LMFDB - Embedded newform 8048.2.a.v.1.24

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:	$28$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.24
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)

$f(q)$	$=$	$q+2.22675 q^{3} -3.90847 q^{5} +0.127234 q^{7} +1.95840 q^{9} +O(q^{10})$	$q+2.22675 q^{3} -3.90847 q^{5} +0.127234 q^{7} +1.95840 q^{9} -1.34859 q^{11} -1.48851 q^{13} -8.70316 q^{15} +1.12665 q^{17} +3.57046 q^{19} +0.283317 q^{21} +6.17751 q^{23} +10.2761 q^{25} -2.31939 q^{27} -0.0940486 q^{29} -1.35429 q^{31} -3.00297 q^{33} -0.497289 q^{35} +2.38774 q^{37} -3.31452 q^{39} +9.91105 q^{41} -2.37943 q^{43} -7.65433 q^{45} -12.8539 q^{47} -6.98381 q^{49} +2.50877 q^{51} +0.100296 q^{53} +5.27093 q^{55} +7.95051 q^{57} -10.4855 q^{59} -8.89059 q^{61} +0.249174 q^{63} +5.81778 q^{65} -5.58516 q^{67} +13.7557 q^{69} +9.40758 q^{71} +4.22594 q^{73} +22.8823 q^{75} -0.171587 q^{77} -12.2627 q^{79} -11.0399 q^{81} +0.492027 q^{83} -4.40349 q^{85} -0.209422 q^{87} +8.78902 q^{89} -0.189388 q^{91} -3.01567 q^{93} -13.9550 q^{95} +18.6955 q^{97} -2.64108 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * 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
$3$	2.22675	1.28561	0.642806	−	0.766029i	$-0.277770\pi$
$3$	2.22675	1.28561	0.642806	+	0.766029i	$0.277770\pi$
$4$	0	0
$5$	−3.90847	−1.74792	−0.873960	−	0.485998i	$-0.838456\pi$
$5$	−3.90847	−1.74792	−0.873960	+	0.485998i	$0.838456\pi$
$6$	0	0
$7$	0.127234	0.0480898	0.0240449	−	0.999711i	$-0.492346\pi$
$7$	0.127234	0.0480898	0.0240449	+	0.999711i	$0.492346\pi$
$8$	0	0
$9$	1.95840	0.652799
$10$	0	0
$11$	−1.34859	−0.406616	−0.203308	−	0.979115i	$-0.565169\pi$
$11$	−1.34859	−0.406616	−0.203308	+	0.979115i	$0.565169\pi$
$12$	0	0
$13$	−1.48851	−0.412837	−0.206419	−	0.978464i	$-0.566181\pi$
$13$	−1.48851	−0.412837	−0.206419	+	0.978464i	$0.566181\pi$
$14$	0	0
$15$	−8.70316	−2.24715
$16$	0	0
$17$	1.12665	0.273254	0.136627	−	0.990623i	$-0.456374\pi$
$17$	1.12665	0.273254	0.136627	+	0.990623i	$0.456374\pi$
$18$	0	0
$19$	3.57046	0.819120	0.409560	−	0.912283i	$-0.365682\pi$
$19$	3.57046	0.819120	0.409560	+	0.912283i	$0.365682\pi$
$20$	0	0
$21$	0.283317	0.0618249
$22$	0	0
$23$	6.17751	1.28810	0.644050	−	0.764983i	$-0.277253\pi$
$23$	6.17751	1.28810	0.644050	+	0.764983i	$0.277253\pi$
$24$	0	0
$25$	10.2761	2.05522
$26$	0	0
$27$	−2.31939	−0.446366
$28$	0	0
$29$	−0.0940486	−0.0174644	−0.00873220	−	0.999962i	$-0.502780\pi$
$29$	−0.0940486	−0.0174644	−0.00873220	+	0.999962i	$0.502780\pi$
$30$	0	0
$31$	−1.35429	−0.243238	−0.121619	−	0.992577i	$-0.538809\pi$
$31$	−1.35429	−0.243238	−0.121619	+	0.992577i	$0.538809\pi$
$32$	0	0
$33$	−3.00297	−0.522751
$34$	0	0
$35$	−0.497289	−0.0840571
$36$	0	0
$37$	2.38774	0.392542	0.196271	−	0.980550i	$-0.437117\pi$
$37$	2.38774	0.392542	0.196271	+	0.980550i	$0.437117\pi$
$38$	0	0
$39$	−3.31452	−0.530749
$40$	0	0
$41$	9.91105	1.54785	0.773923	−	0.633280i	$-0.218292\pi$
$41$	9.91105	1.54785	0.773923	+	0.633280i	$0.218292\pi$
$42$	0	0
$43$	−2.37943	−0.362860	−0.181430	−	0.983404i	$-0.558073\pi$
$43$	−2.37943	−0.362860	−0.181430	+	0.983404i	$0.558073\pi$
$44$	0	0
$45$	−7.65433	−1.14104
$46$	0	0
$47$	−12.8539	−1.87493	−0.937467	−	0.348074i	$-0.886836\pi$
$47$	−12.8539	−1.87493	−0.937467	+	0.348074i	$0.886836\pi$
$48$	0	0
$49$	−6.98381	−0.997687
$50$	0	0
$51$	2.50877	0.351298
$52$	0	0
$53$	0.100296	0.0137767	0.00688835	−	0.999976i	$-0.497807\pi$
$53$	0.100296	0.0137767	0.00688835	+	0.999976i	$0.497807\pi$
$54$	0	0
$55$	5.27093	0.710732
$56$	0	0
$57$	7.95051	1.05307
$58$	0	0
$59$	−10.4855	−1.36510	−0.682549	−	0.730840i	$-0.739129\pi$
$59$	−10.4855	−1.36510	−0.682549	+	0.730840i	$0.739129\pi$
$60$	0	0
$61$	−8.89059	−1.13832	−0.569162	−	0.822226i	$-0.692732\pi$
$61$	−8.89059	−1.13832	−0.569162	+	0.822226i	$0.692732\pi$
$62$	0	0
$63$	0.249174	0.0313930
$64$	0	0
$65$	5.81778	0.721606
$66$	0	0
$67$	−5.58516	−0.682336	−0.341168	−	0.940002i	$-0.610823\pi$
$67$	−5.58516	−0.682336	−0.341168	+	0.940002i	$0.610823\pi$
$68$	0	0
$69$	13.7557	1.65600
$70$	0	0
$71$	9.40758	1.11647	0.558237	−	0.829681i	$-0.311478\pi$
$71$	9.40758	1.11647	0.558237	+	0.829681i	$0.311478\pi$
$72$	0	0
$73$	4.22594	0.494609	0.247304	−	0.968938i	$-0.420455\pi$
$73$	4.22594	0.494609	0.247304	+	0.968938i	$0.420455\pi$
$74$	0	0
$75$	22.8823	2.64222
$76$	0	0
$77$	−0.171587	−0.0195541
$78$	0	0
$79$	−12.2627	−1.37967	−0.689833	−	0.723969i	$-0.742316\pi$
$79$	−12.2627	−1.37967	−0.689833	+	0.723969i	$0.742316\pi$
$80$	0	0
$81$	−11.0399	−1.22665
$82$	0	0
$83$	0.492027	0.0540069	0.0270035	−	0.999635i	$-0.491403\pi$
$83$	0.492027	0.0540069	0.0270035	+	0.999635i	$0.491403\pi$
$84$	0	0
$85$	−4.40349	−0.477625
$86$	0	0
$87$	−0.209422	−0.0224524
$88$	0	0
$89$	8.78902	0.931634	0.465817	−	0.884881i	$-0.345760\pi$
$89$	8.78902	0.931634	0.465817	+	0.884881i	$0.345760\pi$
$90$	0	0
$91$	−0.189388	−0.0198533
$92$	0	0
$93$	−3.01567	−0.312710
$94$	0	0
$95$	−13.9550	−1.43176
$96$	0	0
$97$	18.6955	1.89824	0.949122	−	0.314908i	$-0.101974\pi$
$97$	18.6955	1.89824	0.949122	+	0.314908i	$0.101974\pi$
$98$	0	0
$99$	−2.64108	−0.265439
$100$	0	0
$101$	−3.41811	−0.340115	−0.170057	−	0.985434i	$-0.554395\pi$
$101$	−3.41811	−0.340115	−0.170057	+	0.985434i	$0.554395\pi$
$102$	0	0
$103$	−0.0306739	−0.00302239	−0.00151119	−	0.999999i	$-0.500481\pi$
$103$	−0.0306739	−0.00302239	−0.00151119	+	0.999999i	$0.500481\pi$
$104$	0	0
$105$	−1.10734	−0.108065
$106$	0	0
$107$	16.7393	1.61825	0.809126	−	0.587635i	$-0.199941\pi$
$107$	16.7393	1.61825	0.809126	+	0.587635i	$0.199941\pi$
$108$	0	0
$109$	−6.73233	−0.644840	−0.322420	−	0.946597i	$-0.604496\pi$
$109$	−6.73233	−0.644840	−0.322420	+	0.946597i	$0.604496\pi$
$110$	0	0
$111$	5.31689	0.504657
$112$	0	0
$113$	−11.0582	−1.04027	−0.520134	−	0.854085i	$-0.674118\pi$
$113$	−11.0582	−1.04027	−0.520134	+	0.854085i	$0.674118\pi$
$114$	0	0
$115$	−24.1446	−2.25150
$116$	0	0
$117$	−2.91508	−0.269500
$118$	0	0
$119$	0.143348	0.0131407
$120$	0	0
$121$	−9.18130	−0.834663
$122$	0	0
$123$	22.0694	1.98993
$124$	0	0
$125$	−20.6215	−1.84444
$126$	0	0
$127$	−1.72214	−0.152815	−0.0764076	−	0.997077i	$-0.524345\pi$
$127$	−1.72214	−0.152815	−0.0764076	+	0.997077i	$0.524345\pi$
$128$	0	0
$129$	−5.29839	−0.466497
$130$	0	0
$131$	14.9999	1.31055	0.655273	−	0.755393i	$-0.272554\pi$
$131$	14.9999	1.31055	0.655273	+	0.755393i	$0.272554\pi$
$132$	0	0
$133$	0.454283	0.0393913
$134$	0	0
$135$	9.06525	0.780212
$136$	0	0
$137$	−4.23865	−0.362132	−0.181066	−	0.983471i	$-0.557955\pi$
$137$	−4.23865	−0.362132	−0.181066	+	0.983471i	$0.557955\pi$
$138$	0	0
$139$	−20.2272	−1.71565	−0.857824	−	0.513944i	$-0.828184\pi$
$139$	−20.2272	−1.71565	−0.857824	+	0.513944i	$0.828184\pi$
$140$	0	0
$141$	−28.6224	−2.41044
$142$	0	0
$143$	2.00739	0.167866
$144$	0	0
$145$	0.367586	0.0305264
$146$	0	0
$147$	−15.5512	−1.28264
$148$	0	0
$149$	−20.4272	−1.67346	−0.836732	−	0.547612i	$-0.815537\pi$
$149$	−20.4272	−1.67346	−0.836732	+	0.547612i	$0.815537\pi$
$150$	0	0
$151$	−20.9022	−1.70100	−0.850500	−	0.525975i	$-0.823701\pi$
$151$	−20.9022	−1.70100	−0.850500	+	0.525975i	$0.823701\pi$
$152$	0	0
$153$	2.20643	0.178380
$154$	0	0
$155$	5.29321	0.425161
$156$	0	0
$157$	−4.56927	−0.364667	−0.182334	−	0.983237i	$-0.558365\pi$
$157$	−4.56927	−0.364667	−0.182334	+	0.983237i	$0.558365\pi$
$158$	0	0
$159$	0.223333	0.0177115
$160$	0	0
$161$	0.785988	0.0619445
$162$	0	0
$163$	−7.31476	−0.572936	−0.286468	−	0.958090i	$-0.592481\pi$
$163$	−7.31476	−0.572936	−0.286468	+	0.958090i	$0.592481\pi$
$164$	0	0
$165$	11.7370	0.913726
$166$	0	0
$167$	14.9467	1.15661	0.578303	−	0.815822i	$-0.303715\pi$
$167$	14.9467	1.15661	0.578303	+	0.815822i	$0.303715\pi$
$168$	0	0
$169$	−10.7844	−0.829565
$170$	0	0
$171$	6.99238	0.534721
$172$	0	0
$173$	−18.2864	−1.39029	−0.695145	−	0.718870i	$-0.744660\pi$
$173$	−18.2864	−1.39029	−0.695145	+	0.718870i	$0.744660\pi$
$174$	0	0
$175$	1.30747	0.0988353
$176$	0	0
$177$	−23.3486	−1.75499
$178$	0	0
$179$	14.6387	1.09415	0.547075	−	0.837084i	$-0.315741\pi$
$179$	14.6387	1.09415	0.547075	+	0.837084i	$0.315741\pi$
$180$	0	0
$181$	−18.6879	−1.38906	−0.694531	−	0.719463i	$-0.744388\pi$
$181$	−18.6879	−1.38906	−0.694531	+	0.719463i	$0.744388\pi$
$182$	0	0
$183$	−19.7971	−1.46344
$184$	0	0
$185$	−9.33240	−0.686132
$186$	0	0
$187$	−1.51940	−0.111109
$188$	0	0
$189$	−0.295104	−0.0214657
$190$	0	0
$191$	−7.95144	−0.575346	−0.287673	−	0.957729i	$-0.592882\pi$
$191$	−7.95144	−0.575346	−0.287673	+	0.957729i	$0.592882\pi$
$192$	0	0
$193$	−7.37180	−0.530634	−0.265317	−	0.964161i	$-0.585477\pi$
$193$	−7.37180	−0.530634	−0.265317	+	0.964161i	$0.585477\pi$
$194$	0	0
$195$	12.9547	0.927706
$196$	0	0
$197$	−12.7159	−0.905969	−0.452984	−	0.891519i	$-0.649641\pi$
$197$	−12.7159	−0.905969	−0.452984	+	0.891519i	$0.649641\pi$
$198$	0	0
$199$	−3.41792	−0.242290	−0.121145	−	0.992635i	$-0.538657\pi$
$199$	−3.41792	−0.242290	−0.121145	+	0.992635i	$0.538657\pi$
$200$	0	0
$201$	−12.4367	−0.877220
$202$	0	0
$203$	−0.0119662	−0.000839860 0
$204$	0	0
$205$	−38.7370	−2.70551
$206$	0	0
$207$	12.0980	0.840870
$208$	0	0
$209$	−4.81510	−0.333067
$210$	0	0
$211$	15.1230	1.04111	0.520555	−	0.853828i	$-0.325725\pi$
$211$	15.1230	1.04111	0.520555	+	0.853828i	$0.325725\pi$
$212$	0	0
$213$	20.9483	1.43535
$214$	0	0
$215$	9.29993	0.634250
$216$	0	0
$217$	−0.172312	−0.0116973
$218$	0	0
$219$	9.41009	0.635875
$220$	0	0
$221$	−1.67703	−0.112809
$222$	0	0
$223$	−28.0583	−1.87892	−0.939461	−	0.342657i	$-0.888673\pi$
$223$	−28.0583	−1.87892	−0.939461	+	0.342657i	$0.888673\pi$
$224$	0	0
$225$	20.1247	1.34165
$226$	0	0
$227$	−4.52803	−0.300536	−0.150268	−	0.988645i	$-0.548014\pi$
$227$	−4.52803	−0.300536	−0.150268	+	0.988645i	$0.548014\pi$
$228$	0	0
$229$	4.88767	0.322986	0.161493	−	0.986874i	$-0.448369\pi$
$229$	4.88767	0.322986	0.161493	+	0.986874i	$0.448369\pi$
$230$	0	0
$231$	−0.382080	−0.0251390
$232$	0	0
$233$	10.8902	0.713442	0.356721	−	0.934211i	$-0.383895\pi$
$233$	10.8902	0.713442	0.356721	+	0.934211i	$0.383895\pi$
$234$	0	0
$235$	50.2390	3.27723
$236$	0	0
$237$	−27.3060	−1.77371
$238$	0	0
$239$	0.468879	0.0303293	0.0151646	−	0.999885i	$-0.495173\pi$
$239$	0.468879	0.0303293	0.0151646	+	0.999885i	$0.495173\pi$
$240$	0	0
$241$	7.11565	0.458359	0.229180	−	0.973384i	$-0.426396\pi$
$241$	7.11565	0.458359	0.229180	+	0.973384i	$0.426396\pi$
$242$	0	0
$243$	−17.6248	−1.13063
$244$	0	0
$245$	27.2960	1.74388
$246$	0	0
$247$	−5.31465	−0.338163
$248$	0	0
$249$	1.09562	0.0694320
$250$	0	0
$251$	−16.4814	−1.04030	−0.520150	−	0.854075i	$-0.674124\pi$
$251$	−16.4814	−1.04030	−0.520150	+	0.854075i	$0.674124\pi$
$252$	0	0
$253$	−8.33095	−0.523762
$254$	0	0
$255$	−9.80545	−0.614041
$256$	0	0
$257$	−18.7437	−1.16920	−0.584599	−	0.811322i	$-0.698748\pi$
$257$	−18.7437	−1.16920	−0.584599	+	0.811322i	$0.698748\pi$
$258$	0	0
$259$	0.303801	0.0188773
$260$	0	0
$261$	−0.184185	−0.0114007
$262$	0	0
$263$	3.45247	0.212888	0.106444	−	0.994319i	$-0.466053\pi$
$263$	3.45247	0.212888	0.106444	+	0.994319i	$0.466053\pi$
$264$	0	0
$265$	−0.392003	−0.0240806
$266$	0	0
$267$	19.5709	1.19772
$268$	0	0
$269$	8.54448	0.520966	0.260483	−	0.965478i	$-0.416118\pi$
$269$	8.54448	0.520966	0.260483	+	0.965478i	$0.416118\pi$
$270$	0	0
$271$	12.7868	0.776743	0.388372	−	0.921503i	$-0.373038\pi$
$271$	12.7868	0.776743	0.388372	+	0.921503i	$0.373038\pi$
$272$	0	0
$273$	−0.421719	−0.0255236
$274$	0	0
$275$	−13.8583	−0.835687
$276$	0	0
$277$	26.4273	1.58786	0.793932	−	0.608006i	$-0.208030\pi$
$277$	26.4273	1.58786	0.793932	+	0.608006i	$0.208030\pi$
$278$	0	0
$279$	−2.65224	−0.158786
$280$	0	0
$281$	−6.28222	−0.374766	−0.187383	−	0.982287i	$-0.560001\pi$
$281$	−6.28222	−0.374766	−0.187383	+	0.982287i	$0.560001\pi$
$282$	0	0
$283$	−28.1857	−1.67547	−0.837734	−	0.546078i	$-0.816120\pi$
$283$	−28.1857	−1.67547	−0.837734	+	0.546078i	$0.816120\pi$
$284$	0	0
$285$	−31.0743	−1.84068
$286$	0	0
$287$	1.26102	0.0744356
$288$	0	0
$289$	−15.7307	−0.925332
$290$	0	0
$291$	41.6302	2.44041
$292$	0	0
$293$	−25.2689	−1.47622	−0.738112	−	0.674678i	$-0.764282\pi$
$293$	−25.2689	−1.47622	−0.738112	+	0.674678i	$0.764282\pi$
$294$	0	0
$295$	40.9823	2.38608
$296$	0	0
$297$	3.12791	0.181500
$298$	0	0
$299$	−9.19526	−0.531776
$300$	0	0
$301$	−0.302744	−0.0174499
$302$	0	0
$303$	−7.61126	−0.437255
$304$	0	0
$305$	34.7486	1.98970
$306$	0	0
$307$	−6.65104	−0.379595	−0.189798	−	0.981823i	$-0.560783\pi$
$307$	−6.65104	−0.379595	−0.189798	+	0.981823i	$0.560783\pi$
$308$	0	0
$309$	−0.0683029	−0.00388562
$310$	0	0
$311$	19.2914	1.09391	0.546957	−	0.837161i	$-0.315786\pi$
$311$	19.2914	1.09391	0.546957	+	0.837161i	$0.315786\pi$
$312$	0	0
$313$	11.9686	0.676507	0.338253	−	0.941055i	$-0.390164\pi$
$313$	11.9686	0.676507	0.338253	+	0.941055i	$0.390164\pi$
$314$	0	0
$315$	−0.973888	−0.0548724
$316$	0	0
$317$	−3.53040	−0.198287	−0.0991435	−	0.995073i	$-0.531610\pi$
$317$	−3.53040	−0.198287	−0.0991435	+	0.995073i	$0.531610\pi$
$318$	0	0
$319$	0.126833	0.00710131
$320$	0	0
$321$	37.2742	2.08045
$322$	0	0
$323$	4.02267	0.223827
$324$	0	0
$325$	−15.2961	−0.848472
$326$	0	0
$327$	−14.9912	−0.829014
$328$	0	0
$329$	−1.63545	−0.0901652
$330$	0	0
$331$	3.15944	0.173659	0.0868293	−	0.996223i	$-0.472327\pi$
$331$	3.15944	0.173659	0.0868293	+	0.996223i	$0.472327\pi$
$332$	0	0
$333$	4.67614	0.256251
$334$	0	0
$335$	21.8294	1.19267
$336$	0	0
$337$	−2.86571	−0.156105	−0.0780525	−	0.996949i	$-0.524870\pi$
$337$	−2.86571	−0.156105	−0.0780525	+	0.996949i	$0.524870\pi$
$338$	0	0
$339$	−24.6238	−1.33738
$340$	0	0
$341$	1.82639	0.0989046
$342$	0	0
$343$	−1.77921	−0.0960684
$344$	0	0
$345$	−53.7639	−2.89455
$346$	0	0
$347$	1.87206	0.100497	0.0502486	−	0.998737i	$-0.483999\pi$
$347$	1.87206	0.100497	0.0502486	+	0.998737i	$0.483999\pi$
$348$	0	0
$349$	−19.1333	−1.02418	−0.512091	−	0.858931i	$-0.671129\pi$
$349$	−19.1333	−1.02418	−0.512091	+	0.858931i	$0.671129\pi$
$350$	0	0
$351$	3.45242	0.184277
$352$	0	0
$353$	17.9121	0.953365	0.476683	−	0.879075i	$-0.341839\pi$
$353$	17.9121	0.953365	0.476683	+	0.879075i	$0.341839\pi$
$354$	0	0
$355$	−36.7692	−1.95151
$356$	0	0
$357$	0.319200	0.0168939
$358$	0	0
$359$	−18.6237	−0.982920	−0.491460	−	0.870900i	$-0.663537\pi$
$359$	−18.6237	−0.982920	−0.491460	+	0.870900i	$0.663537\pi$
$360$	0	0
$361$	−6.25180	−0.329042
$362$	0	0
$363$	−20.4444	−1.07305
$364$	0	0
$365$	−16.5169	−0.864536
$366$	0	0
$367$	−14.1199	−0.737051	−0.368526	−	0.929618i	$-0.620137\pi$
$367$	−14.1199	−0.737051	−0.368526	+	0.929618i	$0.620137\pi$
$368$	0	0
$369$	19.4098	1.01043
$370$	0	0
$371$	0.0127610	0.000662519 0
$372$	0	0
$373$	−17.4991	−0.906071	−0.453035	−	0.891493i	$-0.649659\pi$
$373$	−17.4991	−0.906071	−0.453035	+	0.891493i	$0.649659\pi$
$374$	0	0
$375$	−45.9189	−2.37124
$376$	0	0
$377$	0.139992	0.00720995
$378$	0	0
$379$	−1.86167	−0.0956278	−0.0478139	−	0.998856i	$-0.515225\pi$
$379$	−1.86167	−0.0956278	−0.0478139	+	0.998856i	$0.515225\pi$
$380$	0	0
$381$	−3.83477	−0.196461
$382$	0	0
$383$	6.59670	0.337075	0.168538	−	0.985695i	$-0.446096\pi$
$383$	6.59670	0.337075	0.168538	+	0.985695i	$0.446096\pi$
$384$	0	0
$385$	0.670640	0.0341790
$386$	0	0
$387$	−4.65987	−0.236875
$388$	0	0
$389$	−1.58169	−0.0801950	−0.0400975	−	0.999196i	$-0.512767\pi$
$389$	−1.58169	−0.0801950	−0.0400975	+	0.999196i	$0.512767\pi$
$390$	0	0
$391$	6.95992	0.351978
$392$	0	0
$393$	33.4009	1.68485
$394$	0	0
$395$	47.9285	2.41154
$396$	0	0
$397$	25.3966	1.27462	0.637309	−	0.770608i	$-0.280047\pi$
$397$	25.3966	1.27462	0.637309	+	0.770608i	$0.280047\pi$
$398$	0	0
$399$	1.01157	0.0506420
$400$	0	0
$401$	−7.54246	−0.376652	−0.188326	−	0.982107i	$-0.560306\pi$
$401$	−7.54246	−0.376652	−0.188326	+	0.982107i	$0.560306\pi$
$402$	0	0
$403$	2.01587	0.100418
$404$	0	0
$405$	43.1490	2.14409
$406$	0	0
$407$	−3.22009	−0.159614
$408$	0	0
$409$	−4.76455	−0.235592	−0.117796	−	0.993038i	$-0.537583\pi$
$409$	−4.76455	−0.235592	−0.117796	+	0.993038i	$0.537583\pi$
$410$	0	0
$411$	−9.43840	−0.465562
$412$	0	0
$413$	−1.33411	−0.0656473
$414$	0	0
$415$	−1.92307	−0.0943998
$416$	0	0
$417$	−45.0408	−2.20566
$418$	0	0
$419$	13.6082	0.664806	0.332403	−	0.943137i	$-0.392141\pi$
$419$	13.6082	0.664806	0.332403	+	0.943137i	$0.392141\pi$
$420$	0	0
$421$	−12.8645	−0.626978	−0.313489	−	0.949592i	$-0.601498\pi$
$421$	−12.8645	−0.626978	−0.313489	+	0.949592i	$0.601498\pi$
$422$	0	0
$423$	−25.1730	−1.22395
$424$	0	0
$425$	11.5776	0.561597
$426$	0	0
$427$	−1.13118	−0.0547418
$428$	0	0
$429$	4.46994	0.215811
$430$	0	0
$431$	30.4285	1.46569	0.732844	−	0.680397i	$-0.238193\pi$
$431$	30.4285	1.46569	0.732844	+	0.680397i	$0.238193\pi$
$432$	0	0
$433$	8.49495	0.408241	0.204121	−	0.978946i	$-0.434567\pi$
$433$	8.49495	0.408241	0.204121	+	0.978946i	$0.434567\pi$
$434$	0	0
$435$	0.818521	0.0392451
$436$	0	0
$437$	22.0566	1.05511
$438$	0	0
$439$	15.8520	0.756576	0.378288	−	0.925688i	$-0.376513\pi$
$439$	15.8520	0.756576	0.378288	+	0.925688i	$0.376513\pi$
$440$	0	0
$441$	−13.6771	−0.651289
$442$	0	0
$443$	5.52175	0.262346	0.131173	−	0.991359i	$-0.458126\pi$
$443$	5.52175	0.262346	0.131173	+	0.991359i	$0.458126\pi$
$444$	0	0
$445$	−34.3516	−1.62842
$446$	0	0
$447$	−45.4863	−2.15143
$448$	0	0
$449$	31.5859	1.49063	0.745314	−	0.666713i	$-0.232299\pi$
$449$	31.5859	1.49063	0.745314	+	0.666713i	$0.232299\pi$
$450$	0	0
$451$	−13.3660	−0.629379
$452$	0	0
$453$	−46.5440	−2.18683
$454$	0	0
$455$	0.740217	0.0347019
$456$	0	0
$457$	8.14835	0.381164	0.190582	−	0.981671i	$-0.438963\pi$
$457$	8.14835	0.381164	0.190582	+	0.981671i	$0.438963\pi$
$458$	0	0
$459$	−2.61314	−0.121971
$460$	0	0
$461$	−20.2099	−0.941269	−0.470634	−	0.882328i	$-0.655975\pi$
$461$	−20.2099	−0.941269	−0.470634	+	0.882328i	$0.655975\pi$
$462$	0	0
$463$	1.03697	0.0481922	0.0240961	−	0.999710i	$-0.492329\pi$
$463$	1.03697	0.0481922	0.0240961	+	0.999710i	$0.492329\pi$
$464$	0	0
$465$	11.7866	0.546592
$466$	0	0
$467$	0.734354	0.0339819	0.0169909	−	0.999856i	$-0.494591\pi$
$467$	0.734354	0.0339819	0.0169909	+	0.999856i	$0.494591\pi$
$468$	0	0
$469$	−0.710621	−0.0328134
$470$	0	0
$471$	−10.1746	−0.468821
$472$	0	0
$473$	3.20889	0.147545
$474$	0	0
$475$	36.6905	1.68347
$476$	0	0
$477$	0.196419	0.00899341
$478$	0	0
$479$	29.1901	1.33373	0.666865	−	0.745179i	$-0.267636\pi$
$479$	29.1901	1.33373	0.666865	+	0.745179i	$0.267636\pi$
$480$	0	0
$481$	−3.55416	−0.162056
$482$	0	0
$483$	1.75019	0.0796366
$484$	0	0
$485$	−73.0709	−3.31798
$486$	0	0
$487$	−30.1353	−1.36556	−0.682781	−	0.730623i	$-0.739230\pi$
$487$	−30.1353	−1.36556	−0.682781	+	0.730623i	$0.739230\pi$
$488$	0	0
$489$	−16.2881	−0.736574
$490$	0	0
$491$	−29.5021	−1.33141	−0.665706	−	0.746214i	$-0.731870\pi$
$491$	−29.5021	−1.33141	−0.665706	+	0.746214i	$0.731870\pi$
$492$	0	0
$493$	−0.105960	−0.00477221
$494$	0	0
$495$	10.3226	0.463965
$496$	0	0
$497$	1.19696	0.0536910
$498$	0	0
$499$	5.49945	0.246189	0.123095	−	0.992395i	$-0.460718\pi$
$499$	5.49945	0.246189	0.123095	+	0.992395i	$0.460718\pi$
$500$	0	0
$501$	33.2824	1.48695
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	13.3596	0.594493
$506$	0	0
$507$	−24.0140	−1.06650
$508$	0	0
$509$	34.6106	1.53409	0.767043	−	0.641595i	$-0.221727\pi$
$509$	34.6106	1.53409	0.767043	+	0.641595i	$0.221727\pi$
$510$	0	0
$511$	0.537682	0.0237856
$512$	0	0
$513$	−8.28128	−0.365628
$514$	0	0
$515$	0.119888	0.00528289
$516$	0	0
$517$	17.3347	0.762379
$518$	0	0
$519$	−40.7192	−1.78737
$520$	0	0
$521$	−21.6347	−0.947834	−0.473917	−	0.880570i	$-0.657160\pi$
$521$	−21.6347	−0.947834	−0.473917	+	0.880570i	$0.657160\pi$
$522$	0	0
$523$	−9.56540	−0.418266	−0.209133	−	0.977887i	$-0.567064\pi$
$523$	−9.56540	−0.418266	−0.209133	+	0.977887i	$0.567064\pi$
$524$	0	0
$525$	2.91140	0.127064
$526$	0	0
$527$	−1.52582	−0.0664657
$528$	0	0
$529$	15.1617	0.659203
$530$	0	0
$531$	−20.5348	−0.891134
$532$	0	0
$533$	−14.7527	−0.639009
$534$	0	0
$535$	−65.4251	−2.82857
$536$	0	0
$537$	32.5967	1.40665
$538$	0	0
$539$	9.41832	0.405676
$540$	0	0
$541$	−7.05411	−0.303280	−0.151640	−	0.988436i	$-0.548455\pi$
$541$	−7.05411	−0.303280	−0.151640	+	0.988436i	$0.548455\pi$
$542$	0	0
$543$	−41.6132	−1.78579
$544$	0	0
$545$	26.3131	1.12713
$546$	0	0
$547$	4.00855	0.171393	0.0856966	−	0.996321i	$-0.472688\pi$
$547$	4.00855	0.171393	0.0856966	+	0.996321i	$0.472688\pi$
$548$	0	0
$549$	−17.4113	−0.743096
$550$	0	0
$551$	−0.335797	−0.0143054
$552$	0	0
$553$	−1.56023	−0.0663479
$554$	0	0
$555$	−20.7809	−0.882099
$556$	0	0
$557$	22.1913	0.940275	0.470137	−	0.882593i	$-0.344204\pi$
$557$	22.1913	0.940275	0.470137	+	0.882593i	$0.344204\pi$
$558$	0	0
$559$	3.54180	0.149802
$560$	0	0
$561$	−3.38331	−0.142844
$562$	0	0
$563$	−36.2816	−1.52909	−0.764545	−	0.644571i	$-0.777036\pi$
$563$	−36.2816	−1.52909	−0.764545	+	0.644571i	$0.777036\pi$
$564$	0	0
$565$	43.2206	1.81830
$566$	0	0
$567$	−1.40464	−0.0589895
$568$	0	0
$569$	−25.9315	−1.08710	−0.543552	−	0.839376i	$-0.682921\pi$
$569$	−25.9315	−1.08710	−0.543552	+	0.839376i	$0.682921\pi$
$570$	0	0
$571$	4.84904	0.202926	0.101463	−	0.994839i	$-0.467648\pi$
$571$	4.84904	0.202926	0.101463	+	0.994839i	$0.467648\pi$
$572$	0	0
$573$	−17.7058	−0.739672
$574$	0	0
$575$	63.4808	2.64733
$576$	0	0
$577$	−1.13047	−0.0470620	−0.0235310	−	0.999723i	$-0.507491\pi$
$577$	−1.13047	−0.0470620	−0.0235310	+	0.999723i	$0.507491\pi$
$578$	0	0
$579$	−16.4151	−0.682189
$580$	0	0
$581$	0.0626024	0.00259718
$582$	0	0
$583$	−0.135258	−0.00560183
$584$	0	0
$585$	11.3935	0.471064
$586$	0	0
$587$	−16.5250	−0.682060	−0.341030	−	0.940052i	$-0.610776\pi$
$587$	−16.5250	−0.682060	−0.341030	+	0.940052i	$0.610776\pi$
$588$	0	0
$589$	−4.83545	−0.199241
$590$	0	0
$591$	−28.3150	−1.16472
$592$	0	0
$593$	−33.9678	−1.39489	−0.697445	−	0.716638i	$-0.745680\pi$
$593$	−33.9678	−1.39489	−0.697445	+	0.716638i	$0.745680\pi$
$594$	0	0
$595$	−0.560272	−0.0229689
$596$	0	0
$597$	−7.61083	−0.311491
$598$	0	0
$599$	27.4333	1.12089	0.560447	−	0.828190i	$-0.310629\pi$
$599$	27.4333	1.12089	0.560447	+	0.828190i	$0.310629\pi$
$600$	0	0
$601$	15.8279	0.645634	0.322817	−	0.946461i	$-0.395370\pi$
$601$	15.8279	0.645634	0.322817	+	0.946461i	$0.395370\pi$
$602$	0	0
$603$	−10.9380	−0.445428
$604$	0	0
$605$	35.8848	1.45892
$606$	0	0
$607$	40.3952	1.63959	0.819794	−	0.572658i	$-0.194088\pi$
$607$	40.3952	1.63959	0.819794	+	0.572658i	$0.194088\pi$
$608$	0	0
$609$	−0.0266456	−0.00107973
$610$	0	0
$611$	19.1331	0.774043
$612$	0	0
$613$	8.20436	0.331371	0.165686	−	0.986179i	$-0.447016\pi$
$613$	8.20436	0.331371	0.165686	+	0.986179i	$0.447016\pi$
$614$	0	0
$615$	−86.2575	−3.47824
$616$	0	0
$617$	15.6643	0.630623	0.315311	−	0.948988i	$-0.397891\pi$
$617$	15.6643	0.630623	0.315311	+	0.948988i	$0.397891\pi$
$618$	0	0
$619$	−11.8251	−0.475290	−0.237645	−	0.971352i	$-0.576375\pi$
$619$	−11.8251	−0.475290	−0.237645	+	0.971352i	$0.576375\pi$
$620$	0	0
$621$	−14.3280	−0.574965
$622$	0	0
$623$	1.11826	0.0448021
$624$	0	0
$625$	29.2179	1.16872
$626$	0	0
$627$	−10.7220	−0.428196
$628$	0	0
$629$	2.69015	0.107263
$630$	0	0
$631$	−14.3728	−0.572170	−0.286085	−	0.958204i	$-0.592354\pi$
$631$	−14.3728	−0.572170	−0.286085	+	0.958204i	$0.592354\pi$
$632$	0	0
$633$	33.6751	1.33846
$634$	0	0
$635$	6.73093	0.267109
$636$	0	0
$637$	10.3954	0.411882
$638$	0	0
$639$	18.4238	0.728833
$640$	0	0
$641$	−44.9593	−1.77579	−0.887893	−	0.460051i	$-0.847831\pi$
$641$	−44.9593	−1.77579	−0.887893	+	0.460051i	$0.847831\pi$
$642$	0	0
$643$	33.1775	1.30839	0.654196	−	0.756325i	$-0.273007\pi$
$643$	33.1775	1.30839	0.654196	+	0.756325i	$0.273007\pi$
$644$	0	0
$645$	20.7086	0.815400
$646$	0	0
$647$	3.17050	0.124645	0.0623226	−	0.998056i	$-0.480149\pi$
$647$	3.17050	0.124645	0.0623226	+	0.998056i	$0.480149\pi$
$648$	0	0
$649$	14.1407	0.555071
$650$	0	0
$651$	−0.383694	−0.0150382
$652$	0	0
$653$	9.08500	0.355524	0.177762	−	0.984074i	$-0.443114\pi$
$653$	9.08500	0.355524	0.177762	+	0.984074i	$0.443114\pi$
$654$	0	0
$655$	−58.6265	−2.29073
$656$	0	0
$657$	8.27606	0.322880
$658$	0	0
$659$	0.0644937	0.00251232	0.00125616	−	0.999999i	$-0.499600\pi$
$659$	0.0644937	0.00251232	0.00125616	+	0.999999i	$0.499600\pi$
$660$	0	0
$661$	22.7167	0.883576	0.441788	−	0.897120i	$-0.354344\pi$
$661$	22.7167	0.883576	0.441788	+	0.897120i	$0.354344\pi$
$662$	0	0
$663$	−3.73432	−0.145029
$664$	0	0
$665$	−1.77555	−0.0688529
$666$	0	0
$667$	−0.580987	−0.0224959
$668$	0	0
$669$	−62.4786	−2.41556
$670$	0	0
$671$	11.9898	0.462861
$672$	0	0
$673$	−30.3698	−1.17067	−0.585335	−	0.810792i	$-0.699037\pi$
$673$	−30.3698	−1.17067	−0.585335	+	0.810792i	$0.699037\pi$
$674$	0	0
$675$	−23.8343	−0.917382
$676$	0	0
$677$	−10.1454	−0.389921	−0.194960	−	0.980811i	$-0.562458\pi$
$677$	−10.1454	−0.389921	−0.194960	+	0.980811i	$0.562458\pi$
$678$	0	0
$679$	2.37870	0.0912862
$680$	0	0
$681$	−10.0828	−0.386372
$682$	0	0
$683$	15.3406	0.586993	0.293497	−	0.955960i	$-0.405181\pi$
$683$	15.3406	0.586993	0.293497	+	0.955960i	$0.405181\pi$
$684$	0	0
$685$	16.5666	0.632978
$686$	0	0
$687$	10.8836	0.415235
$688$	0	0
$689$	−0.149291	−0.00568753
$690$	0	0
$691$	−4.81419	−0.183140	−0.0915702	−	0.995799i	$-0.529189\pi$
$691$	−4.81419	−0.183140	−0.0915702	+	0.995799i	$0.529189\pi$
$692$	0	0
$693$	−0.336034	−0.0127649
$694$	0	0
$695$	79.0573	2.99881
$696$	0	0
$697$	11.1663	0.422955
$698$	0	0
$699$	24.2498	0.917210
$700$	0	0
$701$	−16.8248	−0.635465	−0.317733	−	0.948180i	$-0.602921\pi$
$701$	−16.8248	−0.635465	−0.317733	+	0.948180i	$0.602921\pi$
$702$	0	0
$703$	8.52533	0.321539
$704$	0	0
$705$	111.870	4.21325
$706$	0	0
$707$	−0.434899	−0.0163560
$708$	0	0
$709$	25.8837	0.972084	0.486042	−	0.873936i	$-0.338440\pi$
$709$	25.8837	0.972084	0.486042	+	0.873936i	$0.338440\pi$
$710$	0	0
$711$	−24.0153	−0.900644
$712$	0	0
$713$	−8.36616	−0.313315
$714$	0	0
$715$	−7.84581	−0.293417
$716$	0	0
$717$	1.04407	0.0389917
$718$	0	0
$719$	40.4115	1.50709	0.753547	−	0.657394i	$-0.228341\pi$
$719$	40.4115	1.50709	0.753547	+	0.657394i	$0.228341\pi$
$720$	0	0
$721$	−0.00390275	−0.000145346 0
$722$	0	0
$723$	15.8447	0.589272
$724$	0	0
$725$	−0.966455	−0.0358932
$726$	0	0
$727$	23.7913	0.882369	0.441185	−	0.897416i	$-0.354558\pi$
$727$	23.7913	0.882369	0.441185	+	0.897416i	$0.354558\pi$
$728$	0	0
$729$	−6.12639	−0.226903
$730$	0	0
$731$	−2.68080	−0.0991528
$732$	0	0
$733$	26.2965	0.971282	0.485641	−	0.874158i	$-0.338586\pi$
$733$	26.2965	0.971282	0.485641	+	0.874158i	$0.338586\pi$
$734$	0	0
$735$	60.7812	2.24195
$736$	0	0
$737$	7.53212	0.277449
$738$	0	0
$739$	27.1782	0.999766	0.499883	−	0.866093i	$-0.333376\pi$
$739$	27.1782	0.999766	0.499883	+	0.866093i	$0.333376\pi$
$740$	0	0
$741$	−11.8344	−0.434747
$742$	0	0
$743$	35.3208	1.29580	0.647898	−	0.761727i	$-0.275648\pi$
$743$	35.3208	1.29580	0.647898	+	0.761727i	$0.275648\pi$
$744$	0	0
$745$	79.8392	2.92508
$746$	0	0
$747$	0.963583	0.0352557
$748$	0	0
$749$	2.12981	0.0778215
$750$	0	0
$751$	5.11785	0.186753	0.0933765	−	0.995631i	$-0.470234\pi$
$751$	5.11785	0.186753	0.0933765	+	0.995631i	$0.470234\pi$
$752$	0	0
$753$	−36.7000	−1.33742
$754$	0	0
$755$	81.6957	2.97321
$756$	0	0
$757$	−6.49682	−0.236131	−0.118065	−	0.993006i	$-0.537669\pi$
$757$	−6.49682	−0.236131	−0.118065	+	0.993006i	$0.537669\pi$
$758$	0	0
$759$	−18.5509	−0.673355
$760$	0	0
$761$	35.0920	1.27208	0.636041	−	0.771655i	$-0.280571\pi$
$761$	35.0920	1.27208	0.636041	+	0.771655i	$0.280571\pi$
$762$	0	0
$763$	−0.856579	−0.0310102
$764$	0	0
$765$	−8.62377	−0.311793
$766$	0	0
$767$	15.6077	0.563563
$768$	0	0
$769$	−29.6939	−1.07079	−0.535395	−	0.844602i	$-0.679837\pi$
$769$	−29.6939	−1.07079	−0.535395	+	0.844602i	$0.679837\pi$
$770$	0	0
$771$	−41.7374	−1.50314
$772$	0	0
$773$	−22.6109	−0.813258	−0.406629	−	0.913593i	$-0.633296\pi$
$773$	−22.6109	−0.813258	−0.406629	+	0.913593i	$0.633296\pi$
$774$	0	0
$775$	−13.9169	−0.499909
$776$	0	0
$777$	0.676487	0.0242688
$778$	0	0
$779$	35.3870	1.26787
$780$	0	0
$781$	−12.6870	−0.453977
$782$	0	0
$783$	0.218135	0.00779552
$784$	0	0
$785$	17.8588	0.637409
$786$	0	0
$787$	43.9375	1.56620	0.783101	−	0.621895i	$-0.213637\pi$
$787$	43.9375	1.56620	0.783101	+	0.621895i	$0.213637\pi$
$788$	0	0
$789$	7.68777	0.273692
$790$	0	0
$791$	−1.40698	−0.0500263
$792$	0	0
$793$	13.2337	0.469942
$794$	0	0
$795$	−0.872891	−0.0309583
$796$	0	0
$797$	46.7524	1.65605	0.828027	−	0.560688i	$-0.189463\pi$
$797$	46.7524	1.65605	0.828027	+	0.560688i	$0.189463\pi$
$798$	0	0
$799$	−14.4819	−0.512332
$800$	0	0
$801$	17.2124	0.608170
$802$	0	0
$803$	−5.69907	−0.201116
$804$	0	0
$805$	−3.07201	−0.108274
$806$	0	0
$807$	19.0264	0.669760
$808$	0	0
$809$	13.3923	0.470848	0.235424	−	0.971893i	$-0.424352\pi$
$809$	13.3923	0.470848	0.235424	+	0.971893i	$0.424352\pi$
$810$	0	0
$811$	13.3858	0.470037	0.235019	−	0.971991i	$-0.424485\pi$
$811$	13.3858	0.470037	0.235019	+	0.971991i	$0.424485\pi$
$812$	0	0
$813$	28.4730	0.998591
$814$	0	0
$815$	28.5895	1.00145
$816$	0	0
$817$	−8.49567	−0.297226
$818$	0	0
$819$	−0.370897	−0.0129602
$820$	0	0
$821$	4.23562	0.147824	0.0739122	−	0.997265i	$-0.476452\pi$
$821$	4.23562	0.147824	0.0739122	+	0.997265i	$0.476452\pi$
$822$	0	0
$823$	−13.3570	−0.465597	−0.232799	−	0.972525i	$-0.574788\pi$
$823$	−13.3570	−0.465597	−0.232799	+	0.972525i	$0.574788\pi$
$824$	0	0
$825$	−30.8589	−1.07437
$826$	0	0
$827$	−4.13307	−0.143721	−0.0718604	−	0.997415i	$-0.522894\pi$
$827$	−4.13307	−0.143721	−0.0718604	+	0.997415i	$0.522894\pi$
$828$	0	0
$829$	−30.2050	−1.04906	−0.524532	−	0.851391i	$-0.675760\pi$
$829$	−30.2050	−1.04906	−0.524532	+	0.851391i	$0.675760\pi$
$830$	0	0
$831$	58.8469	2.04138
$832$	0	0
$833$	−7.86834	−0.272622
$834$	0	0
$835$	−58.4185	−2.02166
$836$	0	0
$837$	3.14113	0.108573
$838$	0	0
$839$	−34.3993	−1.18760	−0.593798	−	0.804614i	$-0.702372\pi$
$839$	−34.3993	−1.18760	−0.593798	+	0.804614i	$0.702372\pi$
$840$	0	0
$841$	−28.9912	−0.999695
$842$	0	0
$843$	−13.9889	−0.481804
$844$	0	0
$845$	42.1503	1.45001
$846$	0	0
$847$	−1.16817	−0.0401388
$848$	0	0
$849$	−62.7625	−2.15400
$850$	0	0
$851$	14.7503	0.505633
$852$	0	0
$853$	−9.70737	−0.332374	−0.166187	−	0.986094i	$-0.553146\pi$
$853$	−9.70737	−0.332374	−0.166187	+	0.986094i	$0.553146\pi$
$854$	0	0
$855$	−27.3295	−0.934648
$856$	0	0
$857$	−34.7710	−1.18775	−0.593877	−	0.804556i	$-0.702404\pi$
$857$	−34.7710	−1.18775	−0.593877	+	0.804556i	$0.702404\pi$
$858$	0	0
$859$	26.5064	0.904387	0.452194	−	0.891920i	$-0.350642\pi$
$859$	26.5064	0.904387	0.452194	+	0.891920i	$0.350642\pi$
$860$	0	0
$861$	2.80797	0.0956954
$862$	0	0
$863$	7.76135	0.264199	0.132100	−	0.991236i	$-0.457828\pi$
$863$	7.76135	0.264199	0.132100	+	0.991236i	$0.457828\pi$
$864$	0	0
$865$	71.4718	2.43011
$866$	0	0
$867$	−35.0282	−1.18962
$868$	0	0
$869$	16.5374	0.560994
$870$	0	0
$871$	8.31355	0.281694
$872$	0	0
$873$	36.6133	1.23917
$874$	0	0
$875$	−2.62375	−0.0886990
$876$	0	0
$877$	33.4061	1.12804	0.564021	−	0.825760i	$-0.309254\pi$
$877$	33.4061	1.12804	0.564021	+	0.825760i	$0.309254\pi$
$878$	0	0
$879$	−56.2674	−1.89785
$880$	0	0
$881$	45.5473	1.53453	0.767263	−	0.641332i	$-0.221618\pi$
$881$	45.5473	1.53453	0.767263	+	0.641332i	$0.221618\pi$
$882$	0	0
$883$	−8.05533	−0.271084	−0.135542	−	0.990772i	$-0.543278\pi$
$883$	−8.05533	−0.271084	−0.135542	+	0.990772i	$0.543278\pi$
$884$	0	0
$885$	91.2571	3.06757
$886$	0	0
$887$	37.8462	1.27075	0.635376	−	0.772203i	$-0.280845\pi$
$887$	37.8462	1.27075	0.635376	+	0.772203i	$0.280845\pi$
$888$	0	0
$889$	−0.219114	−0.00734886
$890$	0	0
$891$	14.8883	0.498777
$892$	0	0
$893$	−45.8944	−1.53580
$894$	0	0
$895$	−57.2150	−1.91249
$896$	0	0
$897$	−20.4755	−0.683657
$898$	0	0
$899$	0.127369	0.00424801
$900$	0	0
$901$	0.112999	0.00376453
$902$	0	0
$903$	−0.674134	−0.0224338
$904$	0	0
$905$	73.0411	2.42797
$906$	0	0
$907$	0.311482	0.0103426	0.00517129	−	0.999987i	$-0.498354\pi$
$907$	0.311482	0.0103426	0.00517129	+	0.999987i	$0.498354\pi$
$908$	0	0
$909$	−6.69401	−0.222026
$910$	0	0
$911$	−42.8928	−1.42110	−0.710551	−	0.703646i	$-0.751554\pi$
$911$	−42.8928	−1.42110	−0.710551	+	0.703646i	$0.751554\pi$
$912$	0	0
$913$	−0.663544	−0.0219601
$914$	0	0
$915$	77.3762	2.55798
$916$	0	0
$917$	1.90849	0.0630239
$918$	0	0
$919$	−40.2516	−1.32778	−0.663889	−	0.747831i	$-0.731095\pi$
$919$	−40.2516	−1.32778	−0.663889	+	0.747831i	$0.731095\pi$
$920$	0	0
$921$	−14.8102	−0.488012
$922$	0	0
$923$	−14.0032	−0.460922
$924$	0	0
$925$	24.5367	0.806761
$926$	0	0
$927$	−0.0600716	−0.00197301
$928$	0	0
$929$	14.7060	0.482489	0.241244	−	0.970464i	$-0.422444\pi$
$929$	14.7060	0.482489	0.241244	+	0.970464i	$0.422444\pi$
$930$	0	0
$931$	−24.9354	−0.817226
$932$	0	0
$933$	42.9570	1.40635
$934$	0	0
$935$	5.93851	0.194210
$936$	0	0
$937$	26.2273	0.856807	0.428404	−	0.903587i	$-0.359076\pi$
$937$	26.2273	0.856807	0.428404	+	0.903587i	$0.359076\pi$
$938$	0	0
$939$	26.6511	0.869725
$940$	0	0
$941$	−51.2435	−1.67049	−0.835246	−	0.549877i	$-0.814675\pi$
$941$	−51.2435	−1.67049	−0.835246	+	0.549877i	$0.814675\pi$
$942$	0	0
$943$	61.2257	1.99378
$944$	0	0
$945$	1.15340	0.0375203
$946$	0	0
$947$	−31.3479	−1.01867	−0.509334	−	0.860569i	$-0.670108\pi$
$947$	−31.3479	−1.01867	−0.509334	+	0.860569i	$0.670108\pi$
$948$	0	0
$949$	−6.29034	−0.204193
$950$	0	0
$951$	−7.86130	−0.254920
$952$	0	0
$953$	−28.4399	−0.921260	−0.460630	−	0.887592i	$-0.652377\pi$
$953$	−28.4399	−0.921260	−0.460630	+	0.887592i	$0.652377\pi$
$954$	0	0
$955$	31.0779	1.00566
$956$	0	0
$957$	0.282426	0.00912953
$958$	0	0
$959$	−0.539299	−0.0174149
$960$	0	0
$961$	−29.1659	−0.940835
$962$	0	0
$963$	32.7822	1.05639
$964$	0	0
$965$	28.8124	0.927505
$966$	0	0
$967$	−15.0183	−0.482956	−0.241478	−	0.970406i	$-0.577632\pi$
$967$	−15.0183	−0.482956	−0.241478	+	0.970406i	$0.577632\pi$
$968$	0	0
$969$	8.95747	0.287755
$970$	0	0
$971$	4.87993	0.156604	0.0783022	−	0.996930i	$-0.475050\pi$
$971$	4.87993	0.156604	0.0783022	+	0.996930i	$0.475050\pi$
$972$	0	0
$973$	−2.57358	−0.0825052
$974$	0	0
$975$	−34.0604	−1.09081
$976$	0	0
$977$	47.3517	1.51492	0.757458	−	0.652884i	$-0.226441\pi$
$977$	47.3517	1.51492	0.757458	+	0.652884i	$0.226441\pi$
$978$	0	0
$979$	−11.8528	−0.378818
$980$	0	0
$981$	−13.1846	−0.420951
$982$	0	0
$983$	16.7138	0.533087	0.266543	−	0.963823i	$-0.414118\pi$
$983$	16.7138	0.533087	0.266543	+	0.963823i	$0.414118\pi$
$984$	0	0
$985$	49.6996	1.58356
$986$	0	0
$987$	−3.64173	−0.115918
$988$	0	0
$989$	−14.6990	−0.467400
$990$	0	0
$991$	22.3986	0.711516	0.355758	−	0.934578i	$-0.384223\pi$
$991$	22.3986	0.711516	0.355758	+	0.934578i	$0.384223\pi$
$992$	0	0
$993$	7.03527	0.223258
$994$	0	0
$995$	13.3588	0.423503
$996$	0	0
$997$	−52.8835	−1.67484	−0.837418	−	0.546563i	$-0.815936\pi$
$997$	−52.8835	−1.67484	−0.837418	+	0.546563i	$0.815936\pi$
$998$	0	0
$999$	−5.53809	−0.175217

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.v.1.24		28
4.3	odd	2		4024.2.a.d.1.5	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.d.1.5	✓	28	4.3	odd	2
8048.2.a.v.1.24		28	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)

\(f(q)\)	\(=\)	\(q+2.22675 q^{3} -3.90847 q^{5} +0.127234 q^{7} +1.95840 q^{9} +O(q^{10})\)	\(q+2.22675 q^{3} -3.90847 q^{5} +0.127234 q^{7} +1.95840 q^{9} -1.34859 q^{11} -1.48851 q^{13} -8.70316 q^{15} +1.12665 q^{17} +3.57046 q^{19} +0.283317 q^{21} +6.17751 q^{23} +10.2761 q^{25} -2.31939 q^{27} -0.0940486 q^{29} -1.35429 q^{31} -3.00297 q^{33} -0.497289 q^{35} +2.38774 q^{37} -3.31452 q^{39} +9.91105 q^{41} -2.37943 q^{43} -7.65433 q^{45} -12.8539 q^{47} -6.98381 q^{49} +2.50877 q^{51} +0.100296 q^{53} +5.27093 q^{55} +7.95051 q^{57} -10.4855 q^{59} -8.89059 q^{61} +0.249174 q^{63} +5.81778 q^{65} -5.58516 q^{67} +13.7557 q^{69} +9.40758 q^{71} +4.22594 q^{73} +22.8823 q^{75} -0.171587 q^{77} -12.2627 q^{79} -11.0399 q^{81} +0.492027 q^{83} -4.40349 q^{85} -0.209422 q^{87} +8.78902 q^{89} -0.189388 q^{91} -3.01567 q^{93} -13.9550 q^{95} +18.6955 q^{97} -2.64108 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * q^99

Introduction

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