LMFDB - Embedded newform 8048.2.a.t.1.3

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:	$21$
Twist minimal:	no (minimal twist has level 2012)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
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)

$f(q)$	$=$	$q-2.09799 q^{3} -0.681202 q^{5} +4.42487 q^{7} +1.40156 q^{9} +O(q^{10})$	$q-2.09799 q^{3} -0.681202 q^{5} +4.42487 q^{7} +1.40156 q^{9} -3.45149 q^{11} +0.452231 q^{13} +1.42915 q^{15} -4.79463 q^{17} +2.70874 q^{19} -9.28334 q^{21} -1.65478 q^{23} -4.53596 q^{25} +3.35351 q^{27} -5.71291 q^{29} -1.44995 q^{31} +7.24118 q^{33} -3.01423 q^{35} -5.71446 q^{37} -0.948776 q^{39} +3.22730 q^{41} -3.25043 q^{43} -0.954746 q^{45} +12.7465 q^{47} +12.5795 q^{49} +10.0591 q^{51} -6.28520 q^{53} +2.35116 q^{55} -5.68290 q^{57} -3.94534 q^{59} +10.6673 q^{61} +6.20173 q^{63} -0.308060 q^{65} -3.81534 q^{67} +3.47171 q^{69} +5.32500 q^{71} +5.48753 q^{73} +9.51641 q^{75} -15.2724 q^{77} +7.64385 q^{79} -11.2403 q^{81} +12.3732 q^{83} +3.26611 q^{85} +11.9856 q^{87} -6.80548 q^{89} +2.00106 q^{91} +3.04198 q^{93} -1.84520 q^{95} -8.44493 q^{97} -4.83747 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9}+O(q^{10}) $ 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9	$ 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9} + 9 q^{11} - 16 q^{13} + 22 q^{15} + q^{17} + 12 q^{19} - 2 q^{21} + 22 q^{23} + 18 q^{25} + 43 q^{27} - 13 q^{29} + 28 q^{31} + 3 q^{33} + 12 q^{35} - 39 q^{37} + 11 q^{39} + 4 q^{41} + 50 q^{43} - 6 q^{45} + 27 q^{47} + 16 q^{49} + 37 q^{51} - 24 q^{53} + 49 q^{55} - q^{57} + 22 q^{59} - 22 q^{61} + 49 q^{63} - 14 q^{65} + 62 q^{67} - 17 q^{69} + 21 q^{71} - 6 q^{73} + 52 q^{75} - 24 q^{77} + 65 q^{79} + 29 q^{81} + 18 q^{83} - 54 q^{85} + 31 q^{87} + q^{89} + 45 q^{91} - 26 q^{93} + 53 q^{95} - 2 q^{97} + 58 q^{99}+O(q^{100}) $ 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9 + 9 * q^11 - 16 * q^13 + 22 * q^15 + q^17 + 12 * q^19 - 2 * q^21 + 22 * q^23 + 18 * q^25 + 43 * q^27 - 13 * q^29 + 28 * q^31 + 3 * q^33 + 12 * q^35 - 39 * q^37 + 11 * q^39 + 4 * q^41 + 50 * q^43 - 6 * q^45 + 27 * q^47 + 16 * q^49 + 37 * q^51 - 24 * q^53 + 49 * q^55 - q^57 + 22 * q^59 - 22 * q^61 + 49 * q^63 - 14 * q^65 + 62 * q^67 - 17 * q^69 + 21 * q^71 - 6 * q^73 + 52 * q^75 - 24 * q^77 + 65 * q^79 + 29 * q^81 + 18 * q^83 - 54 * q^85 + 31 * q^87 + q^89 + 45 * q^91 - 26 * q^93 + 53 * q^95 - 2 * q^97 + 58 * 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
$3$	−2.09799	−1.21128	−0.605638	−	0.795741i	$-0.707082\pi$
$3$	−2.09799	−1.21128	−0.605638	+	0.795741i	$0.707082\pi$
$4$	0	0
$5$	−0.681202	−0.304643	−0.152321	−	0.988331i	$-0.548675\pi$
$5$	−0.681202	−0.304643	−0.152321	+	0.988331i	$0.548675\pi$
$6$	0	0
$7$	4.42487	1.67244	0.836222	−	0.548391i	$-0.184759\pi$
$7$	4.42487	1.67244	0.836222	+	0.548391i	$0.184759\pi$
$8$	0	0
$9$	1.40156	0.467187
$10$	0	0
$11$	−3.45149	−1.04066	−0.520331	−	0.853965i	$-0.674191\pi$
$11$	−3.45149	−1.04066	−0.520331	+	0.853965i	$0.674191\pi$
$12$	0	0
$13$	0.452231	0.125426	0.0627131	−	0.998032i	$-0.480025\pi$
$13$	0.452231	0.125426	0.0627131	+	0.998032i	$0.480025\pi$
$14$	0	0
$15$	1.42915	0.369006
$16$	0	0
$17$	−4.79463	−1.16287	−0.581435	−	0.813593i	$-0.697508\pi$
$17$	−4.79463	−1.16287	−0.581435	+	0.813593i	$0.697508\pi$
$18$	0	0
$19$	2.70874	0.621427	0.310713	−	0.950504i	$-0.399432\pi$
$19$	2.70874	0.621427	0.310713	+	0.950504i	$0.399432\pi$
$20$	0	0
$21$	−9.28334	−2.02579
$22$	0	0
$23$	−1.65478	−0.345045	−0.172522	−	0.985006i	$-0.555192\pi$
$23$	−1.65478	−0.345045	−0.172522	+	0.985006i	$0.555192\pi$
$24$	0	0
$25$	−4.53596	−0.907193
$26$	0	0
$27$	3.35351	0.645383
$28$	0	0
$29$	−5.71291	−1.06086	−0.530430	−	0.847729i	$-0.677970\pi$
$29$	−5.71291	−1.06086	−0.530430	+	0.847729i	$0.677970\pi$
$30$	0	0
$31$	−1.44995	−0.260418	−0.130209	−	0.991487i	$-0.541565\pi$
$31$	−1.44995	−0.260418	−0.130209	+	0.991487i	$0.541565\pi$
$32$	0	0
$33$	7.24118	1.26053
$34$	0	0
$35$	−3.01423	−0.509498
$36$	0	0
$37$	−5.71446	−0.939451	−0.469726	−	0.882812i	$-0.655647\pi$
$37$	−5.71446	−0.939451	−0.469726	+	0.882812i	$0.655647\pi$
$38$	0	0
$39$	−0.948776	−0.151926
$40$	0	0
$41$	3.22730	0.504020	0.252010	−	0.967725i	$-0.418908\pi$
$41$	3.22730	0.504020	0.252010	+	0.967725i	$0.418908\pi$
$42$	0	0
$43$	−3.25043	−0.495686	−0.247843	−	0.968800i	$-0.579722\pi$
$43$	−3.25043	−0.495686	−0.247843	+	0.968800i	$0.579722\pi$
$44$	0	0
$45$	−0.954746	−0.142325
$46$	0	0
$47$	12.7465	1.85927	0.929636	−	0.368478i	$-0.120121\pi$
$47$	12.7465	1.85927	0.929636	+	0.368478i	$0.120121\pi$
$48$	0	0
$49$	12.5795	1.79707
$50$	0	0
$51$	10.0591	1.40855
$52$	0	0
$53$	−6.28520	−0.863338	−0.431669	−	0.902032i	$-0.642075\pi$
$53$	−6.28520	−0.863338	−0.431669	+	0.902032i	$0.642075\pi$
$54$	0	0
$55$	2.35116	0.317030
$56$	0	0
$57$	−5.68290	−0.752719
$58$	0	0
$59$	−3.94534	−0.513639	−0.256820	−	0.966459i	$-0.582675\pi$
$59$	−3.94534	−0.513639	−0.256820	+	0.966459i	$0.582675\pi$
$60$	0	0
$61$	10.6673	1.36581	0.682907	−	0.730506i	$-0.260716\pi$
$61$	10.6673	1.36581	0.682907	+	0.730506i	$0.260716\pi$
$62$	0	0
$63$	6.20173	0.781345
$64$	0	0
$65$	−0.308060	−0.0382102
$66$	0	0
$67$	−3.81534	−0.466118	−0.233059	−	0.972463i	$-0.574873\pi$
$67$	−3.81534	−0.466118	−0.233059	+	0.972463i	$0.574873\pi$
$68$	0	0
$69$	3.47171	0.417944
$70$	0	0
$71$	5.32500	0.631962	0.315981	−	0.948766i	$-0.397666\pi$
$71$	5.32500	0.631962	0.315981	+	0.948766i	$0.397666\pi$
$72$	0	0
$73$	5.48753	0.642266	0.321133	−	0.947034i	$-0.395936\pi$
$73$	5.48753	0.642266	0.321133	+	0.947034i	$0.395936\pi$
$74$	0	0
$75$	9.51641	1.09886
$76$	0	0
$77$	−15.2724	−1.74045
$78$	0	0
$79$	7.64385	0.860000	0.430000	−	0.902829i	$-0.358514\pi$
$79$	7.64385	0.860000	0.430000	+	0.902829i	$0.358514\pi$
$80$	0	0
$81$	−11.2403	−1.24892
$82$	0	0
$83$	12.3732	1.35813	0.679066	−	0.734077i	$-0.262385\pi$
$83$	12.3732	1.35813	0.679066	+	0.734077i	$0.262385\pi$
$84$	0	0
$85$	3.26611	0.354260
$86$	0	0
$87$	11.9856	1.28499
$88$	0	0
$89$	−6.80548	−0.721379	−0.360690	−	0.932686i	$-0.617459\pi$
$89$	−6.80548	−0.721379	−0.360690	+	0.932686i	$0.617459\pi$
$90$	0	0
$91$	2.00106	0.209768
$92$	0	0
$93$	3.04198	0.315438
$94$	0	0
$95$	−1.84520	−0.189313
$96$	0	0
$97$	−8.44493	−0.857452	−0.428726	−	0.903435i	$-0.641037\pi$
$97$	−8.44493	−0.857452	−0.428726	+	0.903435i	$0.641037\pi$
$98$	0	0
$99$	−4.83747	−0.486184
$100$	0	0
$101$	−7.02239	−0.698754	−0.349377	−	0.936982i	$-0.613607\pi$
$101$	−7.02239	−0.698754	−0.349377	+	0.936982i	$0.613607\pi$
$102$	0	0
$103$	7.54750	0.743678	0.371839	−	0.928297i	$-0.378727\pi$
$103$	7.54750	0.743678	0.371839	+	0.928297i	$0.378727\pi$
$104$	0	0
$105$	6.32383	0.617142
$106$	0	0
$107$	−3.22223	−0.311505	−0.155753	−	0.987796i	$-0.549780\pi$
$107$	−3.22223	−0.311505	−0.155753	+	0.987796i	$0.549780\pi$
$108$	0	0
$109$	0.599897	0.0574597	0.0287299	−	0.999587i	$-0.490854\pi$
$109$	0.599897	0.0574597	0.0287299	+	0.999587i	$0.490854\pi$
$110$	0	0
$111$	11.9889	1.13793
$112$	0	0
$113$	12.2557	1.15292	0.576460	−	0.817125i	$-0.304433\pi$
$113$	12.2557	1.15292	0.576460	+	0.817125i	$0.304433\pi$
$114$	0	0
$115$	1.12724	0.105115
$116$	0	0
$117$	0.633829	0.0585975
$118$	0	0
$119$	−21.2156	−1.94483
$120$	0	0
$121$	0.912759	0.0829780
$122$	0	0
$123$	−6.77085	−0.610507
$124$	0	0
$125$	6.49591	0.581012
$126$	0	0
$127$	6.08450	0.539912	0.269956	−	0.962873i	$-0.412991\pi$
$127$	6.08450	0.539912	0.269956	+	0.962873i	$0.412991\pi$
$128$	0	0
$129$	6.81937	0.600412
$130$	0	0
$131$	−12.9100	−1.12795	−0.563976	−	0.825791i	$-0.690729\pi$
$131$	−12.9100	−1.12795	−0.563976	+	0.825791i	$0.690729\pi$
$132$	0	0
$133$	11.9858	1.03930
$134$	0	0
$135$	−2.28441	−0.196611
$136$	0	0
$137$	5.12526	0.437881	0.218940	−	0.975738i	$-0.429740\pi$
$137$	5.12526	0.437881	0.218940	+	0.975738i	$0.429740\pi$
$138$	0	0
$139$	−1.38025	−0.117071	−0.0585355	−	0.998285i	$-0.518643\pi$
$139$	−1.38025	−0.117071	−0.0585355	+	0.998285i	$0.518643\pi$
$140$	0	0
$141$	−26.7421	−2.25209
$142$	0	0
$143$	−1.56087	−0.130526
$144$	0	0
$145$	3.89164	0.323183
$146$	0	0
$147$	−26.3917	−2.17675
$148$	0	0
$149$	7.39463	0.605791	0.302896	−	0.953024i	$-0.402047\pi$
$149$	7.39463	0.605791	0.302896	+	0.953024i	$0.402047\pi$
$150$	0	0
$151$	−4.88443	−0.397489	−0.198744	−	0.980051i	$-0.563686\pi$
$151$	−4.88443	−0.397489	−0.198744	+	0.980051i	$0.563686\pi$
$152$	0	0
$153$	−6.71997	−0.543278
$154$	0	0
$155$	0.987707	0.0793345
$156$	0	0
$157$	−8.57860	−0.684647	−0.342323	−	0.939582i	$-0.611214\pi$
$157$	−8.57860	−0.684647	−0.342323	+	0.939582i	$0.611214\pi$
$158$	0	0
$159$	13.1863	1.04574
$160$	0	0
$161$	−7.32218	−0.577068
$162$	0	0
$163$	−14.9607	−1.17181	−0.585904	−	0.810380i	$-0.699261\pi$
$163$	−14.9607	−1.17181	−0.585904	+	0.810380i	$0.699261\pi$
$164$	0	0
$165$	−4.93271	−0.384011
$166$	0	0
$167$	−8.78758	−0.680003	−0.340002	−	0.940425i	$-0.610428\pi$
$167$	−8.78758	−0.680003	−0.340002	+	0.940425i	$0.610428\pi$
$168$	0	0
$169$	−12.7955	−0.984268
$170$	0	0
$171$	3.79646	0.290323
$172$	0	0
$173$	9.55449	0.726414	0.363207	−	0.931708i	$-0.381682\pi$
$173$	9.55449	0.726414	0.363207	+	0.931708i	$0.381682\pi$
$174$	0	0
$175$	−20.0711	−1.51723
$176$	0	0
$177$	8.27728	0.622158
$178$	0	0
$179$	−14.3379	−1.07167	−0.535834	−	0.844323i	$-0.680003\pi$
$179$	−14.3379	−1.07167	−0.535834	+	0.844323i	$0.680003\pi$
$180$	0	0
$181$	20.0550	1.49067	0.745337	−	0.666687i	$-0.232288\pi$
$181$	20.0550	1.49067	0.745337	+	0.666687i	$0.232288\pi$
$182$	0	0
$183$	−22.3800	−1.65438
$184$	0	0
$185$	3.89270	0.286197
$186$	0	0
$187$	16.5486	1.21015
$188$	0	0
$189$	14.8388	1.07937
$190$	0	0
$191$	19.4425	1.40681	0.703405	−	0.710789i	$-0.251662\pi$
$191$	19.4425	1.40681	0.703405	+	0.710789i	$0.251662\pi$
$192$	0	0
$193$	−7.34946	−0.529026	−0.264513	−	0.964382i	$-0.585211\pi$
$193$	−7.34946	−0.529026	−0.264513	+	0.964382i	$0.585211\pi$
$194$	0	0
$195$	0.646307	0.0462830
$196$	0	0
$197$	9.44045	0.672604	0.336302	−	0.941754i	$-0.390824\pi$
$197$	9.44045	0.672604	0.336302	+	0.941754i	$0.390824\pi$
$198$	0	0
$199$	18.9893	1.34611	0.673057	−	0.739590i	$-0.264981\pi$
$199$	18.9893	1.34611	0.673057	+	0.739590i	$0.264981\pi$
$200$	0	0
$201$	8.00454	0.564597
$202$	0	0
$203$	−25.2789	−1.77423
$204$	0	0
$205$	−2.19844	−0.153546
$206$	0	0
$207$	−2.31927	−0.161201
$208$	0	0
$209$	−9.34917	−0.646695
$210$	0	0
$211$	−3.85813	−0.265605	−0.132802	−	0.991143i	$-0.542398\pi$
$211$	−3.85813	−0.265605	−0.132802	+	0.991143i	$0.542398\pi$
$212$	0	0
$213$	−11.1718	−0.765479
$214$	0	0
$215$	2.21420	0.151007
$216$	0	0
$217$	−6.41584	−0.435535
$218$	0	0
$219$	−11.5128	−0.777961
$220$	0	0
$221$	−2.16828	−0.145854
$222$	0	0
$223$	25.7409	1.72374	0.861869	−	0.507131i	$-0.169294\pi$
$223$	25.7409	1.72374	0.861869	+	0.507131i	$0.169294\pi$
$224$	0	0
$225$	−6.35743	−0.423829
$226$	0	0
$227$	−9.84413	−0.653378	−0.326689	−	0.945132i	$-0.605933\pi$
$227$	−9.84413	−0.653378	−0.326689	+	0.945132i	$0.605933\pi$
$228$	0	0
$229$	16.4648	1.08802	0.544012	−	0.839078i	$-0.316905\pi$
$229$	16.4648	1.08802	0.544012	+	0.839078i	$0.316905\pi$
$230$	0	0
$231$	32.0413	2.10816
$232$	0	0
$233$	−4.59309	−0.300904	−0.150452	−	0.988617i	$-0.548073\pi$
$233$	−4.59309	−0.300904	−0.150452	+	0.988617i	$0.548073\pi$
$234$	0	0
$235$	−8.68296	−0.566414
$236$	0	0
$237$	−16.0367	−1.04170
$238$	0	0
$239$	11.0419	0.714243	0.357121	−	0.934058i	$-0.383758\pi$
$239$	11.0419	0.714243	0.357121	+	0.934058i	$0.383758\pi$
$240$	0	0
$241$	24.5917	1.58409	0.792046	−	0.610462i	$-0.209016\pi$
$241$	24.5917	1.58409	0.792046	+	0.610462i	$0.209016\pi$
$242$	0	0
$243$	13.5215	0.867407
$244$	0	0
$245$	−8.56917	−0.547464
$246$	0	0
$247$	1.22497	0.0779432
$248$	0	0
$249$	−25.9588	−1.64507
$250$	0	0
$251$	−11.7112	−0.739206	−0.369603	−	0.929190i	$-0.620506\pi$
$251$	−11.7112	−0.739206	−0.369603	+	0.929190i	$0.620506\pi$
$252$	0	0
$253$	5.71144	0.359075
$254$	0	0
$255$	−6.85227	−0.429106
$256$	0	0
$257$	−22.1770	−1.38337	−0.691683	−	0.722201i	$-0.743130\pi$
$257$	−22.1770	−1.38337	−0.691683	+	0.722201i	$0.743130\pi$
$258$	0	0
$259$	−25.2858	−1.57118
$260$	0	0
$261$	−8.00699	−0.495620
$262$	0	0
$263$	2.78255	0.171579	0.0857897	−	0.996313i	$-0.472659\pi$
$263$	2.78255	0.171579	0.0857897	+	0.996313i	$0.472659\pi$
$264$	0	0
$265$	4.28149	0.263010
$266$	0	0
$267$	14.2778	0.873789
$268$	0	0
$269$	13.4102	0.817637	0.408819	−	0.912616i	$-0.365941\pi$
$269$	13.4102	0.817637	0.408819	+	0.912616i	$0.365941\pi$
$270$	0	0
$271$	5.77693	0.350924	0.175462	−	0.984486i	$-0.443858\pi$
$271$	5.77693	0.350924	0.175462	+	0.984486i	$0.443858\pi$
$272$	0	0
$273$	−4.19821	−0.254087
$274$	0	0
$275$	15.6558	0.944081
$276$	0	0
$277$	−0.0438467	−0.00263449	−0.00131725	−	0.999999i	$-0.500419\pi$
$277$	−0.0438467	−0.00263449	−0.00131725	+	0.999999i	$0.500419\pi$
$278$	0	0
$279$	−2.03219	−0.121664
$280$	0	0
$281$	20.1218	1.20036	0.600182	−	0.799863i	$-0.295095\pi$
$281$	20.1218	1.20036	0.600182	+	0.799863i	$0.295095\pi$
$282$	0	0
$283$	−6.33197	−0.376397	−0.188198	−	0.982131i	$-0.560265\pi$
$283$	−6.33197	−0.376397	−0.188198	+	0.982131i	$0.560265\pi$
$284$	0	0
$285$	3.87120	0.229310
$286$	0	0
$287$	14.2804	0.842946
$288$	0	0
$289$	5.98851	0.352265
$290$	0	0
$291$	17.7174	1.03861
$292$	0	0
$293$	4.65151	0.271744	0.135872	−	0.990726i	$-0.456616\pi$
$293$	4.65151	0.271744	0.135872	+	0.990726i	$0.456616\pi$
$294$	0	0
$295$	2.68757	0.156476
$296$	0	0
$297$	−11.5746	−0.671626
$298$	0	0
$299$	−0.748341	−0.0432777
$300$	0	0
$301$	−14.3827	−0.829007
$302$	0	0
$303$	14.7329	0.846383
$304$	0	0
$305$	−7.26661	−0.416085
$306$	0	0
$307$	23.7296	1.35432	0.677161	−	0.735835i	$-0.263210\pi$
$307$	23.7296	1.35432	0.677161	+	0.735835i	$0.263210\pi$
$308$	0	0
$309$	−15.8346	−0.900798
$310$	0	0
$311$	−22.5736	−1.28003	−0.640017	−	0.768361i	$-0.721073\pi$
$311$	−22.5736	−1.28003	−0.640017	+	0.768361i	$0.721073\pi$
$312$	0	0
$313$	4.43396	0.250622	0.125311	−	0.992117i	$-0.460007\pi$
$313$	4.43396	0.250622	0.125311	+	0.992117i	$0.460007\pi$
$314$	0	0
$315$	−4.22463	−0.238031
$316$	0	0
$317$	−23.2316	−1.30481	−0.652407	−	0.757869i	$-0.726241\pi$
$317$	−23.2316	−1.30481	−0.652407	+	0.757869i	$0.726241\pi$
$318$	0	0
$319$	19.7180	1.10400
$320$	0	0
$321$	6.76021	0.377318
$322$	0	0
$323$	−12.9874	−0.722638
$324$	0	0
$325$	−2.05130	−0.113786
$326$	0	0
$327$	−1.25858	−0.0695995
$328$	0	0
$329$	56.4018	3.10953
$330$	0	0
$331$	25.4283	1.39766	0.698832	−	0.715285i	$-0.253703\pi$
$331$	25.4283	1.39766	0.698832	+	0.715285i	$0.253703\pi$
$332$	0	0
$333$	−8.00917	−0.438900
$334$	0	0
$335$	2.59901	0.141999
$336$	0	0
$337$	26.1612	1.42509	0.712546	−	0.701625i	$-0.247542\pi$
$337$	26.1612	1.42509	0.712546	+	0.701625i	$0.247542\pi$
$338$	0	0
$339$	−25.7124	−1.39650
$340$	0	0
$341$	5.00448	0.271008
$342$	0	0
$343$	24.6886	1.33306
$344$	0	0
$345$	−2.36493	−0.127324
$346$	0	0
$347$	−13.6095	−0.730596	−0.365298	−	0.930891i	$-0.619033\pi$
$347$	−13.6095	−0.730596	−0.365298	+	0.930891i	$0.619033\pi$
$348$	0	0
$349$	9.06564	0.485273	0.242636	−	0.970117i	$-0.421988\pi$
$349$	9.06564	0.485273	0.242636	+	0.970117i	$0.421988\pi$
$350$	0	0
$351$	1.51656	0.0809480
$352$	0	0
$353$	−6.18442	−0.329164	−0.164582	−	0.986363i	$-0.552627\pi$
$353$	−6.18442	−0.329164	−0.164582	+	0.986363i	$0.552627\pi$
$354$	0	0
$355$	−3.62740	−0.192522
$356$	0	0
$357$	44.5102	2.35573
$358$	0	0
$359$	0.776302	0.0409717	0.0204858	−	0.999790i	$-0.493479\pi$
$359$	0.776302	0.0409717	0.0204858	+	0.999790i	$0.493479\pi$
$360$	0	0
$361$	−11.6627	−0.613829
$362$	0	0
$363$	−1.91496	−0.100509
$364$	0	0
$365$	−3.73811	−0.195662
$366$	0	0
$367$	23.9407	1.24969	0.624846	−	0.780748i	$-0.285162\pi$
$367$	23.9407	1.24969	0.624846	+	0.780748i	$0.285162\pi$
$368$	0	0
$369$	4.52326	0.235472
$370$	0	0
$371$	−27.8112	−1.44389
$372$	0	0
$373$	−30.9301	−1.60150	−0.800750	−	0.598998i	$-0.795566\pi$
$373$	−30.9301	−1.60150	−0.800750	+	0.598998i	$0.795566\pi$
$374$	0	0
$375$	−13.6284	−0.703766
$376$	0	0
$377$	−2.58355	−0.133060
$378$	0	0
$379$	−5.03719	−0.258743	−0.129371	−	0.991596i	$-0.541296\pi$
$379$	−5.03719	−0.258743	−0.129371	+	0.991596i	$0.541296\pi$
$380$	0	0
$381$	−12.7652	−0.653982
$382$	0	0
$383$	−7.46213	−0.381297	−0.190649	−	0.981658i	$-0.561059\pi$
$383$	−7.46213	−0.381297	−0.190649	+	0.981658i	$0.561059\pi$
$384$	0	0
$385$	10.4036	0.530215
$386$	0	0
$387$	−4.55568	−0.231578
$388$	0	0
$389$	−22.0065	−1.11578	−0.557888	−	0.829916i	$-0.688388\pi$
$389$	−22.0065	−1.11578	−0.557888	+	0.829916i	$0.688388\pi$
$390$	0	0
$391$	7.93405	0.401242
$392$	0	0
$393$	27.0851	1.36626
$394$	0	0
$395$	−5.20700	−0.261993
$396$	0	0
$397$	22.4626	1.12736	0.563682	−	0.825992i	$-0.309384\pi$
$397$	22.4626	1.12736	0.563682	+	0.825992i	$0.309384\pi$
$398$	0	0
$399$	−25.1461	−1.25888
$400$	0	0
$401$	30.0992	1.50308	0.751541	−	0.659687i	$-0.229311\pi$
$401$	30.0992	1.50308	0.751541	+	0.659687i	$0.229311\pi$
$402$	0	0
$403$	−0.655711	−0.0326633
$404$	0	0
$405$	7.65692	0.380475
$406$	0	0
$407$	19.7234	0.977651
$408$	0	0
$409$	20.1455	0.996130	0.498065	−	0.867140i	$-0.334044\pi$
$409$	20.1455	0.996130	0.498065	+	0.867140i	$0.334044\pi$
$410$	0	0
$411$	−10.7527	−0.530394
$412$	0	0
$413$	−17.4576	−0.859033
$414$	0	0
$415$	−8.42863	−0.413745
$416$	0	0
$417$	2.89574	0.141805
$418$	0	0
$419$	−16.5091	−0.806524	−0.403262	−	0.915085i	$-0.632124\pi$
$419$	−16.5091	−0.806524	−0.403262	+	0.915085i	$0.632124\pi$
$420$	0	0
$421$	4.15528	0.202516	0.101258	−	0.994860i	$-0.467713\pi$
$421$	4.15528	0.202516	0.101258	+	0.994860i	$0.467713\pi$
$422$	0	0
$423$	17.8650	0.868628
$424$	0	0
$425$	21.7483	1.05495
$426$	0	0
$427$	47.2016	2.28425
$428$	0	0
$429$	3.27469	0.158103
$430$	0	0
$431$	26.0301	1.25383	0.626914	−	0.779089i	$-0.284318\pi$
$431$	26.0301	1.25383	0.626914	+	0.779089i	$0.284318\pi$
$432$	0	0
$433$	13.8836	0.667203	0.333601	−	0.942714i	$-0.391736\pi$
$433$	13.8836	0.667203	0.333601	+	0.942714i	$0.391736\pi$
$434$	0	0
$435$	−8.16463	−0.391464
$436$	0	0
$437$	−4.48235	−0.214420
$438$	0	0
$439$	30.9964	1.47938	0.739689	−	0.672949i	$-0.234972\pi$
$439$	30.9964	1.47938	0.739689	+	0.672949i	$0.234972\pi$
$440$	0	0
$441$	17.6309	0.839569
$442$	0	0
$443$	−18.7380	−0.890267	−0.445134	−	0.895464i	$-0.646844\pi$
$443$	−18.7380	−0.890267	−0.445134	+	0.895464i	$0.646844\pi$
$444$	0	0
$445$	4.63590	0.219763
$446$	0	0
$447$	−15.5139	−0.733780
$448$	0	0
$449$	−26.1379	−1.23352	−0.616761	−	0.787150i	$-0.711556\pi$
$449$	−26.1379	−1.23352	−0.616761	+	0.787150i	$0.711556\pi$
$450$	0	0
$451$	−11.1390	−0.524515
$452$	0	0
$453$	10.2475	0.481468
$454$	0	0
$455$	−1.36313	−0.0639044
$456$	0	0
$457$	21.7825	1.01894	0.509470	−	0.860488i	$-0.329841\pi$
$457$	21.7825	1.01894	0.509470	+	0.860488i	$0.329841\pi$
$458$	0	0
$459$	−16.0788	−0.750496
$460$	0	0
$461$	21.7273	1.01194	0.505970	−	0.862551i	$-0.331135\pi$
$461$	21.7273	1.01194	0.505970	+	0.862551i	$0.331135\pi$
$462$	0	0
$463$	14.3914	0.668823	0.334412	−	0.942427i	$-0.391462\pi$
$463$	14.3914	0.668823	0.334412	+	0.942427i	$0.391462\pi$
$464$	0	0
$465$	−2.07220	−0.0960960
$466$	0	0
$467$	22.0500	1.02035	0.510177	−	0.860069i	$-0.329580\pi$
$467$	22.0500	1.02035	0.510177	+	0.860069i	$0.329580\pi$
$468$	0	0
$469$	−16.8824	−0.779556
$470$	0	0
$471$	17.9978	0.829296
$472$	0	0
$473$	11.2188	0.515841
$474$	0	0
$475$	−12.2867	−0.563754
$476$	0	0
$477$	−8.80909	−0.403341
$478$	0	0
$479$	−40.7903	−1.86376	−0.931878	−	0.362771i	$-0.881831\pi$
$479$	−40.7903	−1.86376	−0.931878	+	0.362771i	$0.881831\pi$
$480$	0	0
$481$	−2.58425	−0.117832
$482$	0	0
$483$	15.3619	0.698989
$484$	0	0
$485$	5.75270	0.261217
$486$	0	0
$487$	−18.5169	−0.839083	−0.419541	−	0.907736i	$-0.637809\pi$
$487$	−18.5169	−0.839083	−0.419541	+	0.907736i	$0.637809\pi$
$488$	0	0
$489$	31.3873	1.41938
$490$	0	0
$491$	18.4321	0.831829	0.415915	−	0.909404i	$-0.363462\pi$
$491$	18.4321	0.831829	0.415915	+	0.909404i	$0.363462\pi$
$492$	0	0
$493$	27.3913	1.23364
$494$	0	0
$495$	3.29529	0.148112
$496$	0	0
$497$	23.5625	1.05692
$498$	0	0
$499$	16.9441	0.758524	0.379262	−	0.925289i	$-0.376178\pi$
$499$	16.9441	0.758524	0.379262	+	0.925289i	$0.376178\pi$
$500$	0	0
$501$	18.4363	0.823671
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	4.78366	0.212870
$506$	0	0
$507$	26.8448	1.19222
$508$	0	0
$509$	−0.203451	−0.00901782	−0.00450891	−	0.999990i	$-0.501435\pi$
$509$	−0.203451	−0.00901782	−0.00450891	+	0.999990i	$0.501435\pi$
$510$	0	0
$511$	24.2816	1.07416
$512$	0	0
$513$	9.08377	0.401058
$514$	0	0
$515$	−5.14137	−0.226556
$516$	0	0
$517$	−43.9945	−1.93488
$518$	0	0
$519$	−20.0452	−0.879887
$520$	0	0
$521$	15.8448	0.694173	0.347087	−	0.937833i	$-0.387171\pi$
$521$	15.8448	0.694173	0.347087	+	0.937833i	$0.387171\pi$
$522$	0	0
$523$	35.0360	1.53202	0.766008	−	0.642831i	$-0.222240\pi$
$523$	35.0360	1.53202	0.766008	+	0.642831i	$0.222240\pi$
$524$	0	0
$525$	42.1089	1.83778
$526$	0	0
$527$	6.95197	0.302833
$528$	0	0
$529$	−20.2617	−0.880944
$530$	0	0
$531$	−5.52963	−0.239966
$532$	0	0
$533$	1.45949	0.0632174
$534$	0	0
$535$	2.19499	0.0948977
$536$	0	0
$537$	30.0809	1.29809
$538$	0	0
$539$	−43.4180	−1.87014
$540$	0	0
$541$	17.7098	0.761402	0.380701	−	0.924698i	$-0.375683\pi$
$541$	17.7098	0.761402	0.380701	+	0.924698i	$0.375683\pi$
$542$	0	0
$543$	−42.0751	−1.80562
$544$	0	0
$545$	−0.408651	−0.0175047
$546$	0	0
$547$	43.0735	1.84169	0.920844	−	0.389930i	$-0.127501\pi$
$547$	43.0735	1.84169	0.920844	+	0.389930i	$0.127501\pi$
$548$	0	0
$549$	14.9509	0.638090
$550$	0	0
$551$	−15.4748	−0.659247
$552$	0	0
$553$	33.8230	1.43830
$554$	0	0
$555$	−8.16684	−0.346663
$556$	0	0
$557$	−0.730809	−0.0309654	−0.0154827	−	0.999880i	$-0.504928\pi$
$557$	−0.730809	−0.0309654	−0.0154827	+	0.999880i	$0.504928\pi$
$558$	0	0
$559$	−1.46994	−0.0621720
$560$	0	0
$561$	−34.7188	−1.46583
$562$	0	0
$563$	−4.30838	−0.181576	−0.0907882	−	0.995870i	$-0.528939\pi$
$563$	−4.30838	−0.181576	−0.0907882	+	0.995870i	$0.528939\pi$
$564$	0	0
$565$	−8.34861	−0.351229
$566$	0	0
$567$	−49.7369	−2.08876
$568$	0	0
$569$	12.3488	0.517688	0.258844	−	0.965919i	$-0.416658\pi$
$569$	12.3488	0.517688	0.258844	+	0.965919i	$0.416658\pi$
$570$	0	0
$571$	18.7541	0.784836	0.392418	−	0.919787i	$-0.371639\pi$
$571$	18.7541	0.784836	0.392418	+	0.919787i	$0.371639\pi$
$572$	0	0
$573$	−40.7902	−1.70403
$574$	0	0
$575$	7.50601	0.313022
$576$	0	0
$577$	17.3036	0.720357	0.360179	−	0.932883i	$-0.382716\pi$
$577$	17.3036	0.720357	0.360179	+	0.932883i	$0.382716\pi$
$578$	0	0
$579$	15.4191	0.640796
$580$	0	0
$581$	54.7497	2.27140
$582$	0	0
$583$	21.6933	0.898444
$584$	0	0
$585$	−0.431766	−0.0178513
$586$	0	0
$587$	−2.75603	−0.113754	−0.0568768	−	0.998381i	$-0.518114\pi$
$587$	−2.75603	−0.113754	−0.0568768	+	0.998381i	$0.518114\pi$
$588$	0	0
$589$	−3.92753	−0.161831
$590$	0	0
$591$	−19.8060	−0.814708
$592$	0	0
$593$	−27.9668	−1.14846	−0.574228	−	0.818695i	$-0.694698\pi$
$593$	−27.9668	−1.14846	−0.574228	+	0.818695i	$0.694698\pi$
$594$	0	0
$595$	14.4521	0.592480
$596$	0	0
$597$	−39.8393	−1.63051
$598$	0	0
$599$	30.4869	1.24566	0.622831	−	0.782356i	$-0.285982\pi$
$599$	30.4869	1.24566	0.622831	+	0.782356i	$0.285982\pi$
$600$	0	0
$601$	−29.0400	−1.18457	−0.592284	−	0.805729i	$-0.701774\pi$
$601$	−29.0400	−1.18457	−0.592284	+	0.805729i	$0.701774\pi$
$602$	0	0
$603$	−5.34743	−0.217764
$604$	0	0
$605$	−0.621773	−0.0252786
$606$	0	0
$607$	19.8858	0.807139	0.403569	−	0.914949i	$-0.367769\pi$
$607$	19.8858	0.807139	0.403569	+	0.914949i	$0.367769\pi$
$608$	0	0
$609$	53.0349	2.14908
$610$	0	0
$611$	5.76437	0.233202
$612$	0	0
$613$	20.0464	0.809668	0.404834	−	0.914390i	$-0.367329\pi$
$613$	20.0464	0.809668	0.404834	+	0.914390i	$0.367329\pi$
$614$	0	0
$615$	4.61231	0.185986
$616$	0	0
$617$	−43.4876	−1.75075	−0.875373	−	0.483448i	$-0.839384\pi$
$617$	−43.4876	−1.75075	−0.875373	+	0.483448i	$0.839384\pi$
$618$	0	0
$619$	8.10032	0.325579	0.162790	−	0.986661i	$-0.447951\pi$
$619$	8.10032	0.325579	0.162790	+	0.986661i	$0.447951\pi$
$620$	0	0
$621$	−5.54931	−0.222686
$622$	0	0
$623$	−30.1134	−1.20647
$624$	0	0
$625$	18.2548	0.730192
$626$	0	0
$627$	19.6145	0.783326
$628$	0	0
$629$	27.3987	1.09246
$630$	0	0
$631$	33.8479	1.34746	0.673732	−	0.738976i	$-0.264690\pi$
$631$	33.8479	1.34746	0.673732	+	0.738976i	$0.264690\pi$
$632$	0	0
$633$	8.09432	0.321720
$634$	0	0
$635$	−4.14477	−0.164480
$636$	0	0
$637$	5.68884	0.225400
$638$	0	0
$639$	7.46332	0.295244
$640$	0	0
$641$	13.5058	0.533447	0.266724	−	0.963773i	$-0.414059\pi$
$641$	13.5058	0.533447	0.266724	+	0.963773i	$0.414059\pi$
$642$	0	0
$643$	−17.6685	−0.696777	−0.348388	−	0.937350i	$-0.613271\pi$
$643$	−17.6685	−0.696777	−0.348388	+	0.937350i	$0.613271\pi$
$644$	0	0
$645$	−4.64536	−0.182911
$646$	0	0
$647$	17.1566	0.674494	0.337247	−	0.941416i	$-0.390504\pi$
$647$	17.1566	0.674494	0.337247	+	0.941416i	$0.390504\pi$
$648$	0	0
$649$	13.6173	0.534525
$650$	0	0
$651$	13.4604	0.527553
$652$	0	0
$653$	9.86056	0.385874	0.192937	−	0.981211i	$-0.438199\pi$
$653$	9.86056	0.385874	0.192937	+	0.981211i	$0.438199\pi$
$654$	0	0
$655$	8.79431	0.343622
$656$	0	0
$657$	7.69111	0.300059
$658$	0	0
$659$	8.71179	0.339363	0.169681	−	0.985499i	$-0.445726\pi$
$659$	8.71179	0.339363	0.169681	+	0.985499i	$0.445726\pi$
$660$	0	0
$661$	−23.9164	−0.930239	−0.465120	−	0.885248i	$-0.653989\pi$
$661$	−23.9164	−0.930239	−0.465120	+	0.885248i	$0.653989\pi$
$662$	0	0
$663$	4.54903	0.176670
$664$	0	0
$665$	−8.16476	−0.316616
$666$	0	0
$667$	9.45359	0.366044
$668$	0	0
$669$	−54.0041	−2.08792
$670$	0	0
$671$	−36.8182	−1.42135
$672$	0	0
$673$	−40.8851	−1.57600	−0.788002	−	0.615673i	$-0.788884\pi$
$673$	−40.8851	−1.57600	−0.788002	+	0.615673i	$0.788884\pi$
$674$	0	0
$675$	−15.2114	−0.585487
$676$	0	0
$677$	38.0977	1.46422	0.732108	−	0.681189i	$-0.238537\pi$
$677$	38.0977	1.46422	0.732108	+	0.681189i	$0.238537\pi$
$678$	0	0
$679$	−37.3677	−1.43404
$680$	0	0
$681$	20.6529	0.791420
$682$	0	0
$683$	32.6768	1.25034	0.625171	−	0.780488i	$-0.285029\pi$
$683$	32.6768	1.25034	0.625171	+	0.780488i	$0.285029\pi$
$684$	0	0
$685$	−3.49134	−0.133397
$686$	0	0
$687$	−34.5430	−1.31790
$688$	0	0
$689$	−2.84236	−0.108285
$690$	0	0
$691$	−5.62586	−0.214018	−0.107009	−	0.994258i	$-0.534127\pi$
$691$	−5.62586	−0.214018	−0.107009	+	0.994258i	$0.534127\pi$
$692$	0	0
$693$	−21.4052	−0.813116
$694$	0	0
$695$	0.940227	0.0356648
$696$	0	0
$697$	−15.4737	−0.586110
$698$	0	0
$699$	9.63626	0.364477
$700$	0	0
$701$	13.2285	0.499633	0.249817	−	0.968293i	$-0.419630\pi$
$701$	13.2285	0.499633	0.249817	+	0.968293i	$0.419630\pi$
$702$	0	0
$703$	−15.4790	−0.583800
$704$	0	0
$705$	18.2168	0.686083
$706$	0	0
$707$	−31.0732	−1.16863
$708$	0	0
$709$	13.0760	0.491079	0.245539	−	0.969387i	$-0.421035\pi$
$709$	13.0760	0.491079	0.245539	+	0.969387i	$0.421035\pi$
$710$	0	0
$711$	10.7133	0.401781
$712$	0	0
$713$	2.39934	0.0898560
$714$	0	0
$715$	1.06327	0.0397639
$716$	0	0
$717$	−23.1659	−0.865145
$718$	0	0
$719$	41.8942	1.56239	0.781196	−	0.624286i	$-0.214610\pi$
$719$	41.8942	1.56239	0.781196	+	0.624286i	$0.214610\pi$
$720$	0	0
$721$	33.3967	1.24376
$722$	0	0
$723$	−51.5932	−1.91877
$724$	0	0
$725$	25.9136	0.962405
$726$	0	0
$727$	7.96240	0.295309	0.147655	−	0.989039i	$-0.452828\pi$
$727$	7.96240	0.295309	0.147655	+	0.989039i	$0.452828\pi$
$728$	0	0
$729$	5.35289	0.198255
$730$	0	0
$731$	15.5846	0.576418
$732$	0	0
$733$	19.4635	0.718899	0.359450	−	0.933164i	$-0.382964\pi$
$733$	19.4635	0.718899	0.359450	+	0.933164i	$0.382964\pi$
$734$	0	0
$735$	17.9780	0.663130
$736$	0	0
$737$	13.1686	0.485071
$738$	0	0
$739$	−37.4080	−1.37608	−0.688039	−	0.725674i	$-0.741528\pi$
$739$	−37.4080	−1.37608	−0.688039	+	0.725674i	$0.741528\pi$
$740$	0	0
$741$	−2.56998	−0.0944107
$742$	0	0
$743$	6.72891	0.246860	0.123430	−	0.992353i	$-0.460611\pi$
$743$	6.72891	0.246860	0.123430	+	0.992353i	$0.460611\pi$
$744$	0	0
$745$	−5.03723	−0.184550
$746$	0	0
$747$	17.3418	0.634502
$748$	0	0
$749$	−14.2580	−0.520975
$750$	0	0
$751$	52.3079	1.90874	0.954371	−	0.298625i	$-0.0965279\pi$
$751$	52.3079	1.90874	0.954371	+	0.298625i	$0.0965279\pi$
$752$	0	0
$753$	24.5700	0.895382
$754$	0	0
$755$	3.32728	0.121092
$756$	0	0
$757$	34.4673	1.25274	0.626368	−	0.779527i	$-0.284541\pi$
$757$	34.4673	1.25274	0.626368	+	0.779527i	$0.284541\pi$
$758$	0	0
$759$	−11.9825	−0.434939
$760$	0	0
$761$	19.9360	0.722681	0.361340	−	0.932434i	$-0.382319\pi$
$761$	19.9360	0.722681	0.361340	+	0.932434i	$0.382319\pi$
$762$	0	0
$763$	2.65447	0.0960982
$764$	0	0
$765$	4.57766	0.165506
$766$	0	0
$767$	−1.78420	−0.0644238
$768$	0	0
$769$	−10.5076	−0.378915	−0.189457	−	0.981889i	$-0.560673\pi$
$769$	−10.5076	−0.378915	−0.189457	+	0.981889i	$0.560673\pi$
$770$	0	0
$771$	46.5272	1.67564
$772$	0	0
$773$	31.4218	1.13016	0.565082	−	0.825035i	$-0.308845\pi$
$773$	31.4218	1.13016	0.565082	+	0.825035i	$0.308845\pi$
$774$	0	0
$775$	6.57691	0.236250
$776$	0	0
$777$	53.0493	1.90313
$778$	0	0
$779$	8.74191	0.313212
$780$	0	0
$781$	−18.3792	−0.657659
$782$	0	0
$783$	−19.1583	−0.684661
$784$	0	0
$785$	5.84376	0.208573
$786$	0	0
$787$	35.8301	1.27721	0.638603	−	0.769536i	$-0.279513\pi$
$787$	35.8301	1.27721	0.638603	+	0.769536i	$0.279513\pi$
$788$	0	0
$789$	−5.83776	−0.207830
$790$	0	0
$791$	54.2300	1.92820
$792$	0	0
$793$	4.82410	0.171309
$794$	0	0
$795$	−8.98252	−0.318577
$796$	0	0
$797$	−13.4816	−0.477542	−0.238771	−	0.971076i	$-0.576745\pi$
$797$	−13.4816	−0.477542	−0.238771	+	0.971076i	$0.576745\pi$
$798$	0	0
$799$	−61.1149	−2.16209
$800$	0	0
$801$	−9.53830	−0.337019
$802$	0	0
$803$	−18.9401	−0.668382
$804$	0	0
$805$	4.98788	0.175800
$806$	0	0
$807$	−28.1346	−0.990383
$808$	0	0
$809$	15.6537	0.550356	0.275178	−	0.961393i	$-0.411263\pi$
$809$	15.6537	0.550356	0.275178	+	0.961393i	$0.411263\pi$
$810$	0	0
$811$	41.0709	1.44220	0.721098	−	0.692833i	$-0.243638\pi$
$811$	41.0709	1.44220	0.721098	+	0.692833i	$0.243638\pi$
$812$	0	0
$813$	−12.1199	−0.425065
$814$	0	0
$815$	10.1912	0.356983
$816$	0	0
$817$	−8.80455	−0.308032
$818$	0	0
$819$	2.80461	0.0980011
$820$	0	0
$821$	19.4157	0.677614	0.338807	−	0.940856i	$-0.389977\pi$
$821$	19.4157	0.677614	0.338807	+	0.940856i	$0.389977\pi$
$822$	0	0
$823$	−22.2850	−0.776806	−0.388403	−	0.921490i	$-0.626973\pi$
$823$	−22.2850	−0.776806	−0.388403	+	0.921490i	$0.626973\pi$
$824$	0	0
$825$	−32.8458	−1.14354
$826$	0	0
$827$	55.5849	1.93287	0.966437	−	0.256902i	$-0.0827018\pi$
$827$	55.5849	1.93287	0.966437	+	0.256902i	$0.0827018\pi$
$828$	0	0
$829$	−6.80937	−0.236499	−0.118250	−	0.992984i	$-0.537728\pi$
$829$	−6.80937	−0.236499	−0.118250	+	0.992984i	$0.537728\pi$
$830$	0	0
$831$	0.0919899	0.00319109
$832$	0	0
$833$	−60.3141	−2.08976
$834$	0	0
$835$	5.98611	0.207158
$836$	0	0
$837$	−4.86241	−0.168070
$838$	0	0
$839$	−4.63893	−0.160154	−0.0800768	−	0.996789i	$-0.525517\pi$
$839$	−4.63893	−0.160154	−0.0800768	+	0.996789i	$0.525517\pi$
$840$	0	0
$841$	3.63733	0.125425
$842$	0	0
$843$	−42.2153	−1.45397
$844$	0	0
$845$	8.71631	0.299850
$846$	0	0
$847$	4.03884	0.138776
$848$	0	0
$849$	13.2844	0.455920
$850$	0	0
$851$	9.45615	0.324153
$852$	0	0
$853$	10.2208	0.349954	0.174977	−	0.984573i	$-0.444015\pi$
$853$	10.2208	0.349954	0.174977	+	0.984573i	$0.444015\pi$
$854$	0	0
$855$	−2.58616	−0.0884446
$856$	0	0
$857$	−12.5156	−0.427525	−0.213762	−	0.976886i	$-0.568572\pi$
$857$	−12.5156	−0.427525	−0.213762	+	0.976886i	$0.568572\pi$
$858$	0	0
$859$	19.7515	0.673913	0.336956	−	0.941520i	$-0.390603\pi$
$859$	19.7515	0.673913	0.336956	+	0.941520i	$0.390603\pi$
$860$	0	0
$861$	−29.9602	−1.02104
$862$	0	0
$863$	−44.6756	−1.52078	−0.760388	−	0.649470i	$-0.774991\pi$
$863$	−44.6756	−1.52078	−0.760388	+	0.649470i	$0.774991\pi$
$864$	0	0
$865$	−6.50853	−0.221297
$866$	0	0
$867$	−12.5638	−0.426690
$868$	0	0
$869$	−26.3826	−0.894969
$870$	0	0
$871$	−1.72541	−0.0584634
$872$	0	0
$873$	−11.8361	−0.400591
$874$	0	0
$875$	28.7436	0.971711
$876$	0	0
$877$	−55.0161	−1.85776	−0.928880	−	0.370380i	$-0.879228\pi$
$877$	−55.0161	−1.85776	−0.928880	+	0.370380i	$0.879228\pi$
$878$	0	0
$879$	−9.75883	−0.329157
$880$	0	0
$881$	−25.5615	−0.861190	−0.430595	−	0.902545i	$-0.641696\pi$
$881$	−25.5615	−0.861190	−0.430595	+	0.902545i	$0.641696\pi$
$882$	0	0
$883$	43.3242	1.45797	0.728987	−	0.684527i	$-0.239991\pi$
$883$	43.3242	1.45797	0.728987	+	0.684527i	$0.239991\pi$
$884$	0	0
$885$	−5.63849	−0.189536
$886$	0	0
$887$	−48.5547	−1.63031	−0.815154	−	0.579244i	$-0.803348\pi$
$887$	−48.5547	−1.63031	−0.815154	+	0.579244i	$0.803348\pi$
$888$	0	0
$889$	26.9231	0.902973
$890$	0	0
$891$	38.7958	1.29971
$892$	0	0
$893$	34.5270	1.15540
$894$	0	0
$895$	9.76703	0.326476
$896$	0	0
$897$	1.57001	0.0524212
$898$	0	0
$899$	8.28342	0.276268
$900$	0	0
$901$	30.1352	1.00395
$902$	0	0
$903$	30.1748	1.00416
$904$	0	0
$905$	−13.6615	−0.454123
$906$	0	0
$907$	−35.8934	−1.19182	−0.595911	−	0.803050i	$-0.703209\pi$
$907$	−35.8934	−1.19182	−0.595911	+	0.803050i	$0.703209\pi$
$908$	0	0
$909$	−9.84231	−0.326449
$910$	0	0
$911$	53.7750	1.78164	0.890822	−	0.454352i	$-0.150129\pi$
$911$	53.7750	1.78164	0.890822	+	0.454352i	$0.150129\pi$
$912$	0	0
$913$	−42.7058	−1.41336
$914$	0	0
$915$	15.2453	0.503993
$916$	0	0
$917$	−57.1251	−1.88644
$918$	0	0
$919$	−20.1281	−0.663963	−0.331982	−	0.943286i	$-0.607717\pi$
$919$	−20.1281	−0.663963	−0.331982	+	0.943286i	$0.607717\pi$
$920$	0	0
$921$	−49.7845	−1.64046
$922$	0	0
$923$	2.40813	0.0792646
$924$	0	0
$925$	25.9206	0.852263
$926$	0	0
$927$	10.5783	0.347437
$928$	0	0
$929$	−57.1460	−1.87490	−0.937449	−	0.348123i	$-0.886819\pi$
$929$	−57.1460	−1.87490	−0.937449	+	0.348123i	$0.886819\pi$
$930$	0	0
$931$	34.0745	1.11675
$932$	0	0
$933$	47.3593	1.55047
$934$	0	0
$935$	−11.2729	−0.368665
$936$	0	0
$937$	48.6511	1.58936	0.794680	−	0.607028i	$-0.207638\pi$
$937$	48.6511	1.58936	0.794680	+	0.607028i	$0.207638\pi$
$938$	0	0
$939$	−9.30240	−0.303572
$940$	0	0
$941$	14.1974	0.462823	0.231412	−	0.972856i	$-0.425666\pi$
$941$	14.1974	0.462823	0.231412	+	0.972856i	$0.425666\pi$
$942$	0	0
$943$	−5.34047	−0.173910
$944$	0	0
$945$	−10.1082	−0.328821
$946$	0	0
$947$	−50.6126	−1.64469	−0.822344	−	0.568990i	$-0.807334\pi$
$947$	−50.6126	−1.64469	−0.822344	+	0.568990i	$0.807334\pi$
$948$	0	0
$949$	2.48163	0.0805571
$950$	0	0
$951$	48.7396	1.58049
$952$	0	0
$953$	30.0943	0.974850	0.487425	−	0.873165i	$-0.337936\pi$
$953$	30.0943	0.974850	0.487425	+	0.873165i	$0.337936\pi$
$954$	0	0
$955$	−13.2443	−0.428575
$956$	0	0
$957$	−41.3682	−1.33724
$958$	0	0
$959$	22.6786	0.732331
$960$	0	0
$961$	−28.8977	−0.932182
$962$	0	0
$963$	−4.51616	−0.145531
$964$	0	0
$965$	5.00646	0.161164
$966$	0	0
$967$	12.1735	0.391472	0.195736	−	0.980657i	$-0.437290\pi$
$967$	12.1735	0.391472	0.195736	+	0.980657i	$0.437290\pi$
$968$	0	0
$969$	27.2474	0.875313
$970$	0	0
$971$	30.6218	0.982701	0.491351	−	0.870962i	$-0.336503\pi$
$971$	30.6218	0.982701	0.491351	+	0.870962i	$0.336503\pi$
$972$	0	0
$973$	−6.10742	−0.195795
$974$	0	0
$975$	4.30361	0.137826
$976$	0	0
$977$	24.4940	0.783632	0.391816	−	0.920044i	$-0.371847\pi$
$977$	24.4940	0.783632	0.391816	+	0.920044i	$0.371847\pi$
$978$	0	0
$979$	23.4890	0.750712
$980$	0	0
$981$	0.840793	0.0268444
$982$	0	0
$983$	−29.5543	−0.942637	−0.471318	−	0.881963i	$-0.656222\pi$
$983$	−29.5543	−0.942637	−0.471318	+	0.881963i	$0.656222\pi$
$984$	0	0
$985$	−6.43085	−0.204904
$986$	0	0
$987$	−118.330	−3.76650
$988$	0	0
$989$	5.37873	0.171034
$990$	0	0
$991$	−37.1762	−1.18094	−0.590471	−	0.807059i	$-0.701058\pi$
$991$	−37.1762	−1.18094	−0.590471	+	0.807059i	$0.701058\pi$
$992$	0	0
$993$	−53.3483	−1.69296
$994$	0	0
$995$	−12.9355	−0.410084
$996$	0	0
$997$	−14.9555	−0.473646	−0.236823	−	0.971553i	$-0.576106\pi$
$997$	−14.9555	−0.473646	−0.236823	+	0.971553i	$0.576106\pi$
$998$	0	0
$999$	−19.1635	−0.606306

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.t.1.3		21
4.3	odd	2		2012.2.a.a.1.19	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2012.2.a.a.1.19	✓	21	4.3	odd	2
8048.2.a.t.1.3		21	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)

\(f(q)\)	\(=\)	\(q-2.09799 q^{3} -0.681202 q^{5} +4.42487 q^{7} +1.40156 q^{9} +O(q^{10})\)	\(q-2.09799 q^{3} -0.681202 q^{5} +4.42487 q^{7} +1.40156 q^{9} -3.45149 q^{11} +0.452231 q^{13} +1.42915 q^{15} -4.79463 q^{17} +2.70874 q^{19} -9.28334 q^{21} -1.65478 q^{23} -4.53596 q^{25} +3.35351 q^{27} -5.71291 q^{29} -1.44995 q^{31} +7.24118 q^{33} -3.01423 q^{35} -5.71446 q^{37} -0.948776 q^{39} +3.22730 q^{41} -3.25043 q^{43} -0.954746 q^{45} +12.7465 q^{47} +12.5795 q^{49} +10.0591 q^{51} -6.28520 q^{53} +2.35116 q^{55} -5.68290 q^{57} -3.94534 q^{59} +10.6673 q^{61} +6.20173 q^{63} -0.308060 q^{65} -3.81534 q^{67} +3.47171 q^{69} +5.32500 q^{71} +5.48753 q^{73} +9.51641 q^{75} -15.2724 q^{77} +7.64385 q^{79} -11.2403 q^{81} +12.3732 q^{83} +3.26611 q^{85} +11.9856 q^{87} -6.80548 q^{89} +2.00106 q^{91} +3.04198 q^{93} -1.84520 q^{95} -8.44493 q^{97} -4.83747 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9}+O(q^{10}) \) 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9	\( 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9} + 9 q^{11} - 16 q^{13} + 22 q^{15} + q^{17} + 12 q^{19} - 2 q^{21} + 22 q^{23} + 18 q^{25} + 43 q^{27} - 13 q^{29} + 28 q^{31} + 3 q^{33} + 12 q^{35} - 39 q^{37} + 11 q^{39} + 4 q^{41} + 50 q^{43} - 6 q^{45} + 27 q^{47} + 16 q^{49} + 37 q^{51} - 24 q^{53} + 49 q^{55} - q^{57} + 22 q^{59} - 22 q^{61} + 49 q^{63} - 14 q^{65} + 62 q^{67} - 17 q^{69} + 21 q^{71} - 6 q^{73} + 52 q^{75} - 24 q^{77} + 65 q^{79} + 29 q^{81} + 18 q^{83} - 54 q^{85} + 31 q^{87} + q^{89} + 45 q^{91} - 26 q^{93} + 53 q^{95} - 2 q^{97} + 58 q^{99}+O(q^{100}) \) 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9 + 9 * q^11 - 16 * q^13 + 22 * q^15 + q^17 + 12 * q^19 - 2 * q^21 + 22 * q^23 + 18 * q^25 + 43 * q^27 - 13 * q^29 + 28 * q^31 + 3 * q^33 + 12 * q^35 - 39 * q^37 + 11 * q^39 + 4 * q^41 + 50 * q^43 - 6 * q^45 + 27 * q^47 + 16 * q^49 + 37 * q^51 - 24 * q^53 + 49 * q^55 - q^57 + 22 * q^59 - 22 * q^61 + 49 * q^63 - 14 * q^65 + 62 * q^67 - 17 * q^69 + 21 * q^71 - 6 * q^73 + 52 * q^75 - 24 * q^77 + 65 * q^79 + 29 * q^81 + 18 * q^83 - 54 * q^85 + 31 * q^87 + q^89 + 45 * q^91 - 26 * q^93 + 53 * q^95 - 2 * q^97 + 58 * q^99

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