LMFDB - Embedded newform 8048.2.a.s.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.2636035467$
Analytic rank:	$1$
Dimension:	$21$
Twist minimal:	no (minimal twist has level 2012)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.10653 q^{3} +2.64866 q^{5} -3.45308 q^{7} -1.77559 q^{9} +O(q^{10})$	$q-1.10653 q^{3} +2.64866 q^{5} -3.45308 q^{7} -1.77559 q^{9} -5.59487 q^{11} +5.51170 q^{13} -2.93083 q^{15} -0.253321 q^{17} +1.01182 q^{19} +3.82094 q^{21} -0.152164 q^{23} +2.01543 q^{25} +5.28433 q^{27} -5.43830 q^{29} +9.22196 q^{31} +6.19088 q^{33} -9.14606 q^{35} +5.41612 q^{37} -6.09886 q^{39} -2.22997 q^{41} +5.39755 q^{43} -4.70295 q^{45} +8.69161 q^{47} +4.92379 q^{49} +0.280307 q^{51} -4.80243 q^{53} -14.8189 q^{55} -1.11961 q^{57} +6.73599 q^{59} -9.30604 q^{61} +6.13127 q^{63} +14.5987 q^{65} -2.64926 q^{67} +0.168374 q^{69} -4.85818 q^{71} +4.54755 q^{73} -2.23013 q^{75} +19.3195 q^{77} -9.05560 q^{79} -0.520494 q^{81} +7.31730 q^{83} -0.670962 q^{85} +6.01764 q^{87} +13.1095 q^{89} -19.0324 q^{91} -10.2044 q^{93} +2.67998 q^{95} -17.5782 q^{97} +9.93420 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q - 10 q^{3} + 3 q^{5} - 13 q^{7} + 21 q^{9}+O(q^{10}) $ 21 * q - 10 * q^3 + 3 * q^5 - 13 * q^7 + 21 * q^9	$ 21 q - 10 q^{3} + 3 q^{5} - 13 q^{7} + 21 q^{9} - 7 q^{11} + 12 q^{13} - 14 q^{15} + q^{17} - 14 q^{19} + 14 q^{21} - 26 q^{23} + 18 q^{25} - 37 q^{27} + 9 q^{29} - 28 q^{31} + 3 q^{33} - 20 q^{35} + 31 q^{37} - 29 q^{39} + 4 q^{41} - 38 q^{43} + 24 q^{45} - 9 q^{47} + 16 q^{49} - 15 q^{51} + 22 q^{53} - 35 q^{55} - q^{57} - 10 q^{59} + 22 q^{61} - 35 q^{63} - 14 q^{65} - 58 q^{67} + 15 q^{69} - 27 q^{71} - 6 q^{73} - 48 q^{75} + 16 q^{77} - 47 q^{79} + 29 q^{81} - 22 q^{83} + 14 q^{85} - 29 q^{87} + q^{89} - 51 q^{91} + 34 q^{93} - 23 q^{95} - 2 q^{97} - 22 q^{99}+O(q^{100}) $ 21 * q - 10 * q^3 + 3 * q^5 - 13 * q^7 + 21 * q^9 - 7 * q^11 + 12 * q^13 - 14 * q^15 + q^17 - 14 * q^19 + 14 * q^21 - 26 * q^23 + 18 * q^25 - 37 * q^27 + 9 * q^29 - 28 * q^31 + 3 * q^33 - 20 * q^35 + 31 * q^37 - 29 * q^39 + 4 * q^41 - 38 * q^43 + 24 * q^45 - 9 * q^47 + 16 * q^49 - 15 * q^51 + 22 * q^53 - 35 * q^55 - q^57 - 10 * q^59 + 22 * q^61 - 35 * q^63 - 14 * q^65 - 58 * q^67 + 15 * q^69 - 27 * q^71 - 6 * q^73 - 48 * q^75 + 16 * q^77 - 47 * q^79 + 29 * q^81 - 22 * q^83 + 14 * q^85 - 29 * q^87 + q^89 - 51 * q^91 + 34 * q^93 - 23 * q^95 - 2 * 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
$3$	−1.10653	−0.638855	−0.319428	−	0.947611i	$-0.603491\pi$
$3$	−1.10653	−0.638855	−0.319428	+	0.947611i	$0.603491\pi$
$4$	0	0
$5$	2.64866	1.18452	0.592259	−	0.805747i	$-0.298236\pi$
$5$	2.64866	1.18452	0.592259	+	0.805747i	$0.298236\pi$
$6$	0	0
$7$	−3.45308	−1.30514	−0.652572	−	0.757727i	$-0.726310\pi$
$7$	−3.45308	−1.30514	−0.652572	+	0.757727i	$0.726310\pi$
$8$	0	0
$9$	−1.77559	−0.591864
$10$	0	0
$11$	−5.59487	−1.68692	−0.843458	−	0.537196i	$-0.819484\pi$
$11$	−5.59487	−1.68692	−0.843458	+	0.537196i	$0.819484\pi$
$12$	0	0
$13$	5.51170	1.52867	0.764335	−	0.644819i	$-0.223067\pi$
$13$	5.51170	1.52867	0.764335	+	0.644819i	$0.223067\pi$
$14$	0	0
$15$	−2.93083	−0.756736
$16$	0	0
$17$	−0.253321	−0.0614393	−0.0307196	−	0.999528i	$-0.509780\pi$
$17$	−0.253321	−0.0614393	−0.0307196	+	0.999528i	$0.509780\pi$
$18$	0	0
$19$	1.01182	0.232128	0.116064	−	0.993242i	$-0.462972\pi$
$19$	1.01182	0.232128	0.116064	+	0.993242i	$0.462972\pi$
$20$	0	0
$21$	3.82094	0.833797
$22$	0	0
$23$	−0.152164	−0.0317284	−0.0158642	−	0.999874i	$-0.505050\pi$
$23$	−0.152164	−0.0317284	−0.0158642	+	0.999874i	$0.505050\pi$
$24$	0	0
$25$	2.01543	0.403085
$26$	0	0
$27$	5.28433	1.01697
$28$	0	0
$29$	−5.43830	−1.00987	−0.504934	−	0.863158i	$-0.668483\pi$
$29$	−5.43830	−1.00987	−0.504934	+	0.863158i	$0.668483\pi$
$30$	0	0
$31$	9.22196	1.65631	0.828157	−	0.560496i	$-0.189390\pi$
$31$	9.22196	1.65631	0.828157	+	0.560496i	$0.189390\pi$
$32$	0	0
$33$	6.19088	1.07769
$34$	0	0
$35$	−9.14606	−1.54597
$36$	0	0
$37$	5.41612	0.890404	0.445202	−	0.895430i	$-0.353132\pi$
$37$	5.41612	0.890404	0.445202	+	0.895430i	$0.353132\pi$
$38$	0	0
$39$	−6.09886	−0.976599
$40$	0	0
$41$	−2.22997	−0.348262	−0.174131	−	0.984722i	$-0.555712\pi$
$41$	−2.22997	−0.348262	−0.174131	+	0.984722i	$0.555712\pi$
$42$	0	0
$43$	5.39755	0.823119	0.411560	−	0.911383i	$-0.364984\pi$
$43$	5.39755	0.823119	0.411560	+	0.911383i	$0.364984\pi$
$44$	0	0
$45$	−4.70295	−0.701074
$46$	0	0
$47$	8.69161	1.26780	0.633901	−	0.773414i	$-0.281453\pi$
$47$	8.69161	1.26780	0.633901	+	0.773414i	$0.281453\pi$
$48$	0	0
$49$	4.92379	0.703399
$50$	0	0
$51$	0.280307	0.0392508
$52$	0	0
$53$	−4.80243	−0.659664	−0.329832	−	0.944040i	$-0.606992\pi$
$53$	−4.80243	−0.659664	−0.329832	+	0.944040i	$0.606992\pi$
$54$	0	0
$55$	−14.8189	−1.99818
$56$	0	0
$57$	−1.11961	−0.148296
$58$	0	0
$59$	6.73599	0.876952	0.438476	−	0.898743i	$-0.355518\pi$
$59$	6.73599	0.876952	0.438476	+	0.898743i	$0.355518\pi$
$60$	0	0
$61$	−9.30604	−1.19152	−0.595758	−	0.803164i	$-0.703148\pi$
$61$	−9.30604	−1.19152	−0.595758	+	0.803164i	$0.703148\pi$
$62$	0	0
$63$	6.13127	0.772468
$64$	0	0
$65$	14.5987	1.81074
$66$	0	0
$67$	−2.64926	−0.323658	−0.161829	−	0.986819i	$-0.551739\pi$
$67$	−2.64926	−0.323658	−0.161829	+	0.986819i	$0.551739\pi$
$68$	0	0
$69$	0.168374	0.0202699
$70$	0	0
$71$	−4.85818	−0.576560	−0.288280	−	0.957546i	$-0.593083\pi$
$71$	−4.85818	−0.576560	−0.288280	+	0.957546i	$0.593083\pi$
$72$	0	0
$73$	4.54755	0.532250	0.266125	−	0.963938i	$-0.414257\pi$
$73$	4.54755	0.532250	0.266125	+	0.963938i	$0.414257\pi$
$74$	0	0
$75$	−2.23013	−0.257513
$76$	0	0
$77$	19.3195	2.20167
$78$	0	0
$79$	−9.05560	−1.01884	−0.509418	−	0.860519i	$-0.670139\pi$
$79$	−9.05560	−1.01884	−0.509418	+	0.860519i	$0.670139\pi$
$80$	0	0
$81$	−0.520494	−0.0578327
$82$	0	0
$83$	7.31730	0.803178	0.401589	−	0.915820i	$-0.368458\pi$
$83$	7.31730	0.803178	0.401589	+	0.915820i	$0.368458\pi$
$84$	0	0
$85$	−0.670962	−0.0727760
$86$	0	0
$87$	6.01764	0.645159
$88$	0	0
$89$	13.1095	1.38960	0.694802	−	0.719201i	$-0.255492\pi$
$89$	13.1095	1.38960	0.694802	+	0.719201i	$0.255492\pi$
$90$	0	0
$91$	−19.0324	−1.99513
$92$	0	0
$93$	−10.2044	−1.05814
$94$	0	0
$95$	2.67998	0.274960
$96$	0	0
$97$	−17.5782	−1.78480	−0.892398	−	0.451249i	$-0.850979\pi$
$97$	−17.5782	−1.78480	−0.892398	+	0.451249i	$0.850979\pi$
$98$	0	0
$99$	9.93420	0.998425
$100$	0	0
$101$	10.1794	1.01289	0.506443	−	0.862273i	$-0.330960\pi$
$101$	10.1794	1.01289	0.506443	+	0.862273i	$0.330960\pi$
$102$	0	0
$103$	8.70508	0.857737	0.428869	−	0.903367i	$-0.358912\pi$
$103$	8.70508	0.857737	0.428869	+	0.903367i	$0.358912\pi$
$104$	0	0
$105$	10.1204	0.987649
$106$	0	0
$107$	−2.07919	−0.201002	−0.100501	−	0.994937i	$-0.532045\pi$
$107$	−2.07919	−0.201002	−0.100501	+	0.994937i	$0.532045\pi$
$108$	0	0
$109$	−17.5971	−1.68549	−0.842747	−	0.538309i	$-0.819063\pi$
$109$	−17.5971	−1.68549	−0.842747	+	0.538309i	$0.819063\pi$
$110$	0	0
$111$	−5.99309	−0.568839
$112$	0	0
$113$	−11.0849	−1.04278	−0.521392	−	0.853317i	$-0.674587\pi$
$113$	−11.0849	−1.04278	−0.521392	+	0.853317i	$0.674587\pi$
$114$	0	0
$115$	−0.403032	−0.0375829
$116$	0	0
$117$	−9.78653	−0.904765
$118$	0	0
$119$	0.874738	0.0801871
$120$	0	0
$121$	20.3025	1.84568
$122$	0	0
$123$	2.46752	0.222489
$124$	0	0
$125$	−7.90514	−0.707057
$126$	0	0
$127$	−9.83597	−0.872801	−0.436401	−	0.899753i	$-0.643747\pi$
$127$	−9.83597	−0.872801	−0.436401	+	0.899753i	$0.643747\pi$
$128$	0	0
$129$	−5.97255	−0.525854
$130$	0	0
$131$	−5.83301	−0.509632	−0.254816	−	0.966990i	$-0.582015\pi$
$131$	−5.83301	−0.509632	−0.254816	+	0.966990i	$0.582015\pi$
$132$	0	0
$133$	−3.49391	−0.302960
$134$	0	0
$135$	13.9964	1.20462
$136$	0	0
$137$	2.84466	0.243036	0.121518	−	0.992589i	$-0.461224\pi$
$137$	2.84466	0.243036	0.121518	+	0.992589i	$0.461224\pi$
$138$	0	0
$139$	−7.44081	−0.631121	−0.315561	−	0.948905i	$-0.602193\pi$
$139$	−7.44081	−0.631121	−0.315561	+	0.948905i	$0.602193\pi$
$140$	0	0
$141$	−9.61752	−0.809942
$142$	0	0
$143$	−30.8372	−2.57874
$144$	0	0
$145$	−14.4042	−1.19621
$146$	0	0
$147$	−5.44832	−0.449370
$148$	0	0
$149$	−10.5852	−0.867174	−0.433587	−	0.901112i	$-0.642752\pi$
$149$	−10.5852	−0.867174	−0.433587	+	0.901112i	$0.642752\pi$
$150$	0	0
$151$	−15.4500	−1.25730	−0.628651	−	0.777688i	$-0.716392\pi$
$151$	−15.4500	−1.25730	−0.628651	+	0.777688i	$0.716392\pi$
$152$	0	0
$153$	0.449794	0.0363637
$154$	0	0
$155$	24.4259	1.96194
$156$	0	0
$157$	1.77329	0.141524	0.0707619	−	0.997493i	$-0.477457\pi$
$157$	1.77329	0.141524	0.0707619	+	0.997493i	$0.477457\pi$
$158$	0	0
$159$	5.31403	0.421430
$160$	0	0
$161$	0.525436	0.0414102
$162$	0	0
$163$	−23.5048	−1.84104	−0.920520	−	0.390695i	$-0.872235\pi$
$163$	−23.5048	−1.84104	−0.920520	+	0.390695i	$0.872235\pi$
$164$	0	0
$165$	16.3976	1.27655
$166$	0	0
$167$	−7.02957	−0.543964	−0.271982	−	0.962302i	$-0.587679\pi$
$167$	−7.02957	−0.543964	−0.271982	+	0.962302i	$0.587679\pi$
$168$	0	0
$169$	17.3789	1.33683
$170$	0	0
$171$	−1.79658	−0.137388
$172$	0	0
$173$	−23.8661	−1.81451	−0.907254	−	0.420582i	$-0.861826\pi$
$173$	−23.8661	−1.81451	−0.907254	+	0.420582i	$0.861826\pi$
$174$	0	0
$175$	−6.95944	−0.526084
$176$	0	0
$177$	−7.45358	−0.560245
$178$	0	0
$179$	−13.6857	−1.02292	−0.511459	−	0.859308i	$-0.670895\pi$
$179$	−13.6857	−1.02292	−0.511459	+	0.859308i	$0.670895\pi$
$180$	0	0
$181$	−2.99168	−0.222370	−0.111185	−	0.993800i	$-0.535465\pi$
$181$	−2.99168	−0.222370	−0.111185	+	0.993800i	$0.535465\pi$
$182$	0	0
$183$	10.2974	0.761206
$184$	0	0
$185$	14.3455	1.05470
$186$	0	0
$187$	1.41730	0.103643
$188$	0	0
$189$	−18.2473	−1.32729
$190$	0	0
$191$	−3.34287	−0.241882	−0.120941	−	0.992660i	$-0.538591\pi$
$191$	−3.34287	−0.241882	−0.120941	+	0.992660i	$0.538591\pi$
$192$	0	0
$193$	6.80421	0.489778	0.244889	−	0.969551i	$-0.421249\pi$
$193$	6.80421	0.489778	0.244889	+	0.969551i	$0.421249\pi$
$194$	0	0
$195$	−16.1538	−1.15680
$196$	0	0
$197$	7.87607	0.561147	0.280573	−	0.959833i	$-0.409475\pi$
$197$	7.87607	0.561147	0.280573	+	0.959833i	$0.409475\pi$
$198$	0	0
$199$	8.53639	0.605129	0.302564	−	0.953129i	$-0.402157\pi$
$199$	8.53639	0.605129	0.302564	+	0.953129i	$0.402157\pi$
$200$	0	0
$201$	2.93148	0.206771
$202$	0	0
$203$	18.7789	1.31802
$204$	0	0
$205$	−5.90644	−0.412523
$206$	0	0
$207$	0.270182	0.0187789
$208$	0	0
$209$	−5.66101	−0.391580
$210$	0	0
$211$	−27.2506	−1.87601	−0.938004	−	0.346626i	$-0.887327\pi$
$211$	−27.2506	−1.87601	−0.938004	+	0.346626i	$0.887327\pi$
$212$	0	0
$213$	5.37572	0.368338
$214$	0	0
$215$	14.2963	0.975000
$216$	0	0
$217$	−31.8442	−2.16173
$218$	0	0
$219$	−5.03200	−0.340031
$220$	0	0
$221$	−1.39623	−0.0939205
$222$	0	0
$223$	−20.1037	−1.34624	−0.673121	−	0.739532i	$-0.735047\pi$
$223$	−20.1037	−1.34624	−0.673121	+	0.739532i	$0.735047\pi$
$224$	0	0
$225$	−3.57858	−0.238572
$226$	0	0
$227$	15.8933	1.05488	0.527439	−	0.849593i	$-0.323152\pi$
$227$	15.8933	1.05488	0.527439	+	0.849593i	$0.323152\pi$
$228$	0	0
$229$	−28.6025	−1.89011	−0.945054	−	0.326913i	$-0.893991\pi$
$229$	−28.6025	−1.89011	−0.945054	+	0.326913i	$0.893991\pi$
$230$	0	0
$231$	−21.3776	−1.40655
$232$	0	0
$233$	21.1158	1.38335	0.691673	−	0.722211i	$-0.256874\pi$
$233$	21.1158	1.38335	0.691673	+	0.722211i	$0.256874\pi$
$234$	0	0
$235$	23.0212	1.50173
$236$	0	0
$237$	10.0203	0.650888
$238$	0	0
$239$	5.17446	0.334708	0.167354	−	0.985897i	$-0.446478\pi$
$239$	5.17446	0.334708	0.167354	+	0.985897i	$0.446478\pi$
$240$	0	0
$241$	11.8479	0.763189	0.381594	−	0.924330i	$-0.375375\pi$
$241$	11.8479	0.763189	0.381594	+	0.924330i	$0.375375\pi$
$242$	0	0
$243$	−15.2771	−0.980024
$244$	0	0
$245$	13.0415	0.833190
$246$	0	0
$247$	5.57686	0.354847
$248$	0	0
$249$	−8.09681	−0.513115
$250$	0	0
$251$	5.78622	0.365223	0.182611	−	0.983185i	$-0.441545\pi$
$251$	5.78622	0.365223	0.182611	+	0.983185i	$0.441545\pi$
$252$	0	0
$253$	0.851338	0.0535232
$254$	0	0
$255$	0.742439	0.0464933
$256$	0	0
$257$	5.54269	0.345744	0.172872	−	0.984944i	$-0.444695\pi$
$257$	5.54269	0.345744	0.172872	+	0.984944i	$0.444695\pi$
$258$	0	0
$259$	−18.7023	−1.16210
$260$	0	0
$261$	9.65621	0.597705
$262$	0	0
$263$	18.0393	1.11235	0.556175	−	0.831065i	$-0.312268\pi$
$263$	18.0393	1.11235	0.556175	+	0.831065i	$0.312268\pi$
$264$	0	0
$265$	−12.7200	−0.781385
$266$	0	0
$267$	−14.5061	−0.887756
$268$	0	0
$269$	−10.8056	−0.658830	−0.329415	−	0.944185i	$-0.606852\pi$
$269$	−10.8056	−0.658830	−0.329415	+	0.944185i	$0.606852\pi$
$270$	0	0
$271$	−24.7882	−1.50578	−0.752888	−	0.658149i	$-0.771340\pi$
$271$	−24.7882	−1.50578	−0.752888	+	0.658149i	$0.771340\pi$
$272$	0	0
$273$	21.0599	1.27460
$274$	0	0
$275$	−11.2760	−0.679971
$276$	0	0
$277$	22.0565	1.32525	0.662624	−	0.748952i	$-0.269443\pi$
$277$	22.0565	1.32525	0.662624	+	0.748952i	$0.269443\pi$
$278$	0	0
$279$	−16.3745	−0.980313
$280$	0	0
$281$	27.7322	1.65436	0.827181	−	0.561935i	$-0.189943\pi$
$281$	27.7322	1.65436	0.827181	+	0.561935i	$0.189943\pi$
$282$	0	0
$283$	1.85187	0.110082	0.0550411	−	0.998484i	$-0.482471\pi$
$283$	1.85187	0.110082	0.0550411	+	0.998484i	$0.482471\pi$
$284$	0	0
$285$	−2.96547	−0.175660
$286$	0	0
$287$	7.70027	0.454532
$288$	0	0
$289$	−16.9358	−0.996225
$290$	0	0
$291$	19.4508	1.14023
$292$	0	0
$293$	31.6493	1.84897	0.924487	−	0.381213i	$-0.124493\pi$
$293$	31.6493	1.84897	0.924487	+	0.381213i	$0.124493\pi$
$294$	0	0
$295$	17.8414	1.03877
$296$	0	0
$297$	−29.5651	−1.71554
$298$	0	0
$299$	−0.838684	−0.0485023
$300$	0	0
$301$	−18.6382	−1.07429
$302$	0	0
$303$	−11.2638	−0.647088
$304$	0	0
$305$	−24.6486	−1.41137
$306$	0	0
$307$	1.43835	0.0820911	0.0410456	−	0.999157i	$-0.486931\pi$
$307$	1.43835	0.0820911	0.0410456	+	0.999157i	$0.486931\pi$
$308$	0	0
$309$	−9.63243	−0.547970
$310$	0	0
$311$	−11.4755	−0.650718	−0.325359	−	0.945591i	$-0.605485\pi$
$311$	−11.4755	−0.650718	−0.325359	+	0.945591i	$0.605485\pi$
$312$	0	0
$313$	−17.3679	−0.981694	−0.490847	−	0.871246i	$-0.663313\pi$
$313$	−17.3679	−0.981694	−0.490847	+	0.871246i	$0.663313\pi$
$314$	0	0
$315$	16.2397	0.915002
$316$	0	0
$317$	1.08088	0.0607083	0.0303541	−	0.999539i	$-0.490336\pi$
$317$	1.08088	0.0607083	0.0303541	+	0.999539i	$0.490336\pi$
$318$	0	0
$319$	30.4266	1.70356
$320$	0	0
$321$	2.30068	0.128411
$322$	0	0
$323$	−0.256315	−0.0142618
$324$	0	0
$325$	11.1084	0.616185
$326$	0	0
$327$	19.4717	1.07679
$328$	0	0
$329$	−30.0129	−1.65466
$330$	0	0
$331$	5.62253	0.309042	0.154521	−	0.987989i	$-0.450617\pi$
$331$	5.62253	0.309042	0.154521	+	0.987989i	$0.450617\pi$
$332$	0	0
$333$	−9.61681	−0.526998
$334$	0	0
$335$	−7.01700	−0.383380
$336$	0	0
$337$	−3.54088	−0.192884	−0.0964421	−	0.995339i	$-0.530746\pi$
$337$	−3.54088	−0.192884	−0.0964421	+	0.995339i	$0.530746\pi$
$338$	0	0
$339$	12.2658	0.666188
$340$	0	0
$341$	−51.5957	−2.79406
$342$	0	0
$343$	7.16932	0.387107
$344$	0	0
$345$	0.445967	0.0240100
$346$	0	0
$347$	21.2017	1.13816	0.569082	−	0.822281i	$-0.307299\pi$
$347$	21.2017	1.13816	0.569082	+	0.822281i	$0.307299\pi$
$348$	0	0
$349$	−26.5640	−1.42194	−0.710968	−	0.703224i	$-0.751743\pi$
$349$	−26.5640	−1.42194	−0.710968	+	0.703224i	$0.751743\pi$
$350$	0	0
$351$	29.1257	1.55461
$352$	0	0
$353$	−4.42319	−0.235423	−0.117711	−	0.993048i	$-0.537556\pi$
$353$	−4.42319	−0.235423	−0.117711	+	0.993048i	$0.537556\pi$
$354$	0	0
$355$	−12.8677	−0.682946
$356$	0	0
$357$	−0.967923	−0.0512279
$358$	0	0
$359$	11.0678	0.584134	0.292067	−	0.956398i	$-0.405657\pi$
$359$	11.0678	0.584134	0.292067	+	0.956398i	$0.405657\pi$
$360$	0	0
$361$	−17.9762	−0.946117
$362$	0	0
$363$	−22.4653	−1.17912
$364$	0	0
$365$	12.0449	0.630461
$366$	0	0
$367$	27.6518	1.44341	0.721706	−	0.692200i	$-0.243358\pi$
$367$	27.6518	1.44341	0.721706	+	0.692200i	$0.243358\pi$
$368$	0	0
$369$	3.95951	0.206124
$370$	0	0
$371$	16.5832	0.860956
$372$	0	0
$373$	3.17358	0.164322	0.0821608	−	0.996619i	$-0.473818\pi$
$373$	3.17358	0.164322	0.0821608	+	0.996619i	$0.473818\pi$
$374$	0	0
$375$	8.74727	0.451707
$376$	0	0
$377$	−29.9743	−1.54376
$378$	0	0
$379$	−25.3397	−1.30161	−0.650807	−	0.759243i	$-0.725569\pi$
$379$	−25.3397	−1.30161	−0.650807	+	0.759243i	$0.725569\pi$
$380$	0	0
$381$	10.8838	0.557593
$382$	0	0
$383$	−13.0103	−0.664798	−0.332399	−	0.943139i	$-0.607858\pi$
$383$	−13.0103	−0.664798	−0.332399	+	0.943139i	$0.607858\pi$
$384$	0	0
$385$	51.1710	2.60792
$386$	0	0
$387$	−9.58386	−0.487175
$388$	0	0
$389$	5.73547	0.290800	0.145400	−	0.989373i	$-0.453553\pi$
$389$	5.73547	0.290800	0.145400	+	0.989373i	$0.453553\pi$
$390$	0	0
$391$	0.0385463	0.00194937
$392$	0	0
$393$	6.45439	0.325581
$394$	0	0
$395$	−23.9853	−1.20683
$396$	0	0
$397$	28.6674	1.43878	0.719389	−	0.694608i	$-0.244422\pi$
$397$	28.6674	1.43878	0.719389	+	0.694608i	$0.244422\pi$
$398$	0	0
$399$	3.86611	0.193548
$400$	0	0
$401$	−17.7891	−0.888345	−0.444172	−	0.895941i	$-0.646502\pi$
$401$	−17.7891	−0.888345	−0.444172	+	0.895941i	$0.646502\pi$
$402$	0	0
$403$	50.8287	2.53196
$404$	0	0
$405$	−1.37862	−0.0685040
$406$	0	0
$407$	−30.3024	−1.50204
$408$	0	0
$409$	19.9478	0.986354	0.493177	−	0.869929i	$-0.335836\pi$
$409$	19.9478	0.986354	0.493177	+	0.869929i	$0.335836\pi$
$410$	0	0
$411$	−3.14770	−0.155265
$412$	0	0
$413$	−23.2600	−1.14455
$414$	0	0
$415$	19.3811	0.951380
$416$	0	0
$417$	8.23347	0.403195
$418$	0	0
$419$	−29.6403	−1.44802	−0.724012	−	0.689788i	$-0.757704\pi$
$419$	−29.6403	−1.44802	−0.724012	+	0.689788i	$0.757704\pi$
$420$	0	0
$421$	20.1714	0.983095	0.491548	−	0.870851i	$-0.336431\pi$
$421$	20.1714	0.983095	0.491548	+	0.870851i	$0.336431\pi$
$422$	0	0
$423$	−15.4328	−0.750366
$424$	0	0
$425$	−0.510549	−0.0247653
$426$	0	0
$427$	32.1345	1.55510
$428$	0	0
$429$	34.1223	1.64744
$430$	0	0
$431$	−18.0405	−0.868981	−0.434491	−	0.900676i	$-0.643072\pi$
$431$	−18.0405	−0.868981	−0.434491	+	0.900676i	$0.643072\pi$
$432$	0	0
$433$	−13.4873	−0.648158	−0.324079	−	0.946030i	$-0.605054\pi$
$433$	−13.4873	−0.648158	−0.324079	+	0.946030i	$0.605054\pi$
$434$	0	0
$435$	15.9387	0.764203
$436$	0	0
$437$	−0.153963	−0.00736505
$438$	0	0
$439$	−21.5828	−1.03009	−0.515047	−	0.857162i	$-0.672225\pi$
$439$	−21.5828	−1.03009	−0.515047	+	0.857162i	$0.672225\pi$
$440$	0	0
$441$	−8.74265	−0.416317
$442$	0	0
$443$	24.7293	1.17492	0.587461	−	0.809252i	$-0.300127\pi$
$443$	24.7293	1.17492	0.587461	+	0.809252i	$0.300127\pi$
$444$	0	0
$445$	34.7227	1.64601
$446$	0	0
$447$	11.7128	0.553998
$448$	0	0
$449$	−39.2803	−1.85375	−0.926877	−	0.375366i	$-0.877517\pi$
$449$	−39.2803	−1.85375	−0.926877	+	0.375366i	$0.877517\pi$
$450$	0	0
$451$	12.4764	0.587489
$452$	0	0
$453$	17.0959	0.803233
$454$	0	0
$455$	−50.4104	−2.36327
$456$	0	0
$457$	−25.7173	−1.20300	−0.601501	−	0.798872i	$-0.705431\pi$
$457$	−25.7173	−1.20300	−0.601501	+	0.798872i	$0.705431\pi$
$458$	0	0
$459$	−1.33863	−0.0624820
$460$	0	0
$461$	−1.54173	−0.0718056	−0.0359028	−	0.999355i	$-0.511431\pi$
$461$	−1.54173	−0.0718056	−0.0359028	+	0.999355i	$0.511431\pi$
$462$	0	0
$463$	−13.4971	−0.627264	−0.313632	−	0.949545i	$-0.601546\pi$
$463$	−13.4971	−0.627264	−0.313632	+	0.949545i	$0.601546\pi$
$464$	0	0
$465$	−27.0280	−1.25339
$466$	0	0
$467$	24.1933	1.11953	0.559765	−	0.828651i	$-0.310891\pi$
$467$	24.1933	1.11953	0.559765	+	0.828651i	$0.310891\pi$
$468$	0	0
$469$	9.14811	0.422421
$470$	0	0
$471$	−1.96220	−0.0904133
$472$	0	0
$473$	−30.1986	−1.38853
$474$	0	0
$475$	2.03925	0.0935673
$476$	0	0
$477$	8.52715	0.390432
$478$	0	0
$479$	−37.7815	−1.72628	−0.863141	−	0.504963i	$-0.831506\pi$
$479$	−37.7815	−1.72628	−0.863141	+	0.504963i	$0.831506\pi$
$480$	0	0
$481$	29.8520	1.36113
$482$	0	0
$483$	−0.581410	−0.0264551
$484$	0	0
$485$	−46.5588	−2.11412
$486$	0	0
$487$	10.8779	0.492927	0.246463	−	0.969152i	$-0.420732\pi$
$487$	10.8779	0.492927	0.246463	+	0.969152i	$0.420732\pi$
$488$	0	0
$489$	26.0088	1.17616
$490$	0	0
$491$	42.2880	1.90843	0.954214	−	0.299124i	$-0.0966943\pi$
$491$	42.2880	1.90843	0.954214	+	0.299124i	$0.0966943\pi$
$492$	0	0
$493$	1.37764	0.0620456
$494$	0	0
$495$	26.3124	1.18265
$496$	0	0
$497$	16.7757	0.752493
$498$	0	0
$499$	−4.70100	−0.210446	−0.105223	−	0.994449i	$-0.533556\pi$
$499$	−4.70100	−0.210446	−0.105223	+	0.994449i	$0.533556\pi$
$500$	0	0
$501$	7.77843	0.347514
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	26.9618	1.19978
$506$	0	0
$507$	−19.2302	−0.854044
$508$	0	0
$509$	−19.6910	−0.872790	−0.436395	−	0.899755i	$-0.643745\pi$
$509$	−19.6910	−0.872790	−0.436395	+	0.899755i	$0.643745\pi$
$510$	0	0
$511$	−15.7031	−0.694663
$512$	0	0
$513$	5.34680	0.236067
$514$	0	0
$515$	23.0568	1.01601
$516$	0	0
$517$	−48.6284	−2.13867
$518$	0	0
$519$	26.4086	1.15921
$520$	0	0
$521$	−20.7388	−0.908582	−0.454291	−	0.890853i	$-0.650107\pi$
$521$	−20.7388	−0.908582	−0.454291	+	0.890853i	$0.650107\pi$
$522$	0	0
$523$	−29.5964	−1.29416	−0.647081	−	0.762421i	$-0.724010\pi$
$523$	−29.5964	−1.29416	−0.647081	+	0.762421i	$0.724010\pi$
$524$	0	0
$525$	7.70082	0.336091
$526$	0	0
$527$	−2.33611	−0.101763
$528$	0	0
$529$	−22.9768	−0.998993
$530$	0	0
$531$	−11.9604	−0.519036
$532$	0	0
$533$	−12.2909	−0.532379
$534$	0	0
$535$	−5.50707	−0.238091
$536$	0	0
$537$	15.1436	0.653497
$538$	0	0
$539$	−27.5480	−1.18657
$540$	0	0
$541$	31.5383	1.35594	0.677969	−	0.735091i	$-0.262861\pi$
$541$	31.5383	1.35594	0.677969	+	0.735091i	$0.262861\pi$
$542$	0	0
$543$	3.31039	0.142062
$544$	0	0
$545$	−46.6088	−1.99650
$546$	0	0
$547$	−44.7638	−1.91396	−0.956981	−	0.290151i	$-0.906294\pi$
$547$	−44.7638	−1.91396	−0.956981	+	0.290151i	$0.906294\pi$
$548$	0	0
$549$	16.5237	0.705216
$550$	0	0
$551$	−5.50260	−0.234418
$552$	0	0
$553$	31.2698	1.32973
$554$	0	0
$555$	−15.8737	−0.673801
$556$	0	0
$557$	8.75189	0.370829	0.185415	−	0.982660i	$-0.440637\pi$
$557$	8.75189	0.370829	0.185415	+	0.982660i	$0.440637\pi$
$558$	0	0
$559$	29.7497	1.25828
$560$	0	0
$561$	−1.56828	−0.0662128
$562$	0	0
$563$	27.6105	1.16365	0.581823	−	0.813316i	$-0.302340\pi$
$563$	27.6105	1.16365	0.581823	+	0.813316i	$0.302340\pi$
$564$	0	0
$565$	−29.3603	−1.23520
$566$	0	0
$567$	1.79731	0.0754800
$568$	0	0
$569$	−7.99979	−0.335369	−0.167684	−	0.985841i	$-0.553629\pi$
$569$	−7.99979	−0.335369	−0.167684	+	0.985841i	$0.553629\pi$
$570$	0	0
$571$	33.7524	1.41249	0.706247	−	0.707966i	$-0.250387\pi$
$571$	33.7524	1.41249	0.706247	+	0.707966i	$0.250387\pi$
$572$	0	0
$573$	3.69899	0.154527
$574$	0	0
$575$	−0.306676	−0.0127893
$576$	0	0
$577$	14.6426	0.609578	0.304789	−	0.952420i	$-0.401414\pi$
$577$	14.6426	0.609578	0.304789	+	0.952420i	$0.401414\pi$
$578$	0	0
$579$	−7.52906	−0.312897
$580$	0	0
$581$	−25.2673	−1.04826
$582$	0	0
$583$	26.8689	1.11280
$584$	0	0
$585$	−25.9213	−1.07171
$586$	0	0
$587$	16.8664	0.696150	0.348075	−	0.937467i	$-0.386835\pi$
$587$	16.8664	0.696150	0.348075	+	0.937467i	$0.386835\pi$
$588$	0	0
$589$	9.33099	0.384477
$590$	0	0
$591$	−8.71511	−0.358492
$592$	0	0
$593$	−23.6172	−0.969844	−0.484922	−	0.874557i	$-0.661152\pi$
$593$	−23.6172	−0.969844	−0.484922	+	0.874557i	$0.661152\pi$
$594$	0	0
$595$	2.31689	0.0949831
$596$	0	0
$597$	−9.44577	−0.386590
$598$	0	0
$599$	−38.6256	−1.57820	−0.789100	−	0.614264i	$-0.789453\pi$
$599$	−38.6256	−1.57820	−0.789100	+	0.614264i	$0.789453\pi$
$600$	0	0
$601$	39.7927	1.62318	0.811588	−	0.584230i	$-0.198603\pi$
$601$	39.7927	1.62318	0.811588	+	0.584230i	$0.198603\pi$
$602$	0	0
$603$	4.70400	0.191562
$604$	0	0
$605$	53.7746	2.18625
$606$	0	0
$607$	6.88788	0.279570	0.139785	−	0.990182i	$-0.455359\pi$
$607$	6.88788	0.279570	0.139785	+	0.990182i	$0.455359\pi$
$608$	0	0
$609$	−20.7794	−0.842025
$610$	0	0
$611$	47.9056	1.93805
$612$	0	0
$613$	−16.6795	−0.673679	−0.336840	−	0.941562i	$-0.609358\pi$
$613$	−16.6795	−0.673679	−0.336840	+	0.941562i	$0.609358\pi$
$614$	0	0
$615$	6.53565	0.263543
$616$	0	0
$617$	−34.5797	−1.39213	−0.696064	−	0.717980i	$-0.745067\pi$
$617$	−34.5797	−1.39213	−0.696064	+	0.717980i	$0.745067\pi$
$618$	0	0
$619$	−9.64535	−0.387679	−0.193840	−	0.981033i	$-0.562094\pi$
$619$	−9.64535	−0.387679	−0.193840	+	0.981033i	$0.562094\pi$
$620$	0	0
$621$	−0.804087	−0.0322669
$622$	0	0
$623$	−45.2682	−1.81363
$624$	0	0
$625$	−31.0152	−1.24061
$626$	0	0
$627$	6.26407	0.250163
$628$	0	0
$629$	−1.37201	−0.0547058
$630$	0	0
$631$	−33.7142	−1.34214	−0.671070	−	0.741394i	$-0.734165\pi$
$631$	−33.7142	−1.34214	−0.671070	+	0.741394i	$0.734165\pi$
$632$	0	0
$633$	30.1536	1.19850
$634$	0	0
$635$	−26.0522	−1.03385
$636$	0	0
$637$	27.1385	1.07527
$638$	0	0
$639$	8.62615	0.341245
$640$	0	0
$641$	16.1261	0.636942	0.318471	−	0.947933i	$-0.396831\pi$
$641$	16.1261	0.636942	0.318471	+	0.947933i	$0.396831\pi$
$642$	0	0
$643$	14.3115	0.564390	0.282195	−	0.959357i	$-0.408938\pi$
$643$	14.3115	0.564390	0.282195	+	0.959357i	$0.408938\pi$
$644$	0	0
$645$	−15.8193	−0.622884
$646$	0	0
$647$	−13.9256	−0.547473	−0.273736	−	0.961805i	$-0.588260\pi$
$647$	−13.9256	−0.547473	−0.273736	+	0.961805i	$0.588260\pi$
$648$	0	0
$649$	−37.6870	−1.47934
$650$	0	0
$651$	35.2366	1.38103
$652$	0	0
$653$	−5.08485	−0.198986	−0.0994928	−	0.995038i	$-0.531722\pi$
$653$	−5.08485	−0.198986	−0.0994928	+	0.995038i	$0.531722\pi$
$654$	0	0
$655$	−15.4497	−0.603669
$656$	0	0
$657$	−8.07460	−0.315020
$658$	0	0
$659$	42.4111	1.65210	0.826051	−	0.563595i	$-0.190582\pi$
$659$	42.4111	1.65210	0.826051	+	0.563595i	$0.190582\pi$
$660$	0	0
$661$	9.78266	0.380501	0.190251	−	0.981736i	$-0.439070\pi$
$661$	9.78266	0.380501	0.190251	+	0.981736i	$0.439070\pi$
$662$	0	0
$663$	1.54497	0.0600016
$664$	0	0
$665$	−9.25419	−0.358862
$666$	0	0
$667$	0.827515	0.0320415
$668$	0	0
$669$	22.2453	0.860054
$670$	0	0
$671$	52.0660	2.00999
$672$	0	0
$673$	3.48461	0.134322	0.0671609	−	0.997742i	$-0.478606\pi$
$673$	3.48461	0.134322	0.0671609	+	0.997742i	$0.478606\pi$
$674$	0	0
$675$	10.6502	0.409926
$676$	0	0
$677$	−9.39944	−0.361250	−0.180625	−	0.983552i	$-0.557812\pi$
$677$	−9.39944	−0.361250	−0.180625	+	0.983552i	$0.557812\pi$
$678$	0	0
$679$	60.6990	2.32941
$680$	0	0
$681$	−17.5865	−0.673914
$682$	0	0
$683$	−19.3111	−0.738918	−0.369459	−	0.929247i	$-0.620457\pi$
$683$	−19.3111	−0.738918	−0.369459	+	0.929247i	$0.620457\pi$
$684$	0	0
$685$	7.53456	0.287881
$686$	0	0
$687$	31.6495	1.20751
$688$	0	0
$689$	−26.4695	−1.00841
$690$	0	0
$691$	−38.6501	−1.47032	−0.735160	−	0.677893i	$-0.762893\pi$
$691$	−38.6501	−1.47032	−0.735160	+	0.677893i	$0.762893\pi$
$692$	0	0
$693$	−34.3036	−1.30309
$694$	0	0
$695$	−19.7082	−0.747575
$696$	0	0
$697$	0.564897	0.0213970
$698$	0	0
$699$	−23.3653	−0.883757
$700$	0	0
$701$	33.0541	1.24844	0.624219	−	0.781249i	$-0.285417\pi$
$701$	33.0541	1.24844	0.624219	+	0.781249i	$0.285417\pi$
$702$	0	0
$703$	5.48015	0.206688
$704$	0	0
$705$	−25.4736	−0.959391
$706$	0	0
$707$	−35.1503	−1.32196
$708$	0	0
$709$	30.9658	1.16295	0.581473	−	0.813566i	$-0.302477\pi$
$709$	30.9658	1.16295	0.581473	+	0.813566i	$0.302477\pi$
$710$	0	0
$711$	16.0791	0.603012
$712$	0	0
$713$	−1.40325	−0.0525522
$714$	0	0
$715$	−81.6775	−3.05456
$716$	0	0
$717$	−5.72570	−0.213830
$718$	0	0
$719$	49.1329	1.83235	0.916174	−	0.400781i	$-0.131261\pi$
$719$	49.1329	1.83235	0.916174	+	0.400781i	$0.131261\pi$
$720$	0	0
$721$	−30.0594	−1.11947
$722$	0	0
$723$	−13.1100	−0.487567
$724$	0	0
$725$	−10.9605	−0.407063
$726$	0	0
$727$	−6.24614	−0.231656	−0.115828	−	0.993269i	$-0.536952\pi$
$727$	−6.24614	−0.231656	−0.115828	+	0.993269i	$0.536952\pi$
$728$	0	0
$729$	18.4660	0.683926
$730$	0	0
$731$	−1.36731	−0.0505719
$732$	0	0
$733$	3.23948	0.119653	0.0598264	−	0.998209i	$-0.480945\pi$
$733$	3.23948	0.119653	0.0598264	+	0.998209i	$0.480945\pi$
$734$	0	0
$735$	−14.4308	−0.532287
$736$	0	0
$737$	14.8222	0.545984
$738$	0	0
$739$	16.5475	0.608711	0.304355	−	0.952559i	$-0.401559\pi$
$739$	16.5475	0.608711	0.304355	+	0.952559i	$0.401559\pi$
$740$	0	0
$741$	−6.17096	−0.226696
$742$	0	0
$743$	−27.9860	−1.02671	−0.513353	−	0.858178i	$-0.671597\pi$
$743$	−27.9860	−1.02671	−0.513353	+	0.858178i	$0.671597\pi$
$744$	0	0
$745$	−28.0367	−1.02718
$746$	0	0
$747$	−12.9925	−0.475372
$748$	0	0
$749$	7.17960	0.262337
$750$	0	0
$751$	−36.4017	−1.32832	−0.664159	−	0.747591i	$-0.731210\pi$
$751$	−36.4017	−1.32832	−0.664159	+	0.747591i	$0.731210\pi$
$752$	0	0
$753$	−6.40262	−0.233324
$754$	0	0
$755$	−40.9218	−1.48930
$756$	0	0
$757$	−24.4958	−0.890314	−0.445157	−	0.895453i	$-0.646852\pi$
$757$	−24.4958	−0.890314	−0.445157	+	0.895453i	$0.646852\pi$
$758$	0	0
$759$	−0.942031	−0.0341936
$760$	0	0
$761$	0.165261	0.00599069	0.00299534	−	0.999996i	$-0.499047\pi$
$761$	0.165261	0.00599069	0.00299534	+	0.999996i	$0.499047\pi$
$762$	0	0
$763$	60.7642	2.19981
$764$	0	0
$765$	1.19135	0.0430735
$766$	0	0
$767$	37.1268	1.34057
$768$	0	0
$769$	35.6959	1.28723	0.643613	−	0.765351i	$-0.277435\pi$
$769$	35.6959	1.28723	0.643613	+	0.765351i	$0.277435\pi$
$770$	0	0
$771$	−6.13315	−0.220880
$772$	0	0
$773$	−22.4134	−0.806153	−0.403076	−	0.915166i	$-0.632059\pi$
$773$	−22.4134	−0.806153	−0.403076	+	0.915166i	$0.632059\pi$
$774$	0	0
$775$	18.5862	0.667636
$776$	0	0
$777$	20.6947	0.742417
$778$	0	0
$779$	−2.25633	−0.0808414
$780$	0	0
$781$	27.1809	0.972608
$782$	0	0
$783$	−28.7378	−1.02701
$784$	0	0
$785$	4.69685	0.167638
$786$	0	0
$787$	−13.2635	−0.472793	−0.236396	−	0.971657i	$-0.575966\pi$
$787$	−13.2635	−0.472793	−0.236396	+	0.971657i	$0.575966\pi$
$788$	0	0
$789$	−19.9610	−0.710630
$790$	0	0
$791$	38.2772	1.36098
$792$	0	0
$793$	−51.2921	−1.82144
$794$	0	0
$795$	14.0751	0.499192
$796$	0	0
$797$	25.7119	0.910762	0.455381	−	0.890297i	$-0.349503\pi$
$797$	25.7119	0.910762	0.455381	+	0.890297i	$0.349503\pi$
$798$	0	0
$799$	−2.20176	−0.0778928
$800$	0	0
$801$	−23.2771	−0.822457
$802$	0	0
$803$	−25.4429	−0.897862
$804$	0	0
$805$	1.39170	0.0490511
$806$	0	0
$807$	11.9567	0.420897
$808$	0	0
$809$	1.32334	0.0465261	0.0232630	−	0.999729i	$-0.492594\pi$
$809$	1.32334	0.0465261	0.0232630	+	0.999729i	$0.492594\pi$
$810$	0	0
$811$	7.00504	0.245980	0.122990	−	0.992408i	$-0.460752\pi$
$811$	7.00504	0.245980	0.122990	+	0.992408i	$0.460752\pi$
$812$	0	0
$813$	27.4289	0.961973
$814$	0	0
$815$	−62.2564	−2.18075
$816$	0	0
$817$	5.46136	0.191069
$818$	0	0
$819$	33.7937	1.18085
$820$	0	0
$821$	−3.93207	−0.137230	−0.0686151	−	0.997643i	$-0.521858\pi$
$821$	−3.93207	−0.137230	−0.0686151	+	0.997643i	$0.521858\pi$
$822$	0	0
$823$	−36.6152	−1.27633	−0.638163	−	0.769901i	$-0.720305\pi$
$823$	−36.6152	−1.27633	−0.638163	+	0.769901i	$0.720305\pi$
$824$	0	0
$825$	12.4773	0.434403
$826$	0	0
$827$	−29.8967	−1.03961	−0.519806	−	0.854284i	$-0.673996\pi$
$827$	−29.8967	−1.03961	−0.519806	+	0.854284i	$0.673996\pi$
$828$	0	0
$829$	−19.1278	−0.664335	−0.332168	−	0.943220i	$-0.607780\pi$
$829$	−19.1278	−0.664335	−0.332168	+	0.943220i	$0.607780\pi$
$830$	0	0
$831$	−24.4062	−0.846642
$832$	0	0
$833$	−1.24730	−0.0432163
$834$	0	0
$835$	−18.6190	−0.644336
$836$	0	0
$837$	48.7319	1.68442
$838$	0	0
$839$	−8.80446	−0.303964	−0.151982	−	0.988383i	$-0.548566\pi$
$839$	−8.80446	−0.303964	−0.151982	+	0.988383i	$0.548566\pi$
$840$	0	0
$841$	0.575159	0.0198331
$842$	0	0
$843$	−30.6865	−1.05690
$844$	0	0
$845$	46.0308	1.58351
$846$	0	0
$847$	−70.1063	−2.40888
$848$	0	0
$849$	−2.04915	−0.0703266
$850$	0	0
$851$	−0.824139	−0.0282511
$852$	0	0
$853$	33.6043	1.15059	0.575295	−	0.817946i	$-0.304887\pi$
$853$	33.6043	1.15059	0.575295	+	0.817946i	$0.304887\pi$
$854$	0	0
$855$	−4.75855	−0.162739
$856$	0	0
$857$	−41.6823	−1.42384	−0.711920	−	0.702261i	$-0.752174\pi$
$857$	−41.6823	−1.42384	−0.711920	+	0.702261i	$0.752174\pi$
$858$	0	0
$859$	−10.5881	−0.361263	−0.180631	−	0.983551i	$-0.557814\pi$
$859$	−10.5881	−0.361263	−0.180631	+	0.983551i	$0.557814\pi$
$860$	0	0
$861$	−8.52057	−0.290380
$862$	0	0
$863$	5.84726	0.199043	0.0995216	−	0.995035i	$-0.468269\pi$
$863$	5.84726	0.199043	0.0995216	+	0.995035i	$0.468269\pi$
$864$	0	0
$865$	−63.2134	−2.14932
$866$	0	0
$867$	18.7400	0.636444
$868$	0	0
$869$	50.6649	1.71869
$870$	0	0
$871$	−14.6019	−0.494767
$872$	0	0
$873$	31.2117	1.05636
$874$	0	0
$875$	27.2971	0.922811
$876$	0	0
$877$	−48.8752	−1.65040	−0.825199	−	0.564841i	$-0.808937\pi$
$877$	−48.8752	−1.65040	−0.825199	+	0.564841i	$0.808937\pi$
$878$	0	0
$879$	−35.0209	−1.18123
$880$	0	0
$881$	−49.2748	−1.66011	−0.830055	−	0.557682i	$-0.811691\pi$
$881$	−49.2748	−1.66011	−0.830055	+	0.557682i	$0.811691\pi$
$882$	0	0
$883$	−14.0318	−0.472209	−0.236104	−	0.971728i	$-0.575871\pi$
$883$	−14.0318	−0.472209	−0.236104	+	0.971728i	$0.575871\pi$
$884$	0	0
$885$	−19.7420	−0.663621
$886$	0	0
$887$	−7.88379	−0.264712	−0.132356	−	0.991202i	$-0.542254\pi$
$887$	−7.88379	−0.264712	−0.132356	+	0.991202i	$0.542254\pi$
$888$	0	0
$889$	33.9644	1.13913
$890$	0	0
$891$	2.91210	0.0975589
$892$	0	0
$893$	8.79436	0.294292
$894$	0	0
$895$	−36.2489	−1.21167
$896$	0	0
$897$	0.928028	0.0309860
$898$	0	0
$899$	−50.1519	−1.67266
$900$	0	0
$901$	1.21655	0.0405293
$902$	0	0
$903$	20.6237	0.686315
$904$	0	0
$905$	−7.92397	−0.263402
$906$	0	0
$907$	−30.8725	−1.02510	−0.512552	−	0.858656i	$-0.671300\pi$
$907$	−30.8725	−1.02510	−0.512552	+	0.858656i	$0.671300\pi$
$908$	0	0
$909$	−18.0744	−0.599491
$910$	0	0
$911$	46.5712	1.54297	0.771486	−	0.636247i	$-0.219514\pi$
$911$	46.5712	1.54297	0.771486	+	0.636247i	$0.219514\pi$
$912$	0	0
$913$	−40.9393	−1.35489
$914$	0	0
$915$	27.2744	0.901663
$916$	0	0
$917$	20.1419	0.665143
$918$	0	0
$919$	15.6383	0.515859	0.257930	−	0.966164i	$-0.416960\pi$
$919$	15.6383	0.515859	0.257930	+	0.966164i	$0.416960\pi$
$920$	0	0
$921$	−1.59158	−0.0524443
$922$	0	0
$923$	−26.7768	−0.881370
$924$	0	0
$925$	10.9158	0.358909
$926$	0	0
$927$	−15.4567	−0.507664
$928$	0	0
$929$	−10.2431	−0.336066	−0.168033	−	0.985781i	$-0.553741\pi$
$929$	−10.2431	−0.336066	−0.168033	+	0.985781i	$0.553741\pi$
$930$	0	0
$931$	4.98200	0.163279
$932$	0	0
$933$	12.6980	0.415715
$934$	0	0
$935$	3.75394	0.122767
$936$	0	0
$937$	−24.1407	−0.788642	−0.394321	−	0.918973i	$-0.629020\pi$
$937$	−24.1407	−0.788642	−0.394321	+	0.918973i	$0.629020\pi$
$938$	0	0
$939$	19.2181	0.627160
$940$	0	0
$941$	−27.9796	−0.912110	−0.456055	−	0.889952i	$-0.650738\pi$
$941$	−27.9796	−0.912110	−0.456055	+	0.889952i	$0.650738\pi$
$942$	0	0
$943$	0.339321	0.0110498
$944$	0	0
$945$	−48.3309	−1.57220
$946$	0	0
$947$	−13.9023	−0.451764	−0.225882	−	0.974155i	$-0.572526\pi$
$947$	−13.9023	−0.451764	−0.225882	+	0.974155i	$0.572526\pi$
$948$	0	0
$949$	25.0647	0.813636
$950$	0	0
$951$	−1.19603	−0.0387838
$952$	0	0
$953$	−26.5201	−0.859072	−0.429536	−	0.903050i	$-0.641323\pi$
$953$	−26.5201	−0.859072	−0.429536	+	0.903050i	$0.641323\pi$
$954$	0	0
$955$	−8.85415	−0.286514
$956$	0	0
$957$	−33.6679	−1.08833
$958$	0	0
$959$	−9.82287	−0.317197
$960$	0	0
$961$	54.0446	1.74338
$962$	0	0
$963$	3.69179	0.118966
$964$	0	0
$965$	18.0221	0.580151
$966$	0	0
$967$	17.3056	0.556510	0.278255	−	0.960507i	$-0.410244\pi$
$967$	17.3056	0.556510	0.278255	+	0.960507i	$0.410244\pi$
$968$	0	0
$969$	0.283621	0.00911121
$970$	0	0
$971$	22.5813	0.724668	0.362334	−	0.932048i	$-0.381980\pi$
$971$	22.5813	0.724668	0.362334	+	0.932048i	$0.381980\pi$
$972$	0	0
$973$	25.6937	0.823704
$974$	0	0
$975$	−12.2918	−0.393653
$976$	0	0
$977$	41.5823	1.33034	0.665168	−	0.746694i	$-0.268360\pi$
$977$	41.5823	1.33034	0.665168	+	0.746694i	$0.268360\pi$
$978$	0	0
$979$	−73.3459	−2.34415
$980$	0	0
$981$	31.2452	0.997584
$982$	0	0
$983$	−26.4107	−0.842371	−0.421186	−	0.906974i	$-0.638386\pi$
$983$	−26.4107	−0.842371	−0.421186	+	0.906974i	$0.638386\pi$
$984$	0	0
$985$	20.8611	0.664689
$986$	0	0
$987$	33.2101	1.05709
$988$	0	0
$989$	−0.821315	−0.0261163
$990$	0	0
$991$	−18.4896	−0.587342	−0.293671	−	0.955907i	$-0.594877\pi$
$991$	−18.4896	−0.587342	−0.293671	+	0.955907i	$0.594877\pi$
$992$	0	0
$993$	−6.22150	−0.197433
$994$	0	0
$995$	22.6100	0.716786
$996$	0	0
$997$	36.0827	1.14275	0.571376	−	0.820689i	$-0.306410\pi$
$997$	36.0827	1.14275	0.571376	+	0.820689i	$0.306410\pi$
$998$	0	0
$999$	28.6206	0.905515

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.s.1.10		21
4.3	odd	2		2012.2.a.b.1.12	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2012.2.a.b.1.12	✓	21	4.3	odd	2
8048.2.a.s.1.10		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)

\(f(q)\)	\(=\)	\(q-1.10653 q^{3} +2.64866 q^{5} -3.45308 q^{7} -1.77559 q^{9} +O(q^{10})\)	\(q-1.10653 q^{3} +2.64866 q^{5} -3.45308 q^{7} -1.77559 q^{9} -5.59487 q^{11} +5.51170 q^{13} -2.93083 q^{15} -0.253321 q^{17} +1.01182 q^{19} +3.82094 q^{21} -0.152164 q^{23} +2.01543 q^{25} +5.28433 q^{27} -5.43830 q^{29} +9.22196 q^{31} +6.19088 q^{33} -9.14606 q^{35} +5.41612 q^{37} -6.09886 q^{39} -2.22997 q^{41} +5.39755 q^{43} -4.70295 q^{45} +8.69161 q^{47} +4.92379 q^{49} +0.280307 q^{51} -4.80243 q^{53} -14.8189 q^{55} -1.11961 q^{57} +6.73599 q^{59} -9.30604 q^{61} +6.13127 q^{63} +14.5987 q^{65} -2.64926 q^{67} +0.168374 q^{69} -4.85818 q^{71} +4.54755 q^{73} -2.23013 q^{75} +19.3195 q^{77} -9.05560 q^{79} -0.520494 q^{81} +7.31730 q^{83} -0.670962 q^{85} +6.01764 q^{87} +13.1095 q^{89} -19.0324 q^{91} -10.2044 q^{93} +2.67998 q^{95} -17.5782 q^{97} +9.93420 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q - 10 q^{3} + 3 q^{5} - 13 q^{7} + 21 q^{9}+O(q^{10}) \) 21 * q - 10 * q^3 + 3 * q^5 - 13 * q^7 + 21 * q^9	\( 21 q - 10 q^{3} + 3 q^{5} - 13 q^{7} + 21 q^{9} - 7 q^{11} + 12 q^{13} - 14 q^{15} + q^{17} - 14 q^{19} + 14 q^{21} - 26 q^{23} + 18 q^{25} - 37 q^{27} + 9 q^{29} - 28 q^{31} + 3 q^{33} - 20 q^{35} + 31 q^{37} - 29 q^{39} + 4 q^{41} - 38 q^{43} + 24 q^{45} - 9 q^{47} + 16 q^{49} - 15 q^{51} + 22 q^{53} - 35 q^{55} - q^{57} - 10 q^{59} + 22 q^{61} - 35 q^{63} - 14 q^{65} - 58 q^{67} + 15 q^{69} - 27 q^{71} - 6 q^{73} - 48 q^{75} + 16 q^{77} - 47 q^{79} + 29 q^{81} - 22 q^{83} + 14 q^{85} - 29 q^{87} + q^{89} - 51 q^{91} + 34 q^{93} - 23 q^{95} - 2 q^{97} - 22 q^{99}+O(q^{100}) \) 21 * q - 10 * q^3 + 3 * q^5 - 13 * q^7 + 21 * q^9 - 7 * q^11 + 12 * q^13 - 14 * q^15 + q^17 - 14 * q^19 + 14 * q^21 - 26 * q^23 + 18 * q^25 - 37 * q^27 + 9 * q^29 - 28 * q^31 + 3 * q^33 - 20 * q^35 + 31 * q^37 - 29 * q^39 + 4 * q^41 - 38 * q^43 + 24 * q^45 - 9 * q^47 + 16 * q^49 - 15 * q^51 + 22 * q^53 - 35 * q^55 - q^57 - 10 * q^59 + 22 * q^61 - 35 * q^63 - 14 * q^65 - 58 * q^67 + 15 * q^69 - 27 * q^71 - 6 * q^73 - 48 * q^75 + 16 * q^77 - 47 * q^79 + 29 * q^81 - 22 * q^83 + 14 * q^85 - 29 * q^87 + q^89 - 51 * q^91 + 34 * q^93 - 23 * q^95 - 2 * q^97 - 22 * q^99

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