LMFDB - Embedded newform 8048.2.a.y.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:	$33$
Twist minimal:	no (minimal twist has level 4024)
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)

$f(q)$	$=$	$q-2.93503 q^{3} +3.00897 q^{5} +2.93938 q^{7} +5.61438 q^{9} +O(q^{10})$	$q-2.93503 q^{3} +3.00897 q^{5} +2.93938 q^{7} +5.61438 q^{9} +3.17090 q^{11} +1.63305 q^{13} -8.83142 q^{15} +0.761300 q^{17} -1.51637 q^{19} -8.62717 q^{21} +0.900836 q^{23} +4.05392 q^{25} -7.67328 q^{27} +4.23077 q^{29} -8.00793 q^{31} -9.30668 q^{33} +8.84453 q^{35} +0.300025 q^{37} -4.79305 q^{39} +12.4695 q^{41} -3.99549 q^{43} +16.8935 q^{45} +2.96374 q^{47} +1.63997 q^{49} -2.23444 q^{51} +10.0644 q^{53} +9.54116 q^{55} +4.45059 q^{57} -1.94748 q^{59} +3.16493 q^{61} +16.5028 q^{63} +4.91381 q^{65} +3.68902 q^{67} -2.64398 q^{69} +5.44913 q^{71} +3.11076 q^{73} -11.8984 q^{75} +9.32049 q^{77} -0.450924 q^{79} +5.67814 q^{81} +3.31332 q^{83} +2.29073 q^{85} -12.4174 q^{87} -0.156449 q^{89} +4.80017 q^{91} +23.5035 q^{93} -4.56272 q^{95} -18.2119 q^{97} +17.8027 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9}+O(q^{10}) $ 33 * q + 2 * q^3 + 12 * q^5 - 4 * q^7 + 43 * q^9	$ 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9} - 22 q^{11} + 25 q^{13} + 4 q^{15} + 17 q^{17} - 6 q^{19} + 18 q^{21} - 16 q^{23} + 47 q^{25} + 20 q^{27} + 47 q^{29} + 7 q^{31} - 6 q^{33} - 19 q^{35} + 75 q^{37} - 21 q^{39} + 22 q^{41} + 5 q^{43} + 33 q^{45} - 10 q^{47} + 31 q^{49} - 9 q^{51} + 64 q^{53} + 3 q^{55} + 5 q^{57} - 28 q^{59} + 49 q^{61} + 10 q^{63} + 46 q^{65} + 14 q^{67} + 30 q^{69} - 35 q^{71} + 19 q^{73} + 33 q^{75} + 32 q^{77} + 12 q^{79} + 57 q^{81} + 82 q^{85} + 5 q^{87} + 42 q^{89} + 15 q^{91} + 55 q^{93} - 33 q^{95} + 4 q^{97} - 22 q^{99}+O(q^{100}) $ 33 * q + 2 * q^3 + 12 * q^5 - 4 * q^7 + 43 * q^9 - 22 * q^11 + 25 * q^13 + 4 * q^15 + 17 * q^17 - 6 * q^19 + 18 * q^21 - 16 * q^23 + 47 * q^25 + 20 * q^27 + 47 * q^29 + 7 * q^31 - 6 * q^33 - 19 * q^35 + 75 * q^37 - 21 * q^39 + 22 * q^41 + 5 * q^43 + 33 * q^45 - 10 * q^47 + 31 * q^49 - 9 * q^51 + 64 * q^53 + 3 * q^55 + 5 * q^57 - 28 * q^59 + 49 * q^61 + 10 * q^63 + 46 * q^65 + 14 * q^67 + 30 * q^69 - 35 * q^71 + 19 * q^73 + 33 * q^75 + 32 * q^77 + 12 * q^79 + 57 * q^81 + 82 * q^85 + 5 * q^87 + 42 * q^89 + 15 * q^91 + 55 * q^93 - 33 * q^95 + 4 * q^97 - 22 * 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
$3$	−2.93503	−1.69454	−0.847269	−	0.531164i	$-0.821755\pi$
$3$	−2.93503	−1.69454	−0.847269	+	0.531164i	$0.821755\pi$
$4$	0	0
$5$	3.00897	1.34565	0.672827	−	0.739800i	$-0.265080\pi$
$5$	3.00897	1.34565	0.672827	+	0.739800i	$0.265080\pi$
$6$	0	0
$7$	2.93938	1.11098	0.555491	−	0.831522i	$-0.312530\pi$
$7$	2.93938	1.11098	0.555491	+	0.831522i	$0.312530\pi$
$8$	0	0
$9$	5.61438	1.87146
$10$	0	0
$11$	3.17090	0.956063	0.478031	−	0.878343i	$-0.341351\pi$
$11$	3.17090	0.956063	0.478031	+	0.878343i	$0.341351\pi$
$12$	0	0
$13$	1.63305	0.452927	0.226464	−	0.974020i	$-0.427284\pi$
$13$	1.63305	0.452927	0.226464	+	0.974020i	$0.427284\pi$
$14$	0	0
$15$	−8.83142	−2.28026
$16$	0	0
$17$	0.761300	0.184642	0.0923212	−	0.995729i	$-0.470571\pi$
$17$	0.761300	0.184642	0.0923212	+	0.995729i	$0.470571\pi$
$18$	0	0
$19$	−1.51637	−0.347879	−0.173940	−	0.984756i	$-0.555650\pi$
$19$	−1.51637	−0.347879	−0.173940	+	0.984756i	$0.555650\pi$
$20$	0	0
$21$	−8.62717	−1.88260
$22$	0	0
$23$	0.900836	0.187837	0.0939187	−	0.995580i	$-0.470061\pi$
$23$	0.900836	0.187837	0.0939187	+	0.995580i	$0.470061\pi$
$24$	0	0
$25$	4.05392	0.810784
$26$	0	0
$27$	−7.67328	−1.47672
$28$	0	0
$29$	4.23077	0.785634	0.392817	−	0.919617i	$-0.371501\pi$
$29$	4.23077	0.785634	0.392817	+	0.919617i	$0.371501\pi$
$30$	0	0
$31$	−8.00793	−1.43827	−0.719133	−	0.694872i	$-0.755461\pi$
$31$	−8.00793	−1.43827	−0.719133	+	0.694872i	$0.755461\pi$
$32$	0	0
$33$	−9.30668	−1.62009
$34$	0	0
$35$	8.84453	1.49500
$36$	0	0
$37$	0.300025	0.0493238	0.0246619	−	0.999696i	$-0.492149\pi$
$37$	0.300025	0.0493238	0.0246619	+	0.999696i	$0.492149\pi$
$38$	0	0
$39$	−4.79305	−0.767502
$40$	0	0
$41$	12.4695	1.94741	0.973703	−	0.227823i	$-0.0731608\pi$
$41$	12.4695	1.94741	0.973703	+	0.227823i	$0.0731608\pi$
$42$	0	0
$43$	−3.99549	−0.609306	−0.304653	−	0.952463i	$-0.598541\pi$
$43$	−3.99549	−0.609306	−0.304653	+	0.952463i	$0.598541\pi$
$44$	0	0
$45$	16.8935	2.51834
$46$	0	0
$47$	2.96374	0.432305	0.216153	−	0.976360i	$-0.430649\pi$
$47$	2.96374	0.432305	0.216153	+	0.976360i	$0.430649\pi$
$48$	0	0
$49$	1.63997	0.234282
$50$	0	0
$51$	−2.23444	−0.312884
$52$	0	0
$53$	10.0644	1.38246	0.691228	−	0.722636i	$-0.257070\pi$
$53$	10.0644	1.38246	0.691228	+	0.722636i	$0.257070\pi$
$54$	0	0
$55$	9.54116	1.28653
$56$	0	0
$57$	4.45059	0.589495
$58$	0	0
$59$	−1.94748	−0.253541	−0.126770	−	0.991932i	$-0.540461\pi$
$59$	−1.94748	−0.253541	−0.126770	+	0.991932i	$0.540461\pi$
$60$	0	0
$61$	3.16493	0.405228	0.202614	−	0.979259i	$-0.435056\pi$
$61$	3.16493	0.405228	0.202614	+	0.979259i	$0.435056\pi$
$62$	0	0
$63$	16.5028	2.07916
$64$	0	0
$65$	4.91381	0.609483
$66$	0	0
$67$	3.68902	0.450686	0.225343	−	0.974280i	$-0.427650\pi$
$67$	3.68902	0.450686	0.225343	+	0.974280i	$0.427650\pi$
$68$	0	0
$69$	−2.64398	−0.318298
$70$	0	0
$71$	5.44913	0.646693	0.323346	−	0.946281i	$-0.395192\pi$
$71$	5.44913	0.646693	0.323346	+	0.946281i	$0.395192\pi$
$72$	0	0
$73$	3.11076	0.364087	0.182044	−	0.983290i	$-0.441729\pi$
$73$	3.11076	0.364087	0.182044	+	0.983290i	$0.441729\pi$
$74$	0	0
$75$	−11.8984	−1.37391
$76$	0	0
$77$	9.32049	1.06217
$78$	0	0
$79$	−0.450924	−0.0507330	−0.0253665	−	0.999678i	$-0.508075\pi$
$79$	−0.450924	−0.0507330	−0.0253665	+	0.999678i	$0.508075\pi$
$80$	0	0
$81$	5.67814	0.630904
$82$	0	0
$83$	3.31332	0.363684	0.181842	−	0.983328i	$-0.441794\pi$
$83$	3.31332	0.363684	0.181842	+	0.983328i	$0.441794\pi$
$84$	0	0
$85$	2.29073	0.248465
$86$	0	0
$87$	−12.4174	−1.33129
$88$	0	0
$89$	−0.156449	−0.0165836	−0.00829180	−	0.999966i	$-0.502639\pi$
$89$	−0.156449	−0.0165836	−0.00829180	+	0.999966i	$0.502639\pi$
$90$	0	0
$91$	4.80017	0.503194
$92$	0	0
$93$	23.5035	2.43720
$94$	0	0
$95$	−4.56272	−0.468125
$96$	0	0
$97$	−18.2119	−1.84914	−0.924570	−	0.381013i	$-0.875575\pi$
$97$	−18.2119	−1.84914	−0.924570	+	0.381013i	$0.875575\pi$
$98$	0	0
$99$	17.8027	1.78923
$100$	0	0
$101$	7.69532	0.765713	0.382857	−	0.923808i	$-0.374940\pi$
$101$	7.69532	0.765713	0.382857	+	0.923808i	$0.374940\pi$
$102$	0	0
$103$	4.19774	0.413615	0.206808	−	0.978382i	$-0.433693\pi$
$103$	4.19774	0.413615	0.206808	+	0.978382i	$0.433693\pi$
$104$	0	0
$105$	−25.9589	−2.53333
$106$	0	0
$107$	5.84073	0.564645	0.282323	−	0.959320i	$-0.408895\pi$
$107$	5.84073	0.564645	0.282323	+	0.959320i	$0.408895\pi$
$108$	0	0
$109$	7.96263	0.762682	0.381341	−	0.924435i	$-0.375462\pi$
$109$	7.96263	0.762682	0.381341	+	0.924435i	$0.375462\pi$
$110$	0	0
$111$	−0.880582	−0.0835811
$112$	0	0
$113$	7.36415	0.692761	0.346380	−	0.938094i	$-0.387411\pi$
$113$	7.36415	0.692761	0.346380	+	0.938094i	$0.387411\pi$
$114$	0	0
$115$	2.71059	0.252764
$116$	0	0
$117$	9.16858	0.847635
$118$	0	0
$119$	2.23775	0.205134
$120$	0	0
$121$	−0.945385	−0.0859441
$122$	0	0
$123$	−36.5982	−3.29995
$124$	0	0
$125$	−2.84673	−0.254619
$126$	0	0
$127$	1.00893	0.0895277	0.0447638	−	0.998998i	$-0.485746\pi$
$127$	1.00893	0.0895277	0.0447638	+	0.998998i	$0.485746\pi$
$128$	0	0
$129$	11.7269	1.03249
$130$	0	0
$131$	0.872962	0.0762710	0.0381355	−	0.999273i	$-0.487858\pi$
$131$	0.872962	0.0762710	0.0381355	+	0.999273i	$0.487858\pi$
$132$	0	0
$133$	−4.45719	−0.386488
$134$	0	0
$135$	−23.0887	−1.98716
$136$	0	0
$137$	−12.2191	−1.04395	−0.521975	−	0.852961i	$-0.674805\pi$
$137$	−12.2191	−1.04395	−0.521975	+	0.852961i	$0.674805\pi$
$138$	0	0
$139$	−4.88576	−0.414405	−0.207202	−	0.978298i	$-0.566436\pi$
$139$	−4.88576	−0.414405	−0.207202	+	0.978298i	$0.566436\pi$
$140$	0	0
$141$	−8.69864	−0.732558
$142$	0	0
$143$	5.17825	0.433027
$144$	0	0
$145$	12.7303	1.05719
$146$	0	0
$147$	−4.81336	−0.396999
$148$	0	0
$149$	4.83152	0.395814	0.197907	−	0.980221i	$-0.436586\pi$
$149$	4.83152	0.395814	0.197907	+	0.980221i	$0.436586\pi$
$150$	0	0
$151$	−15.7231	−1.27953	−0.639766	−	0.768570i	$-0.720969\pi$
$151$	−15.7231	−1.27953	−0.639766	+	0.768570i	$0.720969\pi$
$152$	0	0
$153$	4.27423	0.345551
$154$	0	0
$155$	−24.0956	−1.93541
$156$	0	0
$157$	−8.13310	−0.649092	−0.324546	−	0.945870i	$-0.605211\pi$
$157$	−8.13310	−0.649092	−0.324546	+	0.945870i	$0.605211\pi$
$158$	0	0
$159$	−29.5394	−2.34263
$160$	0	0
$161$	2.64790	0.208684
$162$	0	0
$163$	−1.52941	−0.119793	−0.0598963	−	0.998205i	$-0.519077\pi$
$163$	−1.52941	−0.119793	−0.0598963	+	0.998205i	$0.519077\pi$
$164$	0	0
$165$	−28.0036	−2.18007
$166$	0	0
$167$	20.9475	1.62097	0.810484	−	0.585761i	$-0.199204\pi$
$167$	20.9475	1.62097	0.810484	+	0.585761i	$0.199204\pi$
$168$	0	0
$169$	−10.3331	−0.794857
$170$	0	0
$171$	−8.51348	−0.651042
$172$	0	0
$173$	−11.4091	−0.867414	−0.433707	−	0.901054i	$-0.642795\pi$
$173$	−11.4091	−0.867414	−0.433707	+	0.901054i	$0.642795\pi$
$174$	0	0
$175$	11.9160	0.900767
$176$	0	0
$177$	5.71591	0.429634
$178$	0	0
$179$	20.8098	1.55539	0.777697	−	0.628639i	$-0.216388\pi$
$179$	20.8098	1.55539	0.777697	+	0.628639i	$0.216388\pi$
$180$	0	0
$181$	−13.5892	−1.01008	−0.505039	−	0.863097i	$-0.668522\pi$
$181$	−13.5892	−1.01008	−0.505039	+	0.863097i	$0.668522\pi$
$182$	0	0
$183$	−9.28916	−0.686674
$184$	0	0
$185$	0.902768	0.0663728
$186$	0	0
$187$	2.41401	0.176530
$188$	0	0
$189$	−22.5547	−1.64061
$190$	0	0
$191$	3.82218	0.276564	0.138282	−	0.990393i	$-0.455842\pi$
$191$	3.82218	0.276564	0.138282	+	0.990393i	$0.455842\pi$
$192$	0	0
$193$	4.95251	0.356490	0.178245	−	0.983986i	$-0.442958\pi$
$193$	4.95251	0.356490	0.178245	+	0.983986i	$0.442958\pi$
$194$	0	0
$195$	−14.4222	−1.03279
$196$	0	0
$197$	−11.8069	−0.841208	−0.420604	−	0.907244i	$-0.638182\pi$
$197$	−11.8069	−0.841208	−0.420604	+	0.907244i	$0.638182\pi$
$198$	0	0
$199$	−13.0769	−0.926997	−0.463499	−	0.886098i	$-0.653406\pi$
$199$	−13.0769	−0.926997	−0.463499	+	0.886098i	$0.653406\pi$
$200$	0	0
$201$	−10.8274	−0.763704
$202$	0	0
$203$	12.4359	0.872826
$204$	0	0
$205$	37.5203	2.62053
$206$	0	0
$207$	5.05764	0.351530
$208$	0	0
$209$	−4.80826	−0.332594
$210$	0	0
$211$	26.2118	1.80450	0.902248	−	0.431217i	$-0.141916\pi$
$211$	26.2118	1.80450	0.902248	+	0.431217i	$0.141916\pi$
$212$	0	0
$213$	−15.9933	−1.09585
$214$	0	0
$215$	−12.0223	−0.819915
$216$	0	0
$217$	−23.5384	−1.59789
$218$	0	0
$219$	−9.13018	−0.616960
$220$	0	0
$221$	1.24324	0.0836295
$222$	0	0
$223$	8.74843	0.585838	0.292919	−	0.956137i	$-0.405373\pi$
$223$	8.74843	0.585838	0.292919	+	0.956137i	$0.405373\pi$
$224$	0	0
$225$	22.7603	1.51735
$226$	0	0
$227$	4.93759	0.327719	0.163860	−	0.986484i	$-0.447606\pi$
$227$	4.93759	0.327719	0.163860	+	0.986484i	$0.447606\pi$
$228$	0	0
$229$	8.18438	0.540839	0.270420	−	0.962743i	$-0.412838\pi$
$229$	8.18438	0.540839	0.270420	+	0.962743i	$0.412838\pi$
$230$	0	0
$231$	−27.3559	−1.79989
$232$	0	0
$233$	−3.51257	−0.230116	−0.115058	−	0.993359i	$-0.536705\pi$
$233$	−3.51257	−0.230116	−0.115058	+	0.993359i	$0.536705\pi$
$234$	0	0
$235$	8.91780	0.581733
$236$	0	0
$237$	1.32348	0.0859689
$238$	0	0
$239$	−18.7696	−1.21411	−0.607053	−	0.794661i	$-0.707648\pi$
$239$	−18.7696	−1.21411	−0.607053	+	0.794661i	$0.707648\pi$
$240$	0	0
$241$	3.83273	0.246888	0.123444	−	0.992352i	$-0.460606\pi$
$241$	3.83273	0.246888	0.123444	+	0.992352i	$0.460606\pi$
$242$	0	0
$243$	6.35436	0.407632
$244$	0	0
$245$	4.93463	0.315262
$246$	0	0
$247$	−2.47631	−0.157564
$248$	0	0
$249$	−9.72467	−0.616276
$250$	0	0
$251$	−2.17850	−0.137506	−0.0687529	−	0.997634i	$-0.521902\pi$
$251$	−2.17850	−0.137506	−0.0687529	+	0.997634i	$0.521902\pi$
$252$	0	0
$253$	2.85646	0.179584
$254$	0	0
$255$	−6.72336	−0.421033
$256$	0	0
$257$	−20.9215	−1.30505	−0.652525	−	0.757767i	$-0.726290\pi$
$257$	−20.9215	−1.30505	−0.652525	+	0.757767i	$0.726290\pi$
$258$	0	0
$259$	0.881889	0.0547979
$260$	0	0
$261$	23.7532	1.47028
$262$	0	0
$263$	28.7454	1.77252	0.886259	−	0.463189i	$-0.153295\pi$
$263$	28.7454	1.77252	0.886259	+	0.463189i	$0.153295\pi$
$264$	0	0
$265$	30.2836	1.86031
$266$	0	0
$267$	0.459183	0.0281016
$268$	0	0
$269$	−23.7219	−1.44635	−0.723175	−	0.690665i	$-0.757318\pi$
$269$	−23.7219	−1.44635	−0.723175	+	0.690665i	$0.757318\pi$
$270$	0	0
$271$	−21.2497	−1.29083	−0.645413	−	0.763834i	$-0.723315\pi$
$271$	−21.2497	−1.29083	−0.645413	+	0.763834i	$0.723315\pi$
$272$	0	0
$273$	−14.0886	−0.852682
$274$	0	0
$275$	12.8546	0.775161
$276$	0	0
$277$	−5.60213	−0.336599	−0.168300	−	0.985736i	$-0.553828\pi$
$277$	−5.60213	−0.336599	−0.168300	+	0.985736i	$0.553828\pi$
$278$	0	0
$279$	−44.9596	−2.69166
$280$	0	0
$281$	−21.5491	−1.28551	−0.642754	−	0.766072i	$-0.722208\pi$
$281$	−21.5491	−1.28551	−0.642754	+	0.766072i	$0.722208\pi$
$282$	0	0
$283$	−10.1441	−0.603004	−0.301502	−	0.953466i	$-0.597488\pi$
$283$	−10.1441	−0.603004	−0.301502	+	0.953466i	$0.597488\pi$
$284$	0	0
$285$	13.3917	0.793256
$286$	0	0
$287$	36.6526	2.16353
$288$	0	0
$289$	−16.4204	−0.965907
$290$	0	0
$291$	53.4524	3.13344
$292$	0	0
$293$	13.5838	0.793575	0.396787	−	0.917911i	$-0.370125\pi$
$293$	13.5838	0.793575	0.396787	+	0.917911i	$0.370125\pi$
$294$	0	0
$295$	−5.85992	−0.341178
$296$	0	0
$297$	−24.3312	−1.41184
$298$	0	0
$299$	1.47111	0.0850766
$300$	0	0
$301$	−11.7443	−0.676928
$302$	0	0
$303$	−22.5860	−1.29753
$304$	0	0
$305$	9.52319	0.545296
$306$	0	0
$307$	22.3107	1.27334	0.636669	−	0.771137i	$-0.280312\pi$
$307$	22.3107	1.27334	0.636669	+	0.771137i	$0.280312\pi$
$308$	0	0
$309$	−12.3205	−0.700887
$310$	0	0
$311$	7.04764	0.399635	0.199817	−	0.979833i	$-0.435965\pi$
$311$	7.04764	0.399635	0.199817	+	0.979833i	$0.435965\pi$
$312$	0	0
$313$	−22.6194	−1.27852	−0.639261	−	0.768990i	$-0.720759\pi$
$313$	−22.6194	−1.27852	−0.639261	+	0.768990i	$0.720759\pi$
$314$	0	0
$315$	49.6565	2.79783
$316$	0	0
$317$	2.22814	0.125145	0.0625725	−	0.998040i	$-0.480070\pi$
$317$	2.22814	0.125145	0.0625725	+	0.998040i	$0.480070\pi$
$318$	0	0
$319$	13.4154	0.751116
$320$	0	0
$321$	−17.1427	−0.956813
$322$	0	0
$323$	−1.15441	−0.0642332
$324$	0	0
$325$	6.62026	0.367226
$326$	0	0
$327$	−23.3705	−1.29239
$328$	0	0
$329$	8.71155	0.480284
$330$	0	0
$331$	13.7899	0.757963	0.378982	−	0.925404i	$-0.376274\pi$
$331$	13.7899	0.757963	0.378982	+	0.925404i	$0.376274\pi$
$332$	0	0
$333$	1.68446	0.0923076
$334$	0	0
$335$	11.1002	0.606467
$336$	0	0
$337$	30.5471	1.66401	0.832004	−	0.554769i	$-0.187193\pi$
$337$	30.5471	1.66401	0.832004	+	0.554769i	$0.187193\pi$
$338$	0	0
$339$	−21.6140	−1.17391
$340$	0	0
$341$	−25.3923	−1.37507
$342$	0	0
$343$	−15.7552	−0.850699
$344$	0	0
$345$	−7.95566	−0.428318
$346$	0	0
$347$	−11.5432	−0.619671	−0.309835	−	0.950790i	$-0.600274\pi$
$347$	−11.5432	−0.619671	−0.309835	+	0.950790i	$0.600274\pi$
$348$	0	0
$349$	7.40585	0.396426	0.198213	−	0.980159i	$-0.436486\pi$
$349$	7.40585	0.396426	0.198213	+	0.980159i	$0.436486\pi$
$350$	0	0
$351$	−12.5309	−0.668848
$352$	0	0
$353$	6.12210	0.325846	0.162923	−	0.986639i	$-0.447908\pi$
$353$	6.12210	0.325846	0.162923	+	0.986639i	$0.447908\pi$
$354$	0	0
$355$	16.3963	0.870225
$356$	0	0
$357$	−6.56786	−0.347608
$358$	0	0
$359$	−0.330116	−0.0174229	−0.00871144	−	0.999962i	$-0.502773\pi$
$359$	−0.330116	−0.0174229	−0.00871144	+	0.999962i	$0.502773\pi$
$360$	0	0
$361$	−16.7006	−0.878980
$362$	0	0
$363$	2.77473	0.145636
$364$	0	0
$365$	9.36021	0.489936
$366$	0	0
$367$	12.7160	0.663768	0.331884	−	0.943320i	$-0.392316\pi$
$367$	12.7160	0.663768	0.331884	+	0.943320i	$0.392316\pi$
$368$	0	0
$369$	70.0084	3.64449
$370$	0	0
$371$	29.5832	1.53588
$372$	0	0
$373$	11.7362	0.607676	0.303838	−	0.952724i	$-0.401732\pi$
$373$	11.7362	0.607676	0.303838	+	0.952724i	$0.401732\pi$
$374$	0	0
$375$	8.35521	0.431461
$376$	0	0
$377$	6.90907	0.355835
$378$	0	0
$379$	17.6405	0.906131	0.453066	−	0.891477i	$-0.350330\pi$
$379$	17.6405	0.906131	0.453066	+	0.891477i	$0.350330\pi$
$380$	0	0
$381$	−2.96122	−0.151708
$382$	0	0
$383$	−28.6554	−1.46422	−0.732110	−	0.681186i	$-0.761464\pi$
$383$	−28.6554	−1.46422	−0.732110	+	0.681186i	$0.761464\pi$
$384$	0	0
$385$	28.0451	1.42931
$386$	0	0
$387$	−22.4322	−1.14029
$388$	0	0
$389$	−9.80537	−0.497152	−0.248576	−	0.968612i	$-0.579963\pi$
$389$	−9.80537	−0.497152	−0.248576	+	0.968612i	$0.579963\pi$
$390$	0	0
$391$	0.685807	0.0346827
$392$	0	0
$393$	−2.56217	−0.129244
$394$	0	0
$395$	−1.35682	−0.0682690
$396$	0	0
$397$	5.86248	0.294229	0.147115	−	0.989119i	$-0.453001\pi$
$397$	5.86248	0.294229	0.147115	+	0.989119i	$0.453001\pi$
$398$	0	0
$399$	13.0820	0.654918
$400$	0	0
$401$	−1.16360	−0.0581073	−0.0290536	−	0.999578i	$-0.509249\pi$
$401$	−1.16360	−0.0581073	−0.0290536	+	0.999578i	$0.509249\pi$
$402$	0	0
$403$	−13.0774	−0.651430
$404$	0	0
$405$	17.0854	0.848978
$406$	0	0
$407$	0.951350	0.0471567
$408$	0	0
$409$	26.6005	1.31531	0.657656	−	0.753318i	$-0.271548\pi$
$409$	26.6005	1.31531	0.657656	+	0.753318i	$0.271548\pi$
$410$	0	0
$411$	35.8635	1.76901
$412$	0	0
$413$	−5.72440	−0.281679
$414$	0	0
$415$	9.96968	0.489392
$416$	0	0
$417$	14.3398	0.702225
$418$	0	0
$419$	8.61780	0.421007	0.210503	−	0.977593i	$-0.432490\pi$
$419$	8.61780	0.421007	0.210503	+	0.977593i	$0.432490\pi$
$420$	0	0
$421$	−11.9571	−0.582755	−0.291378	−	0.956608i	$-0.594114\pi$
$421$	−11.9571	−0.582755	−0.291378	+	0.956608i	$0.594114\pi$
$422$	0	0
$423$	16.6395	0.809042
$424$	0	0
$425$	3.08625	0.149705
$426$	0	0
$427$	9.30294	0.450201
$428$	0	0
$429$	−15.1983	−0.733780
$430$	0	0
$431$	−9.79238	−0.471682	−0.235841	−	0.971792i	$-0.575785\pi$
$431$	−9.79238	−0.471682	−0.235841	+	0.971792i	$0.575785\pi$
$432$	0	0
$433$	3.98552	0.191532	0.0957658	−	0.995404i	$-0.469470\pi$
$433$	3.98552	0.191532	0.0957658	+	0.995404i	$0.469470\pi$
$434$	0	0
$435$	−37.3637	−1.79145
$436$	0	0
$437$	−1.36600	−0.0653447
$438$	0	0
$439$	−11.5479	−0.551153	−0.275576	−	0.961279i	$-0.588869\pi$
$439$	−11.5479	−0.551153	−0.275576	+	0.961279i	$0.588869\pi$
$440$	0	0
$441$	9.20743	0.438449
$442$	0	0
$443$	−25.0903	−1.19208	−0.596039	−	0.802955i	$-0.703260\pi$
$443$	−25.0903	−1.19208	−0.596039	+	0.802955i	$0.703260\pi$
$444$	0	0
$445$	−0.470752	−0.0223158
$446$	0	0
$447$	−14.1807	−0.670722
$448$	0	0
$449$	31.9102	1.50594	0.752968	−	0.658058i	$-0.228622\pi$
$449$	31.9102	1.50594	0.752968	+	0.658058i	$0.228622\pi$
$450$	0	0
$451$	39.5395	1.86184
$452$	0	0
$453$	46.1478	2.16821
$454$	0	0
$455$	14.4436	0.677125
$456$	0	0
$457$	22.2967	1.04299	0.521497	−	0.853253i	$-0.325374\pi$
$457$	22.2967	1.04299	0.521497	+	0.853253i	$0.325374\pi$
$458$	0	0
$459$	−5.84167	−0.272666
$460$	0	0
$461$	38.7639	1.80541	0.902707	−	0.430255i	$-0.141576\pi$
$461$	38.7639	1.80541	0.902707	+	0.430255i	$0.141576\pi$
$462$	0	0
$463$	−33.3699	−1.55083	−0.775415	−	0.631453i	$-0.782459\pi$
$463$	−33.3699	−1.55083	−0.775415	+	0.631453i	$0.782459\pi$
$464$	0	0
$465$	70.7213	3.27962
$466$	0	0
$467$	13.1321	0.607679	0.303839	−	0.952723i	$-0.401731\pi$
$467$	13.1321	0.607679	0.303839	+	0.952723i	$0.401731\pi$
$468$	0	0
$469$	10.8434	0.500704
$470$	0	0
$471$	23.8709	1.09991
$472$	0	0
$473$	−12.6693	−0.582535
$474$	0	0
$475$	−6.14725	−0.282055
$476$	0	0
$477$	56.5056	2.58721
$478$	0	0
$479$	−18.5394	−0.847088	−0.423544	−	0.905875i	$-0.639214\pi$
$479$	−18.5394	−0.847088	−0.423544	+	0.905875i	$0.639214\pi$
$480$	0	0
$481$	0.489957	0.0223401
$482$	0	0
$483$	−7.77167	−0.353623
$484$	0	0
$485$	−54.7992	−2.48830
$486$	0	0
$487$	8.06317	0.365377	0.182689	−	0.983171i	$-0.441520\pi$
$487$	8.06317	0.365377	0.182689	+	0.983171i	$0.441520\pi$
$488$	0	0
$489$	4.48886	0.202993
$490$	0	0
$491$	1.43054	0.0645593	0.0322796	−	0.999479i	$-0.489723\pi$
$491$	1.43054	0.0645593	0.0322796	+	0.999479i	$0.489723\pi$
$492$	0	0
$493$	3.22089	0.145061
$494$	0	0
$495$	53.5677	2.40769
$496$	0	0
$497$	16.0171	0.718464
$498$	0	0
$499$	−27.6228	−1.23657	−0.618283	−	0.785956i	$-0.712171\pi$
$499$	−27.6228	−1.23657	−0.618283	+	0.785956i	$0.712171\pi$
$500$	0	0
$501$	−61.4815	−2.74679
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	23.1550	1.03038
$506$	0	0
$507$	30.3280	1.34692
$508$	0	0
$509$	−6.83557	−0.302981	−0.151491	−	0.988459i	$-0.548407\pi$
$509$	−6.83557	−0.302981	−0.151491	+	0.988459i	$0.548407\pi$
$510$	0	0
$511$	9.14373	0.404495
$512$	0	0
$513$	11.6355	0.513721
$514$	0	0
$515$	12.6309	0.556583
$516$	0	0
$517$	9.39771	0.413311
$518$	0	0
$519$	33.4859	1.46987
$520$	0	0
$521$	−2.43432	−0.106649	−0.0533247	−	0.998577i	$-0.516982\pi$
$521$	−2.43432	−0.106649	−0.0533247	+	0.998577i	$0.516982\pi$
$522$	0	0
$523$	37.9287	1.65851	0.829253	−	0.558874i	$-0.188766\pi$
$523$	37.9287	1.65851	0.829253	+	0.558874i	$0.188766\pi$
$524$	0	0
$525$	−34.9739	−1.52638
$526$	0	0
$527$	−6.09644	−0.265565
$528$	0	0
$529$	−22.1885	−0.964717
$530$	0	0
$531$	−10.9339	−0.474491
$532$	0	0
$533$	20.3633	0.882033
$534$	0	0
$535$	17.5746	0.759817
$536$	0	0
$537$	−61.0772	−2.63568
$538$	0	0
$539$	5.20019	0.223988
$540$	0	0
$541$	31.1233	1.33809	0.669047	−	0.743221i	$-0.266703\pi$
$541$	31.1233	1.33809	0.669047	+	0.743221i	$0.266703\pi$
$542$	0	0
$543$	39.8847	1.71162
$544$	0	0
$545$	23.9593	1.02631
$546$	0	0
$547$	11.6426	0.497802	0.248901	−	0.968529i	$-0.419931\pi$
$547$	11.6426	0.497802	0.248901	+	0.968529i	$0.419931\pi$
$548$	0	0
$549$	17.7691	0.758368
$550$	0	0
$551$	−6.41541	−0.273306
$552$	0	0
$553$	−1.32544	−0.0563634
$554$	0	0
$555$	−2.64965	−0.112471
$556$	0	0
$557$	−37.4414	−1.58644	−0.793221	−	0.608934i	$-0.791597\pi$
$557$	−37.4414	−1.58644	−0.793221	+	0.608934i	$0.791597\pi$
$558$	0	0
$559$	−6.52484	−0.275971
$560$	0	0
$561$	−7.08518	−0.299136
$562$	0	0
$563$	17.1633	0.723348	0.361674	−	0.932305i	$-0.382205\pi$
$563$	17.1633	0.723348	0.361674	+	0.932305i	$0.382205\pi$
$564$	0	0
$565$	22.1585	0.932216
$566$	0	0
$567$	16.6902	0.700923
$568$	0	0
$569$	24.3693	1.02161	0.510807	−	0.859695i	$-0.329347\pi$
$569$	24.3693	1.02161	0.510807	+	0.859695i	$0.329347\pi$
$570$	0	0
$571$	29.0864	1.21723	0.608614	−	0.793467i	$-0.291726\pi$
$571$	29.0864	1.21723	0.608614	+	0.793467i	$0.291726\pi$
$572$	0	0
$573$	−11.2182	−0.468648
$574$	0	0
$575$	3.65192	0.152296
$576$	0	0
$577$	31.8916	1.32766	0.663831	−	0.747882i	$-0.268929\pi$
$577$	31.8916	1.32766	0.663831	+	0.747882i	$0.268929\pi$
$578$	0	0
$579$	−14.5358	−0.604085
$580$	0	0
$581$	9.73910	0.404046
$582$	0	0
$583$	31.9133	1.32172
$584$	0	0
$585$	27.5880	1.14062
$586$	0	0
$587$	37.5542	1.55003	0.775013	−	0.631945i	$-0.217743\pi$
$587$	37.5542	1.55003	0.775013	+	0.631945i	$0.217743\pi$
$588$	0	0
$589$	12.1430	0.500343
$590$	0	0
$591$	34.6536	1.42546
$592$	0	0
$593$	−33.2984	−1.36740	−0.683701	−	0.729763i	$-0.739631\pi$
$593$	−33.2984	−1.36740	−0.683701	+	0.729763i	$0.739631\pi$
$594$	0	0
$595$	6.73334	0.276040
$596$	0	0
$597$	38.3811	1.57083
$598$	0	0
$599$	−10.1043	−0.412852	−0.206426	−	0.978462i	$-0.566183\pi$
$599$	−10.1043	−0.412852	−0.206426	+	0.978462i	$0.566183\pi$
$600$	0	0
$601$	−36.0999	−1.47255	−0.736274	−	0.676684i	$-0.763416\pi$
$601$	−36.0999	−1.47255	−0.736274	+	0.676684i	$0.763416\pi$
$602$	0	0
$603$	20.7116	0.843441
$604$	0	0
$605$	−2.84464	−0.115651
$606$	0	0
$607$	−14.4036	−0.584624	−0.292312	−	0.956323i	$-0.594425\pi$
$607$	−14.4036	−0.584624	−0.292312	+	0.956323i	$0.594425\pi$
$608$	0	0
$609$	−36.4996	−1.47904
$610$	0	0
$611$	4.83993	0.195803
$612$	0	0
$613$	17.9049	0.723173	0.361587	−	0.932338i	$-0.382235\pi$
$613$	17.9049	0.723173	0.361587	+	0.932338i	$0.382235\pi$
$614$	0	0
$615$	−110.123	−4.44059
$616$	0	0
$617$	24.7403	0.996007	0.498003	−	0.867175i	$-0.334067\pi$
$617$	24.7403	0.996007	0.498003	+	0.867175i	$0.334067\pi$
$618$	0	0
$619$	5.53990	0.222667	0.111334	−	0.993783i	$-0.464488\pi$
$619$	5.53990	0.222667	0.111334	+	0.993783i	$0.464488\pi$
$620$	0	0
$621$	−6.91237	−0.277384
$622$	0	0
$623$	−0.459865	−0.0184241
$624$	0	0
$625$	−28.8353	−1.15341
$626$	0	0
$627$	14.1124	0.563594
$628$	0	0
$629$	0.228409	0.00910727
$630$	0	0
$631$	34.0849	1.35690	0.678449	−	0.734647i	$-0.262652\pi$
$631$	34.0849	1.35690	0.678449	+	0.734647i	$0.262652\pi$
$632$	0	0
$633$	−76.9324	−3.05779
$634$	0	0
$635$	3.03583	0.120473
$636$	0	0
$637$	2.67816	0.106113
$638$	0	0
$639$	30.5935	1.21026
$640$	0	0
$641$	44.1416	1.74349	0.871744	−	0.489961i	$-0.162989\pi$
$641$	44.1416	1.74349	0.871744	+	0.489961i	$0.162989\pi$
$642$	0	0
$643$	16.8534	0.664632	0.332316	−	0.943168i	$-0.392170\pi$
$643$	16.8534	0.664632	0.332316	+	0.943168i	$0.392170\pi$
$644$	0	0
$645$	35.2858	1.38938
$646$	0	0
$647$	−37.9490	−1.49193	−0.745965	−	0.665985i	$-0.768011\pi$
$647$	−37.9490	−1.49193	−0.745965	+	0.665985i	$0.768011\pi$
$648$	0	0
$649$	−6.17527	−0.242401
$650$	0	0
$651$	69.0857	2.70768
$652$	0	0
$653$	34.5311	1.35131	0.675654	−	0.737219i	$-0.263861\pi$
$653$	34.5311	1.35131	0.675654	+	0.737219i	$0.263861\pi$
$654$	0	0
$655$	2.62672	0.102634
$656$	0	0
$657$	17.4650	0.681375
$658$	0	0
$659$	−27.4980	−1.07117	−0.535584	−	0.844482i	$-0.679909\pi$
$659$	−27.4980	−1.07117	−0.535584	+	0.844482i	$0.679909\pi$
$660$	0	0
$661$	14.9673	0.582159	0.291080	−	0.956699i	$-0.405986\pi$
$661$	14.9673	0.582159	0.291080	+	0.956699i	$0.405986\pi$
$662$	0	0
$663$	−3.64895	−0.141713
$664$	0	0
$665$	−13.4116	−0.520079
$666$	0	0
$667$	3.81123	0.147571
$668$	0	0
$669$	−25.6769	−0.992725
$670$	0	0
$671$	10.0357	0.387423
$672$	0	0
$673$	8.22534	0.317063	0.158532	−	0.987354i	$-0.449324\pi$
$673$	8.22534	0.317063	0.158532	+	0.987354i	$0.449324\pi$
$674$	0	0
$675$	−31.1069	−1.19730
$676$	0	0
$677$	12.3587	0.474984	0.237492	−	0.971389i	$-0.423675\pi$
$677$	12.3587	0.474984	0.237492	+	0.971389i	$0.423675\pi$
$678$	0	0
$679$	−53.5318	−2.05436
$680$	0	0
$681$	−14.4920	−0.555333
$682$	0	0
$683$	−41.0606	−1.57114	−0.785570	−	0.618772i	$-0.787630\pi$
$683$	−41.0606	−1.57114	−0.785570	+	0.618772i	$0.787630\pi$
$684$	0	0
$685$	−36.7670	−1.40480
$686$	0	0
$687$	−24.0214	−0.916473
$688$	0	0
$689$	16.4357	0.626152
$690$	0	0
$691$	−20.4919	−0.779550	−0.389775	−	0.920910i	$-0.627447\pi$
$691$	−20.4919	−0.779550	−0.389775	+	0.920910i	$0.627447\pi$
$692$	0	0
$693$	52.3288	1.98781
$694$	0	0
$695$	−14.7011	−0.557646
$696$	0	0
$697$	9.49301	0.359574
$698$	0	0
$699$	10.3095	0.389941
$700$	0	0
$701$	44.6633	1.68691	0.843454	−	0.537201i	$-0.180518\pi$
$701$	44.6633	1.68691	0.843454	+	0.537201i	$0.180518\pi$
$702$	0	0
$703$	−0.454949	−0.0171587
$704$	0	0
$705$	−26.1740	−0.985769
$706$	0	0
$707$	22.6195	0.850694
$708$	0	0
$709$	12.4657	0.468159	0.234079	−	0.972218i	$-0.424792\pi$
$709$	12.4657	0.468159	0.234079	+	0.972218i	$0.424792\pi$
$710$	0	0
$711$	−2.53166	−0.0949447
$712$	0	0
$713$	−7.21383	−0.270160
$714$	0	0
$715$	15.5812	0.582704
$716$	0	0
$717$	55.0893	2.05735
$718$	0	0
$719$	−36.9637	−1.37851	−0.689257	−	0.724517i	$-0.742063\pi$
$719$	−36.9637	−1.37851	−0.689257	+	0.724517i	$0.742063\pi$
$720$	0	0
$721$	12.3388	0.459519
$722$	0	0
$723$	−11.2492	−0.418361
$724$	0	0
$725$	17.1512	0.636980
$726$	0	0
$727$	−38.6313	−1.43276	−0.716378	−	0.697712i	$-0.754201\pi$
$727$	−38.6313	−1.43276	−0.716378	+	0.697712i	$0.754201\pi$
$728$	0	0
$729$	−35.6846	−1.32165
$730$	0	0
$731$	−3.04177	−0.112504
$732$	0	0
$733$	−25.8256	−0.953891	−0.476945	−	0.878933i	$-0.658256\pi$
$733$	−25.8256	−0.953891	−0.476945	+	0.878933i	$0.658256\pi$
$734$	0	0
$735$	−14.4833	−0.534224
$736$	0	0
$737$	11.6975	0.430884
$738$	0	0
$739$	−21.8590	−0.804095	−0.402048	−	0.915619i	$-0.631701\pi$
$739$	−21.8590	−0.804095	−0.402048	+	0.915619i	$0.631701\pi$
$740$	0	0
$741$	7.26804	0.266998
$742$	0	0
$743$	20.6356	0.757046	0.378523	−	0.925592i	$-0.376432\pi$
$743$	20.6356	0.757046	0.378523	+	0.925592i	$0.376432\pi$
$744$	0	0
$745$	14.5379	0.532628
$746$	0	0
$747$	18.6022	0.680620
$748$	0	0
$749$	17.1682	0.627311
$750$	0	0
$751$	−46.1681	−1.68470	−0.842349	−	0.538933i	$-0.818828\pi$
$751$	−46.1681	−1.68470	−0.842349	+	0.538933i	$0.818828\pi$
$752$	0	0
$753$	6.39396	0.233009
$754$	0	0
$755$	−47.3105	−1.72181
$756$	0	0
$757$	−49.2156	−1.78877	−0.894385	−	0.447298i	$-0.852386\pi$
$757$	−49.2156	−1.78877	−0.894385	+	0.447298i	$0.852386\pi$
$758$	0	0
$759$	−8.38379	−0.304312
$760$	0	0
$761$	31.2493	1.13279	0.566393	−	0.824135i	$-0.308338\pi$
$761$	31.2493	1.13279	0.566393	+	0.824135i	$0.308338\pi$
$762$	0	0
$763$	23.4052	0.847326
$764$	0	0
$765$	12.8610	0.464992
$766$	0	0
$767$	−3.18034	−0.114835
$768$	0	0
$769$	−11.8973	−0.429026	−0.214513	−	0.976721i	$-0.568816\pi$
$769$	−11.8973	−0.429026	−0.214513	+	0.976721i	$0.568816\pi$
$770$	0	0
$771$	61.4053	2.21146
$772$	0	0
$773$	7.07165	0.254350	0.127175	−	0.991880i	$-0.459409\pi$
$773$	7.07165	0.254350	0.127175	+	0.991880i	$0.459409\pi$
$774$	0	0
$775$	−32.4635	−1.16612
$776$	0	0
$777$	−2.58837	−0.0928572
$778$	0	0
$779$	−18.9083	−0.677462
$780$	0	0
$781$	17.2787	0.618279
$782$	0	0
$783$	−32.4639	−1.16016
$784$	0	0
$785$	−24.4723	−0.873453
$786$	0	0
$787$	18.4196	0.656587	0.328294	−	0.944576i	$-0.393526\pi$
$787$	18.4196	0.656587	0.328294	+	0.944576i	$0.393526\pi$
$788$	0	0
$789$	−84.3686	−3.00360
$790$	0	0
$791$	21.6460	0.769645
$792$	0	0
$793$	5.16850	0.183539
$794$	0	0
$795$	−88.8832	−3.15236
$796$	0	0
$797$	5.73696	0.203214	0.101607	−	0.994825i	$-0.467602\pi$
$797$	5.73696	0.203214	0.101607	+	0.994825i	$0.467602\pi$
$798$	0	0
$799$	2.25629	0.0798219
$800$	0	0
$801$	−0.878367	−0.0310356
$802$	0	0
$803$	9.86393	0.348090
$804$	0	0
$805$	7.96747	0.280816
$806$	0	0
$807$	69.6244	2.45090
$808$	0	0
$809$	−39.9833	−1.40574	−0.702869	−	0.711320i	$-0.748098\pi$
$809$	−39.9833	−1.40574	−0.702869	+	0.711320i	$0.748098\pi$
$810$	0	0
$811$	−25.9356	−0.910723	−0.455361	−	0.890307i	$-0.650490\pi$
$811$	−25.9356	−0.910723	−0.455361	+	0.890307i	$0.650490\pi$
$812$	0	0
$813$	62.3683	2.18735
$814$	0	0
$815$	−4.60195	−0.161199
$816$	0	0
$817$	6.05864	0.211965
$818$	0	0
$819$	26.9500	0.941708
$820$	0	0
$821$	−22.2687	−0.777182	−0.388591	−	0.921410i	$-0.627038\pi$
$821$	−22.2687	−0.777182	−0.388591	+	0.921410i	$0.627038\pi$
$822$	0	0
$823$	−2.05300	−0.0715631	−0.0357816	−	0.999360i	$-0.511392\pi$
$823$	−2.05300	−0.0715631	−0.0357816	+	0.999360i	$0.511392\pi$
$824$	0	0
$825$	−37.7285	−1.31354
$826$	0	0
$827$	47.3550	1.64669	0.823347	−	0.567538i	$-0.192104\pi$
$827$	47.3550	1.64669	0.823347	+	0.567538i	$0.192104\pi$
$828$	0	0
$829$	−35.4464	−1.23110	−0.615551	−	0.788097i	$-0.711067\pi$
$829$	−35.4464	−1.23110	−0.615551	+	0.788097i	$0.711067\pi$
$830$	0	0
$831$	16.4424	0.570380
$832$	0	0
$833$	1.24851	0.0432583
$834$	0	0
$835$	63.0305	2.18126
$836$	0	0
$837$	61.4471	2.12392
$838$	0	0
$839$	−27.5208	−0.950122	−0.475061	−	0.879953i	$-0.657574\pi$
$839$	−27.5208	−0.950122	−0.475061	+	0.879953i	$0.657574\pi$
$840$	0	0
$841$	−11.1006	−0.382779
$842$	0	0
$843$	63.2470	2.17834
$844$	0	0
$845$	−31.0921	−1.06960
$846$	0	0
$847$	−2.77885	−0.0954823
$848$	0	0
$849$	29.7732	1.02181
$850$	0	0
$851$	0.270274	0.00926486
$852$	0	0
$853$	−2.57181	−0.0880571	−0.0440286	−	0.999030i	$-0.514019\pi$
$853$	−2.57181	−0.0880571	−0.0440286	+	0.999030i	$0.514019\pi$
$854$	0	0
$855$	−25.6168	−0.876077
$856$	0	0
$857$	7.40817	0.253058	0.126529	−	0.991963i	$-0.459616\pi$
$857$	7.40817	0.253058	0.126529	+	0.991963i	$0.459616\pi$
$858$	0	0
$859$	18.9417	0.646281	0.323141	−	0.946351i	$-0.395261\pi$
$859$	18.9417	0.646281	0.323141	+	0.946351i	$0.395261\pi$
$860$	0	0
$861$	−107.576	−3.66619
$862$	0	0
$863$	32.4155	1.10344	0.551718	−	0.834031i	$-0.313973\pi$
$863$	32.4155	1.10344	0.551718	+	0.834031i	$0.313973\pi$
$864$	0	0
$865$	−34.3295	−1.16724
$866$	0	0
$867$	48.1944	1.63677
$868$	0	0
$869$	−1.42984	−0.0485039
$870$	0	0
$871$	6.02436	0.204128
$872$	0	0
$873$	−102.249	−3.46059
$874$	0	0
$875$	−8.36762	−0.282877
$876$	0	0
$877$	−18.4493	−0.622990	−0.311495	−	0.950248i	$-0.600830\pi$
$877$	−18.4493	−0.622990	−0.311495	+	0.950248i	$0.600830\pi$
$878$	0	0
$879$	−39.8688	−1.34474
$880$	0	0
$881$	24.5280	0.826368	0.413184	−	0.910647i	$-0.364417\pi$
$881$	24.5280	0.826368	0.413184	+	0.910647i	$0.364417\pi$
$882$	0	0
$883$	−15.2990	−0.514854	−0.257427	−	0.966298i	$-0.582875\pi$
$883$	−15.2990	−0.514854	−0.257427	+	0.966298i	$0.582875\pi$
$884$	0	0
$885$	17.1990	0.578139
$886$	0	0
$887$	−8.94801	−0.300445	−0.150222	−	0.988652i	$-0.547999\pi$
$887$	−8.94801	−0.300445	−0.150222	+	0.988652i	$0.547999\pi$
$888$	0	0
$889$	2.96562	0.0994636
$890$	0	0
$891$	18.0048	0.603184
$892$	0	0
$893$	−4.49412	−0.150390
$894$	0	0
$895$	62.6160	2.09302
$896$	0	0
$897$	−4.31775	−0.144166
$898$	0	0
$899$	−33.8797	−1.12995
$900$	0	0
$901$	7.66206	0.255260
$902$	0	0
$903$	34.4697	1.14708
$904$	0	0
$905$	−40.8896	−1.35922
$906$	0	0
$907$	−3.51640	−0.116760	−0.0583800	−	0.998294i	$-0.518594\pi$
$907$	−3.51640	−0.116760	−0.0583800	+	0.998294i	$0.518594\pi$
$908$	0	0
$909$	43.2045	1.43300
$910$	0	0
$911$	24.4315	0.809452	0.404726	−	0.914438i	$-0.367367\pi$
$911$	24.4315	0.809452	0.404726	+	0.914438i	$0.367367\pi$
$912$	0	0
$913$	10.5062	0.347704
$914$	0	0
$915$	−27.9508	−0.924026
$916$	0	0
$917$	2.56597	0.0847357
$918$	0	0
$919$	3.37930	0.111473	0.0557364	−	0.998446i	$-0.482249\pi$
$919$	3.37930	0.111473	0.0557364	+	0.998446i	$0.482249\pi$
$920$	0	0
$921$	−65.4824	−2.15772
$922$	0	0
$923$	8.89871	0.292905
$924$	0	0
$925$	1.21628	0.0399910
$926$	0	0
$927$	23.5677	0.774065
$928$	0	0
$929$	20.8079	0.682684	0.341342	−	0.939939i	$-0.389119\pi$
$929$	20.8079	0.682684	0.341342	+	0.939939i	$0.389119\pi$
$930$	0	0
$931$	−2.48681	−0.0815017
$932$	0	0
$933$	−20.6850	−0.677197
$934$	0	0
$935$	7.26368	0.237548
$936$	0	0
$937$	15.2943	0.499642	0.249821	−	0.968292i	$-0.419628\pi$
$937$	15.2943	0.499642	0.249821	+	0.968292i	$0.419628\pi$
$938$	0	0
$939$	66.3884	2.16650
$940$	0	0
$941$	−49.8728	−1.62581	−0.812903	−	0.582399i	$-0.802114\pi$
$941$	−49.8728	−1.62581	−0.812903	+	0.582399i	$0.802114\pi$
$942$	0	0
$943$	11.2330	0.365795
$944$	0	0
$945$	−67.8665	−2.20770
$946$	0	0
$947$	−7.90640	−0.256923	−0.128462	−	0.991714i	$-0.541004\pi$
$947$	−7.90640	−0.256923	−0.128462	+	0.991714i	$0.541004\pi$
$948$	0	0
$949$	5.08004	0.164905
$950$	0	0
$951$	−6.53966	−0.212063
$952$	0	0
$953$	53.5606	1.73500	0.867499	−	0.497439i	$-0.165726\pi$
$953$	53.5606	1.73500	0.867499	+	0.497439i	$0.165726\pi$
$954$	0	0
$955$	11.5009	0.372159
$956$	0	0
$957$	−39.3744	−1.27279
$958$	0	0
$959$	−35.9167	−1.15981
$960$	0	0
$961$	33.1269	1.06861
$962$	0	0
$963$	32.7921	1.05671
$964$	0	0
$965$	14.9020	0.479712
$966$	0	0
$967$	41.5773	1.33704	0.668519	−	0.743695i	$-0.266929\pi$
$967$	41.5773	1.33704	0.668519	+	0.743695i	$0.266929\pi$
$968$	0	0
$969$	3.38823	0.108846
$970$	0	0
$971$	−10.0689	−0.323126	−0.161563	−	0.986862i	$-0.551653\pi$
$971$	−10.0689	−0.323126	−0.161563	+	0.986862i	$0.551653\pi$
$972$	0	0
$973$	−14.3611	−0.460397
$974$	0	0
$975$	−19.4307	−0.622279
$976$	0	0
$977$	9.15470	0.292885	0.146442	−	0.989219i	$-0.453218\pi$
$977$	9.15470	0.292885	0.146442	+	0.989219i	$0.453218\pi$
$978$	0	0
$979$	−0.496086	−0.0158550
$980$	0	0
$981$	44.7052	1.42733
$982$	0	0
$983$	42.7518	1.36357	0.681785	−	0.731553i	$-0.261204\pi$
$983$	42.7518	1.36357	0.681785	+	0.731553i	$0.261204\pi$
$984$	0	0
$985$	−35.5267	−1.13197
$986$	0	0
$987$	−25.5686	−0.813859
$988$	0	0
$989$	−3.59928	−0.114450
$990$	0	0
$991$	31.0708	0.986997	0.493498	−	0.869747i	$-0.335718\pi$
$991$	31.0708	0.986997	0.493498	+	0.869747i	$0.335718\pi$
$992$	0	0
$993$	−40.4738	−1.28440
$994$	0	0
$995$	−39.3481	−1.24742
$996$	0	0
$997$	46.5965	1.47573	0.737864	−	0.674950i	$-0.235835\pi$
$997$	46.5965	1.47573	0.737864	+	0.674950i	$0.235835\pi$
$998$	0	0
$999$	−2.30218	−0.0728377

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.y.1.3		33
4.3	odd	2		4024.2.a.f.1.31	✓	33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.f.1.31	✓	33	4.3	odd	2
8048.2.a.y.1.3		33	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)

\(f(q)\)	\(=\)	\(q-2.93503 q^{3} +3.00897 q^{5} +2.93938 q^{7} +5.61438 q^{9} +O(q^{10})\)	\(q-2.93503 q^{3} +3.00897 q^{5} +2.93938 q^{7} +5.61438 q^{9} +3.17090 q^{11} +1.63305 q^{13} -8.83142 q^{15} +0.761300 q^{17} -1.51637 q^{19} -8.62717 q^{21} +0.900836 q^{23} +4.05392 q^{25} -7.67328 q^{27} +4.23077 q^{29} -8.00793 q^{31} -9.30668 q^{33} +8.84453 q^{35} +0.300025 q^{37} -4.79305 q^{39} +12.4695 q^{41} -3.99549 q^{43} +16.8935 q^{45} +2.96374 q^{47} +1.63997 q^{49} -2.23444 q^{51} +10.0644 q^{53} +9.54116 q^{55} +4.45059 q^{57} -1.94748 q^{59} +3.16493 q^{61} +16.5028 q^{63} +4.91381 q^{65} +3.68902 q^{67} -2.64398 q^{69} +5.44913 q^{71} +3.11076 q^{73} -11.8984 q^{75} +9.32049 q^{77} -0.450924 q^{79} +5.67814 q^{81} +3.31332 q^{83} +2.29073 q^{85} -12.4174 q^{87} -0.156449 q^{89} +4.80017 q^{91} +23.5035 q^{93} -4.56272 q^{95} -18.2119 q^{97} +17.8027 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9}+O(q^{10}) \) 33 * q + 2 * q^3 + 12 * q^5 - 4 * q^7 + 43 * q^9	\( 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9} - 22 q^{11} + 25 q^{13} + 4 q^{15} + 17 q^{17} - 6 q^{19} + 18 q^{21} - 16 q^{23} + 47 q^{25} + 20 q^{27} + 47 q^{29} + 7 q^{31} - 6 q^{33} - 19 q^{35} + 75 q^{37} - 21 q^{39} + 22 q^{41} + 5 q^{43} + 33 q^{45} - 10 q^{47} + 31 q^{49} - 9 q^{51} + 64 q^{53} + 3 q^{55} + 5 q^{57} - 28 q^{59} + 49 q^{61} + 10 q^{63} + 46 q^{65} + 14 q^{67} + 30 q^{69} - 35 q^{71} + 19 q^{73} + 33 q^{75} + 32 q^{77} + 12 q^{79} + 57 q^{81} + 82 q^{85} + 5 q^{87} + 42 q^{89} + 15 q^{91} + 55 q^{93} - 33 q^{95} + 4 q^{97} - 22 q^{99}+O(q^{100}) \) 33 * q + 2 * q^3 + 12 * q^5 - 4 * q^7 + 43 * q^9 - 22 * q^11 + 25 * q^13 + 4 * q^15 + 17 * q^17 - 6 * q^19 + 18 * q^21 - 16 * q^23 + 47 * q^25 + 20 * q^27 + 47 * q^29 + 7 * q^31 - 6 * q^33 - 19 * q^35 + 75 * q^37 - 21 * q^39 + 22 * q^41 + 5 * q^43 + 33 * q^45 - 10 * q^47 + 31 * q^49 - 9 * q^51 + 64 * q^53 + 3 * q^55 + 5 * q^57 - 28 * q^59 + 49 * q^61 + 10 * q^63 + 46 * q^65 + 14 * q^67 + 30 * q^69 - 35 * q^71 + 19 * q^73 + 33 * q^75 + 32 * q^77 + 12 * q^79 + 57 * q^81 + 82 * q^85 + 5 * q^87 + 42 * q^89 + 15 * q^91 + 55 * q^93 - 33 * q^95 + 4 * q^97 - 22 * q^99

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