837488i	$-0.684024\pi$
$251$	−17.3150	−1.09291	−0.546456	+	0.837488i	$0.684024\pi$
$252$	0	0
$253$	−2.32790	−0.146354
$254$	0	0
$255$	10.9256	0.684187
$256$	0	0
$257$	−17.1205	−1.06795	−0.533974	−	0.845501i	$-0.679302\pi$
$257$	−17.1205	−1.06795	−0.533974	+	0.845501i	$0.679302\pi$
$258$	0	0
$259$	40.6211	2.52407
$260$	0	0
$261$	17.2976	1.07069
$262$	0	0
$263$	18.5769	1.14550	0.572751	−	0.819730i	$-0.305876\pi$
$263$	18.5769	1.14550	0.572751	+	0.819730i	$0.305876\pi$
$264$	0	0
$265$	−12.2997	−0.755562
$266$	0	0
$267$	−4.16785	−0.255068
$268$	0	0
$269$	14.8483	0.905316	0.452658	−	0.891684i	$-0.350476\pi$
$269$	14.8483	0.905316	0.452658	+	0.891684i	$0.350476\pi$
$270$	0	0
$271$	17.7961	1.08104	0.540519	−	0.841332i	$-0.318228\pi$
$271$	17.7961	1.08104	0.540519	+	0.841332i	$0.318228\pi$
$272$	0	0
$273$	−9.87451	−0.597632
$274$	0	0
$275$	4.92697	0.297107
$276$	0	0
$277$	21.5361	1.29398	0.646989	−	0.762499i	$-0.276028\pi$
$277$	21.5361	1.29398	0.646989	+	0.762499i	$0.276028\pi$
$278$	0	0
$279$	−7.22445	−0.432517
$280$	0	0
$281$	−15.2339	−0.908777	−0.454388	−	0.890804i	$-0.650142\pi$
$281$	−15.2339	−0.908777	−0.454388	+	0.890804i	$0.650142\pi$
$282$	0	0
$283$	23.8525	1.41788	0.708942	−	0.705267i	$-0.249173\pi$
$283$	23.8525	1.41788	0.708942	+	0.705267i	$0.249173\pi$
$284$	0	0
$285$	4.10832	0.243356
$286$	0	0
$287$	−4.28809	−0.253118
$288$	0	0
$289$	24.2864	1.42861
$290$	0	0
$291$	−8.35244	−0.489629
$292$	0	0
$293$	24.6958	1.44275	0.721373	−	0.692546i	$-0.243511\pi$
$293$	24.6958	1.44275	0.721373	+	0.692546i	$0.243511\pi$
$294$	0	0
$295$	33.8967	1.97354
$296$	0	0
$297$	−2.20058	−0.127690
$298$	0	0
$299$	−17.6742	−1.02213
$300$	0	0
$301$	41.5376	2.39419
$302$	0	0
$303$	−0.872910	−0.0501474
$304$	0	0
$305$	3.88489	0.222448
$306$	0	0
$307$	26.9084	1.53575	0.767873	−	0.640603i	$-0.221315\pi$
$307$	26.9084	1.53575	0.767873	+	0.640603i	$0.221315\pi$
$308$	0	0
$309$	7.80551	0.444040
$310$	0	0
$311$	1.43997	0.0816532	0.0408266	−	0.999166i	$-0.487001\pi$
$311$	1.43997	0.0816532	0.0408266	+	0.999166i	$0.487001\pi$
$312$	0	0
$313$	−0.829899	−0.0469087	−0.0234543	−	0.999725i	$-0.507466\pi$
$313$	−0.829899	−0.0469087	−0.0234543	+	0.999725i	$0.507466\pi$
$314$	0	0
$315$	−31.6035	−1.78065
$316$	0	0
$317$	−17.8158	−1.00064	−0.500319	−	0.865841i	$-0.666784\pi$
$317$	−17.8158	−1.00064	−0.500319	+	0.865841i	$0.666784\pi$
$318$	0	0
$319$	−4.79897	−0.268691
$320$	0	0
$321$	−2.70119	−0.150766
$322$	0	0
$323$	15.5248	0.863825
$324$	0	0
$325$	37.4072	2.07498
$326$	0	0
$327$	2.91512	0.161207
$328$	0	0
$329$	14.4871	0.798702
$330$	0	0
$331$	−35.1913	−1.93429	−0.967143	−	0.254232i	$-0.918177\pi$
$331$	−35.1913	−1.93429	−0.967143	+	0.254232i	$0.918177\pi$
$332$	0	0
$333$	32.8571	1.80056
$334$	0	0
$335$	−1.65634	−0.0904954
$336$	0	0
$337$	27.9502	1.52255	0.761273	−	0.648432i	$-0.224575\pi$
$337$	27.9502	1.52255	0.761273	+	0.648432i	$0.224575\pi$
$338$	0	0
$339$	−7.25520	−0.394048
$340$	0	0
$341$	2.00432	0.108540
$342$	0	0
$343$	8.35666	0.451217
$344$	0	0
$345$	5.19232	0.279545
$346$	0	0
$347$	3.25742	0.174867	0.0874337	−	0.996170i	$-0.472133\pi$
$347$	3.25742	0.174867	0.0874337	+	0.996170i	$0.472133\pi$
$348$	0	0
$349$	−4.29704	−0.230015	−0.115008	−	0.993365i	$-0.536689\pi$
$349$	−4.29704	−0.230015	−0.115008	+	0.993365i	$0.536689\pi$
$350$	0	0
$351$	−16.7075	−0.891781
$352$	0	0
$353$	9.60904	0.511438	0.255719	−	0.966751i	$-0.417688\pi$
$353$	9.60904	0.511438	0.255719	+	0.966751i	$0.417688\pi$
$354$	0	0
$355$	4.66754	0.247727
$356$	0	0
$357$	10.9622	0.580183
$358$	0	0
$359$	−3.82924	−0.202100	−0.101050	−	0.994881i	$-0.532220\pi$
$359$	−3.82924	−0.202100	−0.101050	+	0.994881i	$0.532220\pi$
$360$	0	0
$361$	−13.1622	−0.692749
$362$	0	0
$363$	−5.23253	−0.274637
$364$	0	0
$365$	−31.8751	−1.66842
$366$	0	0
$367$	31.1840	1.62779	0.813895	−	0.581012i	$-0.197343\pi$
$367$	31.1840	1.62779	0.813895	+	0.581012i	$0.197343\pi$
$368$	0	0
$369$	−3.46850	−0.180563
$370$	0	0
$371$	−12.3409	−0.640708
$372$	0	0
$373$	13.5045	0.699239	0.349619	−	0.936892i	$-0.386311\pi$
$373$	13.5045	0.699239	0.349619	+	0.936892i	$0.386311\pi$
$374$	0	0
$375$	−2.48765	−0.128462
$376$	0	0
$377$	−36.4354	−1.87652
$378$	0	0
$379$	36.9790	1.89948	0.949742	−	0.313035i	$-0.101346\pi$
$379$	36.9790	1.89948	0.949742	+	0.313035i	$0.101346\pi$
$380$	0	0
$381$	5.65275	0.289599
$382$	0	0
$383$	−22.8079	−1.16543	−0.582713	−	0.812678i	$-0.698009\pi$
$383$	−22.8079	−1.16543	−0.582713	+	0.812678i	$0.698009\pi$
$384$	0	0
$385$	8.76794	0.446855
$386$	0	0
$387$	33.5985	1.70791
$388$	0	0
$389$	22.5220	1.14191	0.570956	−	0.820981i	$-0.306573\pi$
$389$	22.5220	1.14191	0.570956	+	0.820981i	$0.306573\pi$
$390$	0	0
$391$	19.6211	0.992282
$392$	0	0
$393$	3.15102	0.158948
$394$	0	0
$395$	−15.3039	−0.770025
$396$	0	0
$397$	−2.65515	−0.133258	−0.0666290	−	0.997778i	$-0.521224\pi$
$397$	−2.65515	−0.133258	−0.0666290	+	0.997778i	$0.521224\pi$
$398$	0	0
$399$	4.12210	0.206363
$400$	0	0
$401$	5.69994	0.284642	0.142321	−	0.989821i	$-0.454544\pi$
$401$	5.69994	0.284642	0.142321	+	0.989821i	$0.454544\pi$
$402$	0	0
$403$	15.2175	0.758038
$404$	0	0
$405$	−23.0012	−1.14294
$406$	0	0
$407$	−9.11575	−0.451851
$408$	0	0
$409$	−16.9307	−0.837168	−0.418584	−	0.908178i	$-0.637473\pi$
$409$	−16.9307	−0.837168	−0.418584	+	0.908178i	$0.637473\pi$
$410$	0	0
$411$	−1.36732	−0.0674449
$412$	0	0
$413$	34.0104	1.67354
$414$	0	0
$415$	−27.5554	−1.35264
$416$	0	0
$417$	9.87665	0.483662
$418$	0	0
$419$	−8.92919	−0.436219	−0.218110	−	0.975924i	$-0.569989\pi$
$419$	−8.92919	−0.436219	−0.218110	+	0.975924i	$0.569989\pi$
$420$	0	0
$421$	23.2974	1.13545	0.567723	−	0.823220i	$-0.307824\pi$
$421$	23.2974	1.13545	0.567723	+	0.823220i	$0.307824\pi$
$422$	0	0
$423$	11.7182	0.569758
$424$	0	0
$425$	−41.5278	−2.01439
$426$	0	0
$427$	3.89792	0.188634
$428$	0	0
$429$	2.21593	0.106986
$430$	0	0
$431$	25.4824	1.22744	0.613722	−	0.789522i	$-0.289672\pi$
$431$	25.4824	1.22744	0.613722	+	0.789522i	$0.289672\pi$
$432$	0	0
$433$	−13.0919	−0.629156	−0.314578	−	0.949232i	$-0.601863\pi$
$433$	−13.0919	−0.629156	−0.314578	+	0.949232i	$0.601863\pi$
$434$	0	0
$435$	10.7040	0.513216
$436$	0	0
$437$	7.37808	0.352941
$438$	0	0
$439$	−18.8198	−0.898220	−0.449110	−	0.893477i	$-0.648259\pi$
$439$	−18.8198	−0.898220	−0.449110	+	0.893477i	$0.648259\pi$
$440$	0	0
$441$	−12.4750	−0.594048
$442$	0	0
$443$	−25.3241	−1.20319	−0.601593	−	0.798803i	$-0.705467\pi$
$443$	−25.3241	−1.20319	−0.601593	+	0.798803i	$0.705467\pi$
$444$	0	0
$445$	28.0976	1.33196
$446$	0	0
$447$	−11.6701	−0.551976
$448$	0	0
$449$	23.1287	1.09151	0.545755	−	0.837945i	$-0.316243\pi$
$449$	23.1287	1.09151	0.545755	+	0.837945i	$0.316243\pi$
$450$	0	0
$451$	0.962288	0.0453124
$452$	0	0
$453$	−9.83935	−0.462293
$454$	0	0
$455$	66.5691	3.12081
$456$	0	0
$457$	−1.79070	−0.0837652	−0.0418826	−	0.999123i	$-0.513336\pi$
$457$	−1.79070	−0.0837652	−0.0418826	+	0.999123i	$0.513336\pi$
$458$	0	0
$459$	18.5479	0.865744
$460$	0	0
$461$	−17.9113	−0.834211	−0.417106	−	0.908858i	$-0.636956\pi$
$461$	−17.9113	−0.834211	−0.417106	+	0.908858i	$0.636956\pi$
$462$	0	0
$463$	21.9867	1.02181	0.510903	−	0.859638i	$-0.329311\pi$
$463$	21.9867	1.02181	0.510903	+	0.859638i	$0.329311\pi$
$464$	0	0
$465$	−4.47059	−0.207319
$466$	0	0
$467$	11.1324	0.515144	0.257572	−	0.966259i	$-0.417078\pi$
$467$	11.1324	0.515144	0.257572	+	0.966259i	$0.417078\pi$
$468$	0	0
$469$	−1.66189	−0.0767391
$470$	0	0
$471$	4.10886	0.189326
$472$	0	0
$473$	−9.32144	−0.428600
$474$	0	0
$475$	−15.6156	−0.716492
$476$	0	0
$477$	−9.98218	−0.457053
$478$	0	0
$479$	−37.2308	−1.70112	−0.850558	−	0.525881i	$-0.823736\pi$
$479$	−37.2308	−1.70112	−0.850558	+	0.525881i	$0.823736\pi$
$480$	0	0
$481$	−69.2099	−3.15570
$482$	0	0
$483$	5.20973	0.237051
$484$	0	0
$485$	56.3081	2.55682
$486$	0	0
$487$	−14.4584	−0.655174	−0.327587	−	0.944821i	$-0.606235\pi$
$487$	−14.4584	−0.655174	−0.327587	+	0.944821i	$0.606235\pi$
$488$	0	0
$489$	1.46777	0.0663748
$490$	0	0
$491$	33.1195	1.49466	0.747331	−	0.664452i	$-0.231335\pi$
$491$	33.1195	1.49466	0.747331	+	0.664452i	$0.231335\pi$
$492$	0	0
$493$	40.4490	1.82173
$494$	0	0
$495$	7.09211	0.318767
$496$	0	0
$497$	4.68319	0.210070
$498$	0	0
$499$	19.5768	0.876379	0.438190	−	0.898883i	$-0.355620\pi$
$499$	19.5768	0.876379	0.438190	+	0.898883i	$0.355620\pi$
$500$	0	0
$501$	11.7060	0.522987
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	5.88473	0.261867
$506$	0	0
$507$	10.2953	0.457229
$508$	0	0
$509$	−10.7482	−0.476404	−0.238202	−	0.971216i	$-0.576558\pi$
$509$	−10.7482	−0.476404	−0.238202	+	0.971216i	$0.576558\pi$
$510$	0	0
$511$	−31.9820	−1.41480
$512$	0	0
$513$	6.97453	0.307933
$514$	0	0
$515$	−52.6209	−2.31876
$516$	0	0
$517$	−3.25105	−0.142981
$518$	0	0
$519$	−4.52337	−0.198554
$520$	0	0
$521$	9.61895	0.421414	0.210707	−	0.977549i	$-0.432423\pi$
$521$	9.61895	0.421414	0.210707	+	0.977549i	$0.432423\pi$
$522$	0	0
$523$	−37.3461	−1.63303	−0.816515	−	0.577324i	$-0.804097\pi$
$523$	−37.3461	−1.63303	−0.816515	+	0.577324i	$0.804097\pi$
$524$	0	0
$525$	−11.0263	−0.481228
$526$	0	0
$527$	−16.8938	−0.735905
$528$	0	0
$529$	−13.6752	−0.594574
$530$	0	0
$531$	27.5099	1.19383
$532$	0	0
$533$	7.30601	0.316459
$534$	0	0
$535$	18.2101	0.787293
$536$	0	0
$537$	−3.29703	−0.142277
$538$	0	0
$539$	3.46102	0.149077
$540$	0	0
$541$	13.0572	0.561372	0.280686	−	0.959800i	$-0.409438\pi$
$541$	13.0572	0.561372	0.280686	+	0.959800i	$0.409438\pi$
$542$	0	0
$543$	−10.3726	−0.445130
$544$	0	0
$545$	−19.6523	−0.841814
$546$	0	0
$547$	29.4516	1.25926	0.629629	−	0.776896i	$-0.283207\pi$
$547$	29.4516	1.25926	0.629629	+	0.776896i	$0.283207\pi$
$548$	0	0
$549$	3.15291	0.134563
$550$	0	0
$551$	15.2099	0.647964
$552$	0	0
$553$	−15.3553	−0.652973
$554$	0	0
$555$	20.3324	0.863064
$556$	0	0
$557$	34.0369	1.44219	0.721095	−	0.692836i	$-0.243639\pi$
$557$	34.0369	1.44219	0.721095	+	0.692836i	$0.243639\pi$
$558$	0	0
$559$	−70.7715	−2.99331
$560$	0	0
$561$	−2.46003	−0.103862
$562$	0	0
$563$	−46.4046	−1.95572	−0.977860	−	0.209260i	$-0.932895\pi$
$563$	−46.4046	−1.95572	−0.977860	+	0.209260i	$0.932895\pi$
$564$	0	0
$565$	48.9110	2.05770
$566$	0	0
$567$	−23.0783	−0.969199
$568$	0	0
$569$	−11.2577	−0.471945	−0.235973	−	0.971760i	$-0.575828\pi$
$569$	−11.2577	−0.471945	−0.235973	+	0.971760i	$0.575828\pi$
$570$	0	0
$571$	−7.62598	−0.319137	−0.159569	−	0.987187i	$-0.551010\pi$
$571$	−7.62598	−0.319137	−0.159569	+	0.987187i	$0.551010\pi$
$572$	0	0
$573$	−0.772904	−0.0322885
$574$	0	0
$575$	−19.7358	−0.823040
$576$	0	0
$577$	−0.918004	−0.0382170	−0.0191085	−	0.999817i	$-0.506083\pi$
$577$	−0.918004	−0.0382170	−0.0191085	+	0.999817i	$0.506083\pi$
$578$	0	0
$579$	−10.1118	−0.420231
$580$	0	0
$581$	−27.6478	−1.14702
$582$	0	0
$583$	2.76942	0.114698
$584$	0	0
$585$	53.8457	2.22624
$586$	0	0
$587$	−30.3074	−1.25092	−0.625460	−	0.780256i	$-0.715089\pi$
$587$	−30.3074	−1.25092	−0.625460	+	0.780256i	$0.715089\pi$
$588$	0	0
$589$	−6.35253	−0.261751
$590$	0	0
$591$	2.21967	0.0913052
$592$	0	0
$593$	−32.5254	−1.33566	−0.667828	−	0.744315i	$-0.732776\pi$
$593$	−32.5254	−1.33566	−0.667828	+	0.744315i	$0.732776\pi$
$594$	0	0
$595$	−73.9021	−3.02969
$596$	0	0
$597$	1.63998	0.0671200
$598$	0	0
$599$	15.6587	0.639797	0.319899	−	0.947452i	$-0.396351\pi$
$599$	15.6587	0.639797	0.319899	+	0.947452i	$0.396351\pi$
$600$	0	0
$601$	−39.8042	−1.62365	−0.811823	−	0.583904i	$-0.801525\pi$
$601$	−39.8042	−1.62365	−0.811823	+	0.583904i	$0.801525\pi$
$602$	0	0
$603$	−1.34425	−0.0547422
$604$	0	0
$605$	35.2752	1.43414
$606$	0	0
$607$	10.0370	0.407388	0.203694	−	0.979035i	$-0.434705\pi$
$607$	10.0370	0.407388	0.203694	+	0.979035i	$0.434705\pi$
$608$	0	0
$609$	10.7399	0.435202
$610$	0	0
$611$	−24.6831	−0.998570
$612$	0	0
$613$	−33.3936	−1.34875	−0.674377	−	0.738387i	$-0.735588\pi$
$613$	−33.3936	−1.34875	−0.674377	+	0.738387i	$0.735588\pi$
$614$	0	0
$615$	−2.14636	−0.0865495
$616$	0	0
$617$	−32.6536	−1.31459	−0.657293	−	0.753635i	$-0.728299\pi$
$617$	−32.6536	−1.31459	−0.657293	+	0.753635i	$0.728299\pi$
$618$	0	0
$619$	−36.5567	−1.46934	−0.734668	−	0.678427i	$-0.762662\pi$
$619$	−36.5567	−1.46934	−0.734668	+	0.678427i	$0.762662\pi$
$620$	0	0
$621$	8.81479	0.353725
$622$	0	0
$623$	28.1919	1.12948
$624$	0	0
$625$	−15.5445	−0.621781
$626$	0	0
$627$	−0.925038	−0.0369425
$628$	0	0
$629$	76.8337	3.06356
$630$	0	0
$631$	−26.0937	−1.03877	−0.519387	−	0.854539i	$-0.673840\pi$
$631$	−26.0937	−1.03877	−0.519387	+	0.854539i	$0.673840\pi$
$632$	0	0
$633$	−4.97270	−0.197647
$634$	0	0
$635$	−38.1081	−1.51227
$636$	0	0
$637$	26.2772	1.04114
$638$	0	0
$639$	3.78809	0.149854
$640$	0	0
$641$	−49.1065	−1.93959	−0.969796	−	0.243918i	$-0.921567\pi$
$641$	−49.1065	−1.93959	−0.969796	+	0.243918i	$0.921567\pi$
$642$	0	0
$643$	−15.2894	−0.602953	−0.301477	−	0.953474i	$-0.597480\pi$
$643$	−15.2894	−0.602953	−0.301477	+	0.953474i	$0.597480\pi$
$644$	0	0
$645$	20.7912	0.818653
$646$	0	0
$647$	−12.4825	−0.490736	−0.245368	−	0.969430i	$-0.578909\pi$
$647$	−12.4825	−0.490736	−0.245368	+	0.969430i	$0.578909\pi$
$648$	0	0
$649$	−7.63224	−0.299592
$650$	0	0
$651$	−4.48558	−0.175804
$652$	0	0
$653$	−34.6400	−1.35557	−0.677785	−	0.735260i	$-0.737060\pi$
$653$	−34.6400	−1.35557	−0.677785	+	0.735260i	$0.737060\pi$
$654$	0	0
$655$	−21.2426	−0.830018
$656$	0	0
$657$	−25.8692	−1.00926
$658$	0	0
$659$	−17.1379	−0.667596	−0.333798	−	0.942645i	$-0.608330\pi$
$659$	−17.1379	−0.667596	−0.333798	+	0.942645i	$0.608330\pi$
$660$	0	0
$661$	9.49574	0.369341	0.184671	−	0.982800i	$-0.440878\pi$
$661$	9.49574	0.369341	0.184671	+	0.982800i	$0.440878\pi$
$662$	0	0
$663$	−18.6774	−0.725369
$664$	0	0
$665$	−27.7892	−1.07762
$666$	0	0
$667$	19.2231	0.744322
$668$	0	0
$669$	−4.47642	−0.173068
$670$	0	0
$671$	−0.874731	−0.0337686
$672$	0	0
$673$	21.7805	0.839578	0.419789	−	0.907622i	$-0.362104\pi$
$673$	21.7805	0.839578	0.419789	+	0.907622i	$0.362104\pi$
$674$	0	0
$675$	−18.6564	−0.718084
$676$	0	0
$677$	1.08553	0.0417203	0.0208602	−	0.999782i	$-0.493360\pi$
$677$	1.08553	0.0417203	0.0208602	+	0.999782i	$0.493360\pi$
$678$	0	0
$679$	56.4970	2.16815
$680$	0	0
$681$	4.78085	0.183203
$682$	0	0
$683$	35.4304	1.35571	0.677853	−	0.735198i	$-0.262911\pi$
$683$	35.4304	1.35571	0.677853	+	0.735198i	$0.262911\pi$
$684$	0	0
$685$	9.21780	0.352194
$686$	0	0
$687$	4.39775	0.167785
$688$	0	0
$689$	21.0263	0.801040
$690$	0	0
$691$	−0.222737	−0.00847333	−0.00423666	−	0.999991i	$-0.501349\pi$
$691$	−0.222737	−0.00847333	−0.00423666	+	0.999991i	$0.501349\pi$
$692$	0	0
$693$	7.11590	0.270311
$694$	0	0
$695$	−66.5836	−2.52566
$696$	0	0
$697$	−8.11081	−0.307219
$698$	0	0
$699$	−3.47241	−0.131339
$700$	0	0
$701$	−25.5391	−0.964599	−0.482299	−	0.876006i	$-0.660198\pi$
$701$	−25.5391	−0.964599	−0.482299	+	0.876006i	$0.660198\pi$
$702$	0	0
$703$	28.8916	1.08967
$704$	0	0
$705$	7.25138	0.273103
$706$	0	0
$707$	5.90447	0.222060
$708$	0	0
$709$	7.71954	0.289913	0.144957	−	0.989438i	$-0.453696\pi$
$709$	7.71954	0.289913	0.144957	+	0.989438i	$0.453696\pi$
$710$	0	0
$711$	−12.4204	−0.465801
$712$	0	0
$713$	−8.02867	−0.300676
$714$	0	0
$715$	−14.9387	−0.558677
$716$	0	0
$717$	12.2586	0.457808
$718$	0	0
$719$	−15.8737	−0.591987	−0.295994	−	0.955190i	$-0.595651\pi$
$719$	−15.8737	−0.591987	−0.295994	+	0.955190i	$0.595651\pi$
$720$	0	0
$721$	−52.7974	−1.96628
$722$	0	0
$723$	5.50170	0.204610
$724$	0	0
$725$	−40.6854	−1.51102
$726$	0	0
$727$	−28.8589	−1.07032	−0.535158	−	0.844752i	$-0.679748\pi$
$727$	−28.8589	−1.07032	−0.535158	+	0.844752i	$0.679748\pi$
$728$	0	0
$729$	−14.3182	−0.530303
$730$	0	0
$731$	78.5673	2.90592
$732$	0	0
$733$	16.4300	0.606857	0.303429	−	0.952854i	$-0.401869\pi$
$733$	16.4300	0.606857	0.303429	+	0.952854i	$0.401869\pi$
$734$	0	0
$735$	−7.71970	−0.284746
$736$	0	0
$737$	0.372944	0.0137376
$738$	0	0
$739$	5.63589	0.207320	0.103660	−	0.994613i	$-0.466945\pi$
$739$	5.63589	0.207320	0.103660	+	0.994613i	$0.466945\pi$
$740$	0	0
$741$	−7.02320	−0.258004
$742$	0	0
$743$	7.19423	0.263931	0.131965	−	0.991254i	$-0.457871\pi$
$743$	7.19423	0.263931	0.131965	+	0.991254i	$0.457871\pi$
$744$	0	0
$745$	78.6740	2.88239
$746$	0	0
$747$	−22.3634	−0.818235
$748$	0	0
$749$	18.2712	0.667616
$750$	0	0
$751$	−36.4126	−1.32872	−0.664358	−	0.747415i	$-0.731295\pi$
$751$	−36.4126	−1.32872	−0.664358	+	0.747415i	$0.731295\pi$
$752$	0	0
$753$	−8.69590	−0.316896
$754$	0	0
$755$	66.3321	2.41407
$756$	0	0
$757$	−3.77942	−0.137365	−0.0686826	−	0.997639i	$-0.521880\pi$
$757$	−3.77942	−0.137365	−0.0686826	+	0.997639i	$0.521880\pi$
$758$	0	0
$759$	−1.16911	−0.0424361
$760$	0	0
$761$	29.0487	1.05301	0.526507	−	0.850171i	$-0.323501\pi$
$761$	29.0487	1.05301	0.526507	+	0.850171i	$0.323501\pi$
$762$	0	0
$763$	−19.7183	−0.713849
$764$	0	0
$765$	−59.7771	−2.16124
$766$	0	0
$767$	−57.9466	−2.09233
$768$	0	0
$769$	10.8125	0.389910	0.194955	−	0.980812i	$-0.437544\pi$
$769$	10.8125	0.389910	0.194955	+	0.980812i	$0.437544\pi$
$770$	0	0
$771$	−8.59822	−0.309657
$772$	0	0
$773$	44.1643	1.58848	0.794239	−	0.607605i	$-0.207870\pi$
$773$	44.1643	1.58848	0.794239	+	0.607605i	$0.207870\pi$
$774$	0	0
$775$	16.9925	0.610390
$776$	0	0
$777$	20.4006	0.731869
$778$	0	0
$779$	−3.04989	−0.109274
$780$	0	0
$781$	−1.05095	−0.0376060
$782$	0	0
$783$	18.1717	0.649404
$784$	0	0
$785$	−27.6999	−0.988653
$786$	0	0
$787$	−10.5035	−0.374409	−0.187205	−	0.982321i	$-0.559943\pi$
$787$	−10.5035	−0.374409	−0.187205	+	0.982321i	$0.559943\pi$
$788$	0	0
$789$	9.32966	0.332145
$790$	0	0
$791$	49.0751	1.74491
$792$	0	0
$793$	−6.64125	−0.235838
$794$	0	0
$795$	−6.17711	−0.219079
$796$	0	0
$797$	33.2294	1.17705	0.588524	−	0.808480i	$-0.299709\pi$
$797$	33.2294	1.17705	0.588524	+	0.808480i	$0.299709\pi$
$798$	0	0
$799$	27.4020	0.969414
$800$	0	0
$801$	22.8035	0.805722
$802$	0	0
$803$	7.17706	0.253273
$804$	0	0
$805$	−35.1215	−1.23787
$806$	0	0
$807$	7.45707	0.262501
$808$	0	0
$809$	−53.0401	−1.86479	−0.932395	−	0.361442i	$-0.882285\pi$
$809$	−53.0401	−1.86479	−0.932395	+	0.361442i	$0.882285\pi$
$810$	0	0
$811$	3.08523	0.108337	0.0541685	−	0.998532i	$-0.482749\pi$
$811$	3.08523	0.108337	0.0541685	+	0.998532i	$0.482749\pi$
$812$	0	0
$813$	8.93753	0.313453
$814$	0	0
$815$	−9.89498	−0.346606
$816$	0	0
$817$	29.5435	1.03360
$818$	0	0
$819$	54.0263	1.88783
$820$	0	0
$821$	−15.4321	−0.538584	−0.269292	−	0.963059i	$-0.586790\pi$
$821$	−15.4321	−0.538584	−0.269292	+	0.963059i	$0.586790\pi$
$822$	0	0
$823$	−1.59217	−0.0554994	−0.0277497	−	0.999615i	$-0.508834\pi$
$823$	−1.59217	−0.0554994	−0.0277497	+	0.999615i	$0.508834\pi$
$824$	0	0
$825$	2.47441	0.0861479
$826$	0	0
$827$	24.3464	0.846606	0.423303	−	0.905988i	$-0.360871\pi$
$827$	24.3464	0.846606	0.423303	+	0.905988i	$0.360871\pi$
$828$	0	0
$829$	17.8671	0.620549	0.310275	−	0.950647i	$-0.399579\pi$
$829$	17.8671	0.620549	0.310275	+	0.950647i	$0.399579\pi$
$830$	0	0
$831$	10.8158	0.375196
$832$	0	0
$833$	−29.1718	−1.01074
$834$	0	0
$835$	−78.9164	−2.73101
$836$	0	0
$837$	−7.58954	−0.262333
$838$	0	0
$839$	−19.8033	−0.683687	−0.341843	−	0.939757i	$-0.611051\pi$
$839$	−19.8033	−0.683687	−0.341843	+	0.939757i	$0.611051\pi$
$840$	0	0
$841$	10.6285	0.366499
$842$	0	0
$843$	−7.65072	−0.263505
$844$	0	0
$845$	−69.4057	−2.38763
$846$	0	0
$847$	35.3935	1.21613
$848$	0	0
$849$	11.9791	0.411123
$850$	0	0
$851$	36.5147	1.25171
$852$	0	0
$853$	−2.16443	−0.0741087	−0.0370544	−	0.999313i	$-0.511797\pi$
$853$	−2.16443	−0.0741087	−0.0370544	+	0.999313i	$0.511797\pi$
$854$	0	0
$855$	−22.4778	−0.768725
$856$	0	0
$857$	−40.3820	−1.37942	−0.689711	−	0.724085i	$-0.742262\pi$
$857$	−40.3820	−1.37942	−0.689711	+	0.724085i	$0.742262\pi$
$858$	0	0
$859$	41.8914	1.42932	0.714659	−	0.699473i	$-0.246582\pi$
$859$	41.8914	1.42932	0.714659	+	0.699473i	$0.246582\pi$
$860$	0	0
$861$	−2.15356	−0.0733930
$862$	0	0
$863$	−11.0329	−0.375564	−0.187782	−	0.982211i	$-0.560130\pi$
$863$	−11.0329	−0.375564	−0.187782	+	0.982211i	$0.560130\pi$
$864$	0	0
$865$	30.4943	1.03684
$866$	0	0
$867$	12.1971	0.414234
$868$	0	0
$869$	3.44587	0.116893
$870$	0	0
$871$	2.83152	0.0959424
$872$	0	0
$873$	45.6986	1.54666
$874$	0	0
$875$	16.8268	0.568849
$876$	0	0
$877$	−50.1302	−1.69278	−0.846389	−	0.532566i	$-0.821228\pi$
$877$	−50.1302	−1.69278	−0.846389	+	0.532566i	$0.821228\pi$
$878$	0	0
$879$	12.4027	0.418332
$880$	0	0
$881$	43.7445	1.47379	0.736894	−	0.676008i	$-0.236292\pi$
$881$	43.7445	1.47379	0.736894	+	0.676008i	$0.236292\pi$
$882$	0	0
$883$	6.85634	0.230734	0.115367	−	0.993323i	$-0.463196\pi$
$883$	6.85634	0.230734	0.115367	+	0.993323i	$0.463196\pi$
$884$	0	0
$885$	17.0235	0.572239
$886$	0	0
$887$	−47.0479	−1.57971	−0.789857	−	0.613291i	$-0.789845\pi$
$887$	−47.0479	−1.57971	−0.789857	+	0.613291i	$0.789845\pi$
$888$	0	0
$889$	−38.2359	−1.28239
$890$	0	0
$891$	5.17899	0.173503
$892$	0	0
$893$	10.3039	0.344808
$894$	0	0
$895$	22.2270	0.742966
$896$	0	0
$897$	−8.87630	−0.296371
$898$	0	0
$899$	−16.5511	−0.552010
$900$	0	0
$901$	−23.3425	−0.777652
$902$	0	0
$903$	20.8609	0.694209
$904$	0	0
$905$	69.9268	2.32445
$906$	0	0
$907$	16.6330	0.552289	0.276145	−	0.961116i	$-0.410943\pi$
$907$	16.6330	0.552289	0.276145	+	0.961116i	$0.410943\pi$
$908$	0	0
$909$	4.77594	0.158408
$910$	0	0
$911$	−58.0134	−1.92207	−0.961034	−	0.276430i	$-0.910849\pi$
$911$	−58.0134	−1.92207	−0.961034	+	0.276430i	$0.910849\pi$
$912$	0	0
$913$	6.20442	0.205336
$914$	0	0
$915$	1.95106	0.0645002
$916$	0	0
$917$	−21.3139	−0.703846
$918$	0	0
$919$	8.66336	0.285778	0.142889	−	0.989739i	$-0.454361\pi$
$919$	8.66336	0.285778	0.142889	+	0.989739i	$0.454361\pi$
$920$	0	0
$921$	13.5139	0.445298
$922$	0	0
$923$	−7.97918	−0.262638
$924$	0	0
$925$	−77.2828	−2.54104
$926$	0	0
$927$	−42.7062	−1.40265
$928$	0	0
$929$	17.1657	0.563188	0.281594	−	0.959534i	$-0.409137\pi$
$929$	17.1657	0.563188	0.281594	+	0.959534i	$0.409137\pi$
$930$	0	0
$931$	−10.9694	−0.359507
$932$	0	0
$933$	0.723178	0.0236758
$934$	0	0
$935$	16.5843	0.542365
$936$	0	0
$937$	−32.9402	−1.07611	−0.538054	−	0.842910i	$-0.680840\pi$
$937$	−32.9402	−1.07611	−0.538054	+	0.842910i	$0.680840\pi$
$938$	0	0
$939$	−0.416790	−0.0136014
$940$	0	0
$941$	9.34516	0.304643	0.152322	−	0.988331i	$-0.451325\pi$
$941$	9.34516	0.304643	0.152322	+	0.988331i	$0.451325\pi$
$942$	0	0
$943$	−3.85461	−0.125523
$944$	0	0
$945$	−33.2005	−1.08001
$946$	0	0
$947$	22.8779	0.743432	0.371716	−	0.928346i	$-0.378769\pi$
$947$	22.8779	0.743432	0.371716	+	0.928346i	$0.378769\pi$
$948$	0	0
$949$	54.4907	1.76884
$950$	0	0
$951$	−8.94743	−0.290140
$952$	0	0
$953$	−0.362386	−0.0117388	−0.00586942	−	0.999983i	$-0.501868\pi$
$953$	−0.362386	−0.0117388	−0.00586942	+	0.999983i	$0.501868\pi$
$954$	0	0
$955$	5.21054	0.168609
$956$	0	0
$957$	−2.41013	−0.0779084
$958$	0	0
$959$	9.24871	0.298657
$960$	0	0
$961$	−24.0873	−0.777010
$962$	0	0
$963$	14.7790	0.476247
$964$	0	0
$965$	68.1687	2.19443
$966$	0	0
$967$	3.81693	0.122744	0.0613722	−	0.998115i	$-0.480452\pi$
$967$	3.81693	0.122744	0.0613722	+	0.998115i	$0.480452\pi$
$968$	0	0
$969$	7.79685	0.250471
$970$	0	0
$971$	17.2799	0.554538	0.277269	−	0.960792i	$-0.410571\pi$
$971$	17.2799	0.554538	0.277269	+	0.960792i	$0.410571\pi$
$972$	0	0
$973$	−66.8069	−2.14173
$974$	0	0
$975$	18.7865	0.601651
$976$	0	0
$977$	−40.2803	−1.28868	−0.644340	−	0.764739i	$-0.722868\pi$
$977$	−40.2803	−1.28868	−0.644340	+	0.764739i	$0.722868\pi$
$978$	0	0
$979$	−6.32652	−0.202196
$980$	0	0
$981$	−15.9495	−0.509228
$982$	0	0
$983$	−3.11974	−0.0995041	−0.0497521	−	0.998762i	$-0.515843\pi$
$983$	−3.11974	−0.0995041	−0.0497521	+	0.998762i	$0.515843\pi$
$984$	0	0
$985$	−14.9640	−0.476792
$986$	0	0
$987$	7.27570	0.231588
$988$	0	0
$989$	37.3386	1.18730
$990$	0	0
$991$	12.2745	0.389912	0.194956	−	0.980812i	$-0.437544\pi$
$991$	12.2745	0.389912	0.194956	+	0.980812i	$0.437544\pi$
$992$	0	0
$993$	−17.6737	−0.560857
$994$	0	0
$995$	−11.0560	−0.350498
$996$	0	0
$997$	−33.0446	−1.04653	−0.523267	−	0.852169i	$-0.675287\pi$
$997$	−33.0446	−1.04653	−0.523267	+	0.852169i	$0.675287\pi$
$998$	0	0
$999$	34.5176	1.09209

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.w.1.15		29
4.3	odd	2		4024.2.a.e.1.15	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.e.1.15	✓	29	4.3	odd	2
8048.2.a.w.1.15		29	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)

\(f(q)\)	\(=\)	\(q+0.502218 q^{3} -3.38571 q^{5} -3.39706 q^{7} -2.74778 q^{9} +O(q^{10})\)	\(q+0.502218 q^{3} -3.38571 q^{5} -3.39706 q^{7} -2.74778 q^{9} +0.762333 q^{11} +5.78788 q^{13} -1.70036 q^{15} -6.42545 q^{17} -2.41615 q^{19} -1.70606 q^{21} -3.05365 q^{23} +6.46301 q^{25} -2.88664 q^{27} -6.29511 q^{29} +2.62920 q^{31} +0.382857 q^{33} +11.5015 q^{35} -11.9577 q^{37} +2.90678 q^{39} +1.26229 q^{41} -12.2275 q^{43} +9.30317 q^{45} -4.26461 q^{47} +4.54003 q^{49} -3.22698 q^{51} +3.63282 q^{53} -2.58103 q^{55} -1.21343 q^{57} -10.0117 q^{59} -1.14744 q^{61} +9.33437 q^{63} -19.5961 q^{65} +0.489215 q^{67} -1.53360 q^{69} -1.37860 q^{71} +9.41461 q^{73} +3.24584 q^{75} -2.58969 q^{77} +4.52016 q^{79} +6.79361 q^{81} +8.13873 q^{83} +21.7547 q^{85} -3.16152 q^{87} -8.29889 q^{89} -19.6618 q^{91} +1.32043 q^{93} +8.18036 q^{95} -16.6311 q^{97} -2.09472 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9}+O(q^{10}) \) 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9	\( 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9} + 27 q^{11} + 16 q^{13} + 14 q^{15} - 15 q^{17} + 14 q^{19} + q^{21} + 25 q^{23} + 21 q^{25} + 25 q^{27} - 13 q^{29} + 27 q^{31} - 9 q^{33} + 29 q^{35} + 35 q^{37} + 38 q^{39} - 30 q^{41} + 38 q^{43} + q^{45} + 35 q^{47} + 14 q^{49} + 21 q^{51} + 2 q^{53} + 25 q^{55} - 25 q^{57} + 40 q^{59} + 10 q^{61} + 56 q^{63} - 50 q^{65} + 31 q^{67} + 11 q^{69} + 65 q^{71} - 23 q^{73} + 32 q^{75} + 13 q^{77} + 44 q^{79} - 7 q^{81} + 41 q^{83} + 26 q^{85} + 25 q^{87} - 48 q^{89} + 44 q^{91} + 25 q^{93} + 75 q^{95} - 18 q^{97} + 80 q^{99}+O(q^{100}) \) 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9 + 27 * q^11 + 16 * q^13 + 14 * q^15 - 15 * q^17 + 14 * q^19 + q^21 + 25 * q^23 + 21 * q^25 + 25 * q^27 - 13 * q^29 + 27 * q^31 - 9 * q^33 + 29 * q^35 + 35 * q^37 + 38 * q^39 - 30 * q^41 + 38 * q^43 + q^45 + 35 * q^47 + 14 * q^49 + 21 * q^51 + 2 * q^53 + 25 * q^55 - 25 * q^57 + 40 * q^59 + 10 * q^61 + 56 * q^63 - 50 * q^65 + 31 * q^67 + 11 * q^69 + 65 * q^71 - 23 * q^73 + 32 * q^75 + 13 * q^77 + 44 * q^79 - 7 * q^81 + 41 * q^83 + 26 * q^85 + 25 * q^87 - 48 * q^89 + 44 * q^91 + 25 * q^93 + 75 * q^95 - 18 * q^97 + 80 * q^99

Introduction

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