LMFDB - Embedded newform 4864.2.a.bk.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4864,2,Mod(1,4864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4864, 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("4864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4864 = 2^{8} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4864.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:	$38.8392355432$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 11x^{2} + 16 $ x^4 - 11*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1216)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.31342$ of defining polynomial
Character	$\chi$	$=$	4864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.31342 q^{5} -4.77753 q^{7} -3.00000 q^{9} +O(q^{10})$	$q+1.31342 q^{5} -4.77753 q^{7} -3.00000 q^{9} -2.27492 q^{11} +6.09095 q^{13} -4.27492 q^{17} -1.00000 q^{19} -3.46410 q^{23} -3.27492 q^{25} +6.09095 q^{29} -2.62685 q^{31} -6.27492 q^{35} -0.837253 q^{37} +10.5498 q^{41} +10.2749 q^{43} -3.94027 q^{45} -4.77753 q^{47} +15.8248 q^{49} -10.3923 q^{53} -2.98793 q^{55} +8.54983 q^{59} -1.31342 q^{61} +14.3326 q^{63} +8.00000 q^{65} -8.54983 q^{67} -14.8087 q^{71} -4.27492 q^{73} +10.8685 q^{77} +4.30136 q^{79} +9.00000 q^{81} +13.0997 q^{83} -5.61478 q^{85} +10.0000 q^{89} -29.0997 q^{91} -1.31342 q^{95} +1.45017 q^{97} +6.82475 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9}+O(q^{10}) $ 4 * q - 12 * q^9	$ 4 q - 12 q^{9} + 6 q^{11} - 2 q^{17} - 4 q^{19} + 2 q^{25} - 10 q^{35} + 12 q^{41} + 26 q^{43} + 18 q^{49} + 4 q^{59} + 32 q^{65} - 4 q^{67} - 2 q^{73} + 36 q^{81} - 8 q^{83} + 40 q^{89} - 56 q^{91} + 36 q^{97} - 18 q^{99}+O(q^{100}) $ 4 * q - 12 * q^9 + 6 * q^11 - 2 * q^17 - 4 * q^19 + 2 * q^25 - 10 * q^35 + 12 * q^41 + 26 * q^43 + 18 * q^49 + 4 * q^59 + 32 * q^65 - 4 * q^67 - 2 * q^73 + 36 * q^81 - 8 * q^83 + 40 * q^89 - 56 * q^91 + 36 * q^97 - 18 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	0	0
$5$	1.31342	0.587381	0.293691	−	0.955901i	$-0.405116\pi$
$5$	1.31342	0.587381	0.293691	+	0.955901i	$0.405116\pi$
$6$	0	0
$7$	−4.77753	−1.80573	−0.902867	−	0.429919i	$-0.858542\pi$
$7$	−4.77753	−1.80573	−0.902867	+	0.429919i	$0.858542\pi$
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	−2.27492	−0.685913	−0.342957	−	0.939351i	$-0.611428\pi$
$11$	−2.27492	−0.685913	−0.342957	+	0.939351i	$0.611428\pi$
$12$	0	0
$13$	6.09095	1.68933	0.844663	−	0.535299i	$-0.179801\pi$
$13$	6.09095	1.68933	0.844663	+	0.535299i	$0.179801\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.27492	−1.03682	−0.518410	−	0.855132i	$-0.673476\pi$
$17$	−4.27492	−1.03682	−0.518410	+	0.855132i	$0.673476\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.46410	−0.722315	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410	−0.722315	−0.361158	+	0.932505i	$0.617618\pi$
$24$	0	0
$25$	−3.27492	−0.654983
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.09095	1.13106	0.565530	−	0.824727i	$-0.308671\pi$
$29$	6.09095	1.13106	0.565530	+	0.824727i	$0.308671\pi$
$30$	0	0
$31$	−2.62685	−0.471796	−0.235898	−	0.971778i	$-0.575803\pi$
$31$	−2.62685	−0.471796	−0.235898	+	0.971778i	$0.575803\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−6.27492	−1.06065
$36$	0	0
$37$	−0.837253	−0.137644	−0.0688218	−	0.997629i	$-0.521924\pi$
$37$	−0.837253	−0.137644	−0.0688218	+	0.997629i	$0.521924\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10.5498	1.64761	0.823804	−	0.566875i	$-0.191848\pi$
$41$	10.5498	1.64761	0.823804	+	0.566875i	$0.191848\pi$
$42$	0	0
$43$	10.2749	1.56691	0.783455	−	0.621448i	$-0.213455\pi$
$43$	10.2749	1.56691	0.783455	+	0.621448i	$0.213455\pi$
$44$	0	0
$45$	−3.94027	−0.587381
$46$	0	0
$47$	−4.77753	−0.696874	−0.348437	−	0.937332i	$-0.613287\pi$
$47$	−4.77753	−0.696874	−0.348437	+	0.937332i	$0.613287\pi$
$48$	0	0
$49$	15.8248	2.26068
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.3923	−1.42749	−0.713746	−	0.700404i	$-0.753003\pi$
$53$	−10.3923	−1.42749	−0.713746	+	0.700404i	$0.753003\pi$
$54$	0	0
$55$	−2.98793	−0.402893
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.54983	1.11309	0.556547	−	0.830816i	$-0.312126\pi$
$59$	8.54983	1.11309	0.556547	+	0.830816i	$0.312126\pi$
$60$	0	0
$61$	−1.31342	−0.168167	−0.0840834	−	0.996459i	$-0.526796\pi$
$61$	−1.31342	−0.168167	−0.0840834	+	0.996459i	$0.526796\pi$
$62$	0	0
$63$	14.3326	1.80573
$64$	0	0
$65$	8.00000	0.992278
$66$	0	0
$67$	−8.54983	−1.04453	−0.522264	−	0.852784i	$-0.674913\pi$
$67$	−8.54983	−1.04453	−0.522264	+	0.852784i	$0.674913\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−14.8087	−1.75748	−0.878738	−	0.477305i	$-0.841614\pi$
$71$	−14.8087	−1.75748	−0.878738	+	0.477305i	$0.841614\pi$
$72$	0	0
$73$	−4.27492	−0.500341	−0.250171	−	0.968202i	$-0.580487\pi$
$73$	−4.27492	−0.500341	−0.250171	+	0.968202i	$0.580487\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.8685	1.23858
$78$	0	0
$79$	4.30136	0.483940	0.241970	−	0.970284i	$-0.422206\pi$
$79$	4.30136	0.483940	0.241970	+	0.970284i	$0.422206\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	13.0997	1.43788	0.718938	−	0.695074i	$-0.244629\pi$
$83$	13.0997	1.43788	0.718938	+	0.695074i	$0.244629\pi$
$84$	0	0
$85$	−5.61478	−0.609008
$86$	0	0
$87$	0	0
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	−29.0997	−3.05047
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.31342	−0.134754
$96$	0	0
$97$	1.45017	0.147242	0.0736210	−	0.997286i	$-0.476544\pi$
$97$	1.45017	0.147242	0.0736210	+	0.997286i	$0.476544\pi$
$98$	0	0
$99$	6.82475	0.685913
$100$	0	0
$101$	13.8564	1.37876	0.689382	−	0.724398i	$-0.257882\pi$
$101$	13.8564	1.37876	0.689382	+	0.724398i	$0.257882\pi$
$102$	0	0
$103$	2.62685	0.258831	0.129416	−	0.991590i	$-0.458690\pi$
$103$	2.62685	0.258831	0.129416	+	0.991590i	$0.458690\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.5498	1.59993	0.799966	−	0.600045i	$-0.204851\pi$
$107$	16.5498	1.59993	0.799966	+	0.600045i	$0.204851\pi$
$108$	0	0
$109$	−3.46410	−0.331801	−0.165900	−	0.986143i	$-0.553053\pi$
$109$	−3.46410	−0.331801	−0.165900	+	0.986143i	$0.553053\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.54983	−0.239868	−0.119934	−	0.992782i	$-0.538268\pi$
$113$	−2.54983	−0.239868	−0.119934	+	0.992782i	$0.538268\pi$
$114$	0	0
$115$	−4.54983	−0.424274
$116$	0	0
$117$	−18.2728	−1.68933
$118$	0	0
$119$	20.4235	1.87222
$120$	0	0
$121$	−5.82475	−0.529523
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−10.8685	−0.972106
$126$	0	0
$127$	17.4356	1.54716	0.773579	−	0.633699i	$-0.218464\pi$
$127$	17.4356	1.54716	0.773579	+	0.633699i	$0.218464\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.27492	0.198760	0.0993802	−	0.995050i	$-0.468314\pi$
$131$	2.27492	0.198760	0.0993802	+	0.995050i	$0.468314\pi$
$132$	0	0
$133$	4.77753	0.414264
$134$	0	0
$135$	0	0
$136$	0	0
$137$	20.2749	1.73220	0.866102	−	0.499868i	$-0.166618\pi$
$137$	20.2749	1.73220	0.866102	+	0.499868i	$0.166618\pi$
$138$	0	0
$139$	10.2749	0.871507	0.435754	−	0.900066i	$-0.356482\pi$
$139$	10.2749	0.871507	0.435754	+	0.900066i	$0.356482\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−13.8564	−1.15873
$144$	0	0
$145$	8.00000	0.664364
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.98793	0.244781	0.122390	−	0.992482i	$-0.460944\pi$
$149$	2.98793	0.244781	0.122390	+	0.992482i	$0.460944\pi$
$150$	0	0
$151$	9.55505	0.777579	0.388790	−	0.921327i	$-0.372893\pi$
$151$	9.55505	0.777579	0.388790	+	0.921327i	$0.372893\pi$
$152$	0	0
$153$	12.8248	1.03682
$154$	0	0
$155$	−3.45017	−0.277124
$156$	0	0
$157$	−5.25370	−0.419291	−0.209645	−	0.977778i	$-0.567231\pi$
$157$	−5.25370	−0.419291	−0.209645	+	0.977778i	$0.567231\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	16.5498	1.30431
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.67451	0.129577	0.0647886	−	0.997899i	$-0.479363\pi$
$167$	1.67451	0.129577	0.0647886	+	0.997899i	$0.479363\pi$
$168$	0	0
$169$	24.0997	1.85382
$170$	0	0
$171$	3.00000	0.229416
$172$	0	0
$173$	8.71780	0.662802	0.331401	−	0.943490i	$-0.392479\pi$
$173$	8.71780	0.662802	0.331401	+	0.943490i	$0.392479\pi$
$174$	0	0
$175$	15.6460	1.18273
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−9.09967	−0.680141	−0.340071	−	0.940400i	$-0.610451\pi$
$179$	−9.09967	−0.680141	−0.340071	+	0.940400i	$0.610451\pi$
$180$	0	0
$181$	−3.46410	−0.257485	−0.128742	−	0.991678i	$-0.541094\pi$
$181$	−3.46410	−0.257485	−0.128742	+	0.991678i	$0.541094\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.09967	−0.0808493
$186$	0	0
$187$	9.72508	0.711168
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−10.0312	−0.725834	−0.362917	−	0.931822i	$-0.618219\pi$
$191$	−10.0312	−0.725834	−0.362917	+	0.931822i	$0.618219\pi$
$192$	0	0
$193$	−19.0997	−1.37482	−0.687412	−	0.726268i	$-0.741253\pi$
$193$	−19.0997	−1.37482	−0.687412	+	0.726268i	$0.741253\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.1819	0.867924	0.433962	−	0.900931i	$-0.357115\pi$
$197$	12.1819	0.867924	0.433962	+	0.900931i	$0.357115\pi$
$198$	0	0
$199$	18.6339	1.32092	0.660462	−	0.750859i	$-0.270360\pi$
$199$	18.6339	1.32092	0.660462	+	0.750859i	$0.270360\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−29.0997	−2.04240
$204$	0	0
$205$	13.8564	0.967773
$206$	0	0
$207$	10.3923	0.722315
$208$	0	0
$209$	2.27492	0.157359
$210$	0	0
$211$	11.4502	0.788262	0.394131	−	0.919054i	$-0.371046\pi$
$211$	11.4502	0.788262	0.394131	+	0.919054i	$0.371046\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	13.4953	0.920373
$216$	0	0
$217$	12.5498	0.851938
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−26.0383	−1.75153
$222$	0	0
$223$	−17.4356	−1.16757	−0.583787	−	0.811907i	$-0.698430\pi$
$223$	−17.4356	−1.16757	−0.583787	+	0.811907i	$0.698430\pi$
$224$	0	0
$225$	9.82475	0.654983
$226$	0	0
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	22.0980	1.46028	0.730140	−	0.683298i	$-0.239455\pi$
$229$	22.0980	1.46028	0.730140	+	0.683298i	$0.239455\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−17.3746	−1.13825	−0.569123	−	0.822252i	$-0.692717\pi$
$233$	−17.3746	−1.13825	−0.569123	+	0.822252i	$0.692717\pi$
$234$	0	0
$235$	−6.27492	−0.409330
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3.82518	−0.247431	−0.123715	−	0.992318i	$-0.539481\pi$
$239$	−3.82518	−0.247431	−0.123715	+	0.992318i	$0.539481\pi$
$240$	0	0
$241$	2.54983	0.164249	0.0821246	−	0.996622i	$-0.473829\pi$
$241$	2.54983	0.164249	0.0821246	+	0.996622i	$0.473829\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	20.7846	1.32788
$246$	0	0
$247$	−6.09095	−0.387558
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−18.2749	−1.15350	−0.576751	−	0.816920i	$-0.695680\pi$
$251$	−18.2749	−1.15350	−0.576751	+	0.816920i	$0.695680\pi$
$252$	0	0
$253$	7.88054	0.495446
$254$	0	0
$255$	0	0
$256$	0	0
$257$	22.5498	1.40662	0.703310	−	0.710883i	$-0.251705\pi$
$257$	22.5498	1.40662	0.703310	+	0.710883i	$0.251705\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	0	0
$261$	−18.2728	−1.13106
$262$	0	0
$263$	−14.3326	−0.883785	−0.441892	−	0.897068i	$-0.645693\pi$
$263$	−14.3326	−0.883785	−0.441892	+	0.897068i	$0.645693\pi$
$264$	0	0
$265$	−13.6495	−0.838482
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−22.5742	−1.37637	−0.688187	−	0.725534i	$-0.741593\pi$
$269$	−22.5742	−1.37637	−0.688187	+	0.725534i	$0.741593\pi$
$270$	0	0
$271$	7.04329	0.427849	0.213925	−	0.976850i	$-0.431375\pi$
$271$	7.04329	0.427849	0.213925	+	0.976850i	$0.431375\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	7.45017	0.449262
$276$	0	0
$277$	−2.98793	−0.179527	−0.0897637	−	0.995963i	$-0.528611\pi$
$277$	−2.98793	−0.179527	−0.0897637	+	0.995963i	$0.528611\pi$
$278$	0	0
$279$	7.88054	0.471796
$280$	0	0
$281$	19.0997	1.13939	0.569695	−	0.821856i	$-0.307061\pi$
$281$	19.0997	1.13939	0.569695	+	0.821856i	$0.307061\pi$
$282$	0	0
$283$	11.3746	0.676149	0.338074	−	0.941119i	$-0.390224\pi$
$283$	11.3746	0.676149	0.338074	+	0.941119i	$0.390224\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−50.4021	−2.97514
$288$	0	0
$289$	1.27492	0.0749951
$290$	0	0
$291$	0	0
$292$	0	0
$293$	21.6219	1.26316	0.631581	−	0.775310i	$-0.282406\pi$
$293$	21.6219	1.26316	0.631581	+	0.775310i	$0.282406\pi$
$294$	0	0
$295$	11.2296	0.653810
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−21.0997	−1.22023
$300$	0	0
$301$	−49.0887	−2.82942
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.72508	−0.0987780
$306$	0	0
$307$	16.5498	0.944549	0.472274	−	0.881452i	$-0.343433\pi$
$307$	16.5498	0.944549	0.472274	+	0.881452i	$0.343433\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−5.72987	−0.324911	−0.162455	−	0.986716i	$-0.551941\pi$
$311$	−5.72987	−0.324911	−0.162455	+	0.986716i	$0.551941\pi$
$312$	0	0
$313$	15.0997	0.853484	0.426742	−	0.904373i	$-0.359661\pi$
$313$	15.0997	0.853484	0.426742	+	0.904373i	$0.359661\pi$
$314$	0	0
$315$	18.8248	1.06065
$316$	0	0
$317$	−8.71780	−0.489640	−0.244820	−	0.969569i	$-0.578729\pi$
$317$	−8.71780	−0.489640	−0.244820	+	0.969569i	$0.578729\pi$
$318$	0	0
$319$	−13.8564	−0.775810
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.27492	0.237863
$324$	0	0
$325$	−19.9474	−1.10648
$326$	0	0
$327$	0	0
$328$	0	0
$329$	22.8248	1.25837
$330$	0	0
$331$	24.5498	1.34938	0.674690	−	0.738101i	$-0.264277\pi$
$331$	24.5498	1.34938	0.674690	+	0.738101i	$0.264277\pi$
$332$	0	0
$333$	2.51176	0.137644
$334$	0	0
$335$	−11.2296	−0.613536
$336$	0	0
$337$	−11.6495	−0.634589	−0.317294	−	0.948327i	$-0.602774\pi$
$337$	−11.6495	−0.634589	−0.317294	+	0.948327i	$0.602774\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.97586	0.323611
$342$	0	0
$343$	−42.1605	−2.27645
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−14.8248	−0.795834	−0.397917	−	0.917421i	$-0.630267\pi$
$347$	−14.8248	−0.795834	−0.397917	+	0.917421i	$0.630267\pi$
$348$	0	0
$349$	35.2323	1.88594	0.942970	−	0.332877i	$-0.108019\pi$
$349$	35.2323	1.88594	0.942970	+	0.332877i	$0.108019\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	0	0
$355$	−19.4502	−1.03231
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−0.476171	−0.0251313	−0.0125657	−	0.999921i	$-0.504000\pi$
$359$	−0.476171	−0.0251313	−0.0125657	+	0.999921i	$0.504000\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−5.61478	−0.293891
$366$	0	0
$367$	−1.78959	−0.0934161	−0.0467080	−	0.998909i	$-0.514873\pi$
$367$	−1.78959	−0.0934161	−0.0467080	+	0.998909i	$0.514873\pi$
$368$	0	0
$369$	−31.6495	−1.64761
$370$	0	0
$371$	49.6495	2.57767
$372$	0	0
$373$	−25.2011	−1.30486	−0.652431	−	0.757849i	$-0.726251\pi$
$373$	−25.2011	−1.30486	−0.652431	+	0.757849i	$0.726251\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	37.0997	1.91073
$378$	0	0
$379$	−21.6495	−1.11206	−0.556030	−	0.831162i	$-0.687676\pi$
$379$	−21.6495	−1.11206	−0.556030	+	0.831162i	$0.687676\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5.97586	−0.305352	−0.152676	−	0.988276i	$-0.548789\pi$
$383$	−5.97586	−0.305352	−0.152676	+	0.988276i	$0.548789\pi$
$384$	0	0
$385$	14.2749	0.727517
$386$	0	0
$387$	−30.8248	−1.56691
$388$	0	0
$389$	−17.7967	−0.902327	−0.451164	−	0.892441i	$-0.648991\pi$
$389$	−17.7967	−0.902327	−0.451164	+	0.892441i	$0.648991\pi$
$390$	0	0
$391$	14.8087	0.748911
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5.64950	0.284257
$396$	0	0
$397$	24.0027	1.20466	0.602331	−	0.798247i	$-0.294239\pi$
$397$	24.0027	1.20466	0.602331	+	0.798247i	$0.294239\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.90033	0.244711	0.122355	−	0.992486i	$-0.460955\pi$
$401$	4.90033	0.244711	0.122355	+	0.992486i	$0.460955\pi$
$402$	0	0
$403$	−16.0000	−0.797017
$404$	0	0
$405$	11.8208	0.587381
$406$	0	0
$407$	1.90468	0.0944116
$408$	0	0
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−40.8471	−2.00995
$414$	0	0
$415$	17.2054	0.844581
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−24.2487	−1.18181	−0.590905	−	0.806741i	$-0.701229\pi$
$421$	−24.2487	−1.18181	−0.590905	+	0.806741i	$0.701229\pi$
$422$	0	0
$423$	14.3326	0.696874
$424$	0	0
$425$	14.0000	0.679100
$426$	0	0
$427$	6.27492	0.303665
$428$	0	0
$429$	0	0
$430$	0	0
$431$	19.1101	0.920501	0.460251	−	0.887789i	$-0.347760\pi$
$431$	19.1101	0.920501	0.460251	+	0.887789i	$0.347760\pi$
$432$	0	0
$433$	27.6495	1.32875	0.664375	−	0.747399i	$-0.268698\pi$
$433$	27.6495	1.32875	0.664375	+	0.747399i	$0.268698\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.46410	0.165710
$438$	0	0
$439$	−3.57919	−0.170825	−0.0854127	−	0.996346i	$-0.527221\pi$
$439$	−3.57919	−0.170825	−0.0854127	+	0.996346i	$0.527221\pi$
$440$	0	0
$441$	−47.4743	−2.26068
$442$	0	0
$443$	27.3746	1.30061	0.650303	−	0.759675i	$-0.274642\pi$
$443$	27.3746	1.30061	0.650303	+	0.759675i	$0.274642\pi$
$444$	0	0
$445$	13.1342	0.622623
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−5.45017	−0.257209	−0.128605	−	0.991696i	$-0.541050\pi$
$449$	−5.45017	−0.257209	−0.128605	+	0.991696i	$0.541050\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−38.2202	−1.79179
$456$	0	0
$457$	7.72508	0.361364	0.180682	−	0.983542i	$-0.442169\pi$
$457$	7.72508	0.361364	0.180682	+	0.983542i	$0.442169\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	35.2323	1.64093	0.820465	−	0.571696i	$-0.193714\pi$
$461$	35.2323	1.64093	0.820465	+	0.571696i	$0.193714\pi$
$462$	0	0
$463$	−33.4427	−1.55421	−0.777107	−	0.629369i	$-0.783313\pi$
$463$	−33.4427	−1.55421	−0.777107	+	0.629369i	$0.783313\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	30.8248	1.42640	0.713200	−	0.700961i	$-0.247245\pi$
$467$	30.8248	1.42640	0.713200	+	0.700961i	$0.247245\pi$
$468$	0	0
$469$	40.8471	1.88614
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−23.3746	−1.07476
$474$	0	0
$475$	3.27492	0.150264
$476$	0	0
$477$	31.1769	1.42749
$478$	0	0
$479$	−17.3205	−0.791394	−0.395697	−	0.918381i	$-0.629497\pi$
$479$	−17.3205	−0.791394	−0.395697	+	0.918381i	$0.629497\pi$
$480$	0	0
$481$	−5.09967	−0.232525
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.90468	0.0864872
$486$	0	0
$487$	−12.9041	−0.584739	−0.292370	−	0.956305i	$-0.594444\pi$
$487$	−12.9041	−0.584739	−0.292370	+	0.956305i	$0.594444\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	−26.0383	−1.17271
$494$	0	0
$495$	8.96379	0.402893
$496$	0	0
$497$	70.7492	3.17353
$498$	0	0
$499$	34.2749	1.53436	0.767178	−	0.641434i	$-0.221660\pi$
$499$	34.2749	1.53436	0.767178	+	0.641434i	$0.221660\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6.81312	0.303782	0.151891	−	0.988397i	$-0.451464\pi$
$503$	6.81312	0.303782	0.151891	+	0.988397i	$0.451464\pi$
$504$	0	0
$505$	18.1993	0.809860
$506$	0	0
$507$	0	0
$508$	0	0
$509$	13.0192	0.577064	0.288532	−	0.957470i	$-0.406833\pi$
$509$	13.0192	0.577064	0.288532	+	0.957470i	$0.406833\pi$
$510$	0	0
$511$	20.4235	0.903484
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.45017	0.152032
$516$	0	0
$517$	10.8685	0.477995
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−3.09967	−0.135799	−0.0678995	−	0.997692i	$-0.521630\pi$
$521$	−3.09967	−0.135799	−0.0678995	+	0.997692i	$0.521630\pi$
$522$	0	0
$523$	−20.5498	−0.898582	−0.449291	−	0.893386i	$-0.648323\pi$
$523$	−20.5498	−0.898582	−0.449291	+	0.893386i	$0.648323\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	11.2296	0.489167
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	−25.6495	−1.11309
$532$	0	0
$533$	64.2585	2.78335
$534$	0	0
$535$	21.7370	0.939770
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−36.0000	−1.55063
$540$	0	0
$541$	7.28929	0.313391	0.156695	−	0.987647i	$-0.449916\pi$
$541$	7.28929	0.313391	0.156695	+	0.987647i	$0.449916\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−4.54983	−0.194893
$546$	0	0
$547$	−32.0000	−1.36822	−0.684111	−	0.729378i	$-0.739809\pi$
$547$	−32.0000	−1.36822	−0.684111	+	0.729378i	$0.739809\pi$
$548$	0	0
$549$	3.94027	0.168167
$550$	0	0
$551$	−6.09095	−0.259483
$552$	0	0
$553$	−20.5498	−0.873868
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16.8443	0.713717	0.356859	−	0.934158i	$-0.383848\pi$
$557$	16.8443	0.713717	0.356859	+	0.934158i	$0.383848\pi$
$558$	0	0
$559$	62.5840	2.64702
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	−3.34901	−0.140894
$566$	0	0
$567$	−42.9977	−1.80573
$568$	0	0
$569$	34.5498	1.44840	0.724202	−	0.689588i	$-0.242208\pi$
$569$	34.5498	1.44840	0.724202	+	0.689588i	$0.242208\pi$
$570$	0	0
$571$	10.9003	0.456165	0.228082	−	0.973642i	$-0.426754\pi$
$571$	10.9003	0.456165	0.228082	+	0.973642i	$0.426754\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	11.3446	0.473104
$576$	0	0
$577$	15.7251	0.654644	0.327322	−	0.944913i	$-0.393854\pi$
$577$	15.7251	0.654644	0.327322	+	0.944913i	$0.393854\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−62.5840	−2.59642
$582$	0	0
$583$	23.6416	0.979136
$584$	0	0
$585$	−24.0000	−0.992278
$586$	0	0
$587$	29.7251	1.22689	0.613443	−	0.789739i	$-0.289784\pi$
$587$	29.7251	1.22689	0.613443	+	0.789739i	$0.289784\pi$
$588$	0	0
$589$	2.62685	0.108237
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−11.0997	−0.455809	−0.227904	−	0.973684i	$-0.573187\pi$
$593$	−11.0997	−0.455809	−0.227904	+	0.973684i	$0.573187\pi$
$594$	0	0
$595$	26.8248	1.09971
$596$	0	0
$597$	0	0
$598$	0	0
$599$	35.5934	1.45431	0.727153	−	0.686476i	$-0.240843\pi$
$599$	35.5934	1.45431	0.727153	+	0.686476i	$0.240843\pi$
$600$	0	0
$601$	19.0997	0.779092	0.389546	−	0.921007i	$-0.372632\pi$
$601$	19.0997	0.779092	0.389546	+	0.921007i	$0.372632\pi$
$602$	0	0
$603$	25.6495	1.04453
$604$	0	0
$605$	−7.65037	−0.311032
$606$	0	0
$607$	46.8229	1.90048	0.950242	−	0.311513i	$-0.100836\pi$
$607$	46.8229	1.90048	0.950242	+	0.311513i	$0.100836\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−29.0997	−1.17725
$612$	0	0
$613$	−16.8443	−0.680336	−0.340168	−	0.940365i	$-0.610484\pi$
$613$	−16.8443	−0.680336	−0.340168	+	0.940365i	$0.610484\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−20.2749	−0.816237	−0.408119	−	0.912929i	$-0.633815\pi$
$617$	−20.2749	−0.816237	−0.408119	+	0.912929i	$0.633815\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−47.7753	−1.91408
$624$	0	0
$625$	2.09967	0.0839868
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.57919	0.142712
$630$	0	0
$631$	−9.07888	−0.361425	−0.180712	−	0.983536i	$-0.557840\pi$
$631$	−9.07888	−0.361425	−0.180712	+	0.983536i	$0.557840\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	22.9003	0.908772
$636$	0	0
$637$	96.3878	3.81902
$638$	0	0
$639$	44.4262	1.75748
$640$	0	0
$641$	2.54983	0.100712	0.0503562	−	0.998731i	$-0.483964\pi$
$641$	2.54983	0.100712	0.0503562	+	0.998731i	$0.483964\pi$
$642$	0	0
$643$	7.92442	0.312509	0.156254	−	0.987717i	$-0.450058\pi$
$643$	7.92442	0.312509	0.156254	+	0.987717i	$0.450058\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3.82518	−0.150384	−0.0751918	−	0.997169i	$-0.523957\pi$
$647$	−3.82518	−0.150384	−0.0751918	+	0.997169i	$0.523957\pi$
$648$	0	0
$649$	−19.4502	−0.763486
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.98793	0.116927	0.0584634	−	0.998290i	$-0.481380\pi$
$653$	2.98793	0.116927	0.0584634	+	0.998290i	$0.481380\pi$
$654$	0	0
$655$	2.98793	0.116748
$656$	0	0
$657$	12.8248	0.500341
$658$	0	0
$659$	−28.5498	−1.11214	−0.556072	−	0.831134i	$-0.687692\pi$
$659$	−28.5498	−1.11214	−0.556072	+	0.831134i	$0.687692\pi$
$660$	0	0
$661$	−0.837253	−0.0325654	−0.0162827	−	0.999867i	$-0.505183\pi$
$661$	−0.837253	−0.0325654	−0.0162827	+	0.999867i	$0.505183\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	6.27492	0.243331
$666$	0	0
$667$	−21.0997	−0.816982
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.98793	0.115348
$672$	0	0
$673$	−16.9003	−0.651460	−0.325730	−	0.945463i	$-0.605610\pi$
$673$	−16.9003	−0.651460	−0.325730	+	0.945463i	$0.605610\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−44.3112	−1.70302	−0.851508	−	0.524342i	$-0.824312\pi$
$677$	−44.3112	−1.70302	−0.851508	+	0.524342i	$0.824312\pi$
$678$	0	0
$679$	−6.92820	−0.265880
$680$	0	0
$681$	0	0
$682$	0	0
$683$	38.1993	1.46166	0.730829	−	0.682561i	$-0.239134\pi$
$683$	38.1993	1.46166	0.730829	+	0.682561i	$0.239134\pi$
$684$	0	0
$685$	26.6296	1.01746
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−63.2990	−2.41150
$690$	0	0
$691$	−44.4743	−1.69188	−0.845940	−	0.533278i	$-0.820960\pi$
$691$	−44.4743	−1.69188	−0.845940	+	0.533278i	$0.820960\pi$
$692$	0	0
$693$	−32.6054	−1.23858
$694$	0	0
$695$	13.4953	0.511907
$696$	0	0
$697$	−45.0997	−1.70827
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−48.7276	−1.84042	−0.920208	−	0.391430i	$-0.871981\pi$
$701$	−48.7276	−1.84042	−0.920208	+	0.391430i	$0.871981\pi$
$702$	0	0
$703$	0.837253	0.0315776
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−66.1993	−2.48968
$708$	0	0
$709$	3.34901	0.125775	0.0628874	−	0.998021i	$-0.479969\pi$
$709$	3.34901	0.125775	0.0628874	+	0.998021i	$0.479969\pi$
$710$	0	0
$711$	−12.9041	−0.483940
$712$	0	0
$713$	9.09967	0.340785
$714$	0	0
$715$	−18.1993	−0.680617
$716$	0	0
$717$	0	0
$718$	0	0
$719$	12.4279	0.463482	0.231741	−	0.972777i	$-0.425558\pi$
$719$	12.4279	0.463482	0.231741	+	0.972777i	$0.425558\pi$
$720$	0	0
$721$	−12.5498	−0.467380
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−19.9474	−0.740826
$726$	0	0
$727$	−19.5863	−0.726415	−0.363207	−	0.931708i	$-0.618318\pi$
$727$	−19.5863	−0.726415	−0.363207	+	0.931708i	$0.618318\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−43.9244	−1.62460
$732$	0	0
$733$	1.67451	0.0618493	0.0309247	−	0.999522i	$-0.490155\pi$
$733$	1.67451	0.0618493	0.0309247	+	0.999522i	$0.490155\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	19.4502	0.716456
$738$	0	0
$739$	−14.8248	−0.545337	−0.272669	−	0.962108i	$-0.587906\pi$
$739$	−14.8248	−0.545337	−0.272669	+	0.962108i	$0.587906\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	26.7605	0.981746	0.490873	−	0.871231i	$-0.336678\pi$
$743$	26.7605	0.981746	0.490873	+	0.871231i	$0.336678\pi$
$744$	0	0
$745$	3.92442	0.143780
$746$	0	0
$747$	−39.2990	−1.43788
$748$	0	0
$749$	−79.0673	−2.88905
$750$	0	0
$751$	−53.7511	−1.96141	−0.980703	−	0.195503i	$-0.937366\pi$
$751$	−53.7511	−1.96141	−0.980703	+	0.195503i	$0.937366\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	12.5498	0.456735
$756$	0	0
$757$	−1.31342	−0.0477372	−0.0238686	−	0.999715i	$-0.507598\pi$
$757$	−1.31342	−0.0477372	−0.0238686	+	0.999715i	$0.507598\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	4.27492	0.154966	0.0774828	−	0.996994i	$-0.475312\pi$
$761$	4.27492	0.154966	0.0774828	+	0.996994i	$0.475312\pi$
$762$	0	0
$763$	16.5498	0.599144
$764$	0	0
$765$	16.8443	0.609008
$766$	0	0
$767$	52.0766	1.88038
$768$	0	0
$769$	8.82475	0.318229	0.159114	−	0.987260i	$-0.449136\pi$
$769$	8.82475	0.318229	0.159114	+	0.987260i	$0.449136\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−0.115088	−0.00413942	−0.00206971	−	0.999998i	$-0.500659\pi$
$773$	−0.115088	−0.00413942	−0.00206971	+	0.999998i	$0.500659\pi$
$774$	0	0
$775$	8.60271	0.309018
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−10.5498	−0.377987
$780$	0	0
$781$	33.6887	1.20548
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−6.90033	−0.246283
$786$	0	0
$787$	46.1993	1.64683	0.823414	−	0.567441i	$-0.192066\pi$
$787$	46.1993	1.64683	0.823414	+	0.567441i	$0.192066\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	12.1819	0.433138
$792$	0	0
$793$	−8.00000	−0.284088
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−24.2487	−0.858933	−0.429467	−	0.903083i	$-0.641298\pi$
$797$	−24.2487	−0.858933	−0.429467	+	0.903083i	$0.641298\pi$
$798$	0	0
$799$	20.4235	0.722532
$800$	0	0
$801$	−30.0000	−1.06000
$802$	0	0
$803$	9.72508	0.343191
$804$	0	0
$805$	21.7370	0.766127
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−16.8248	−0.591527	−0.295763	−	0.955261i	$-0.595574\pi$
$809$	−16.8248	−0.591527	−0.295763	+	0.955261i	$0.595574\pi$
$810$	0	0
$811$	43.2990	1.52043	0.760217	−	0.649669i	$-0.225093\pi$
$811$	43.2990	1.52043	0.760217	+	0.649669i	$0.225093\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−5.25370	−0.184029
$816$	0	0
$817$	−10.2749	−0.359474
$818$	0	0
$819$	87.2990	3.05047
$820$	0	0
$821$	−21.3759	−0.746023	−0.373011	−	0.927827i	$-0.621675\pi$
$821$	−21.3759	−0.746023	−0.373011	+	0.927827i	$0.621675\pi$
$822$	0	0
$823$	−43.9501	−1.53200	−0.766002	−	0.642839i	$-0.777757\pi$
$823$	−43.9501	−1.53200	−0.766002	+	0.642839i	$0.777757\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.35050	−0.0817348	−0.0408674	−	0.999165i	$-0.513012\pi$
$827$	−2.35050	−0.0817348	−0.0408674	+	0.999165i	$0.513012\pi$
$828$	0	0
$829$	−35.7084	−1.24021	−0.620103	−	0.784521i	$-0.712909\pi$
$829$	−35.7084	−1.24021	−0.620103	+	0.784521i	$0.712909\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−67.6495	−2.34392
$834$	0	0
$835$	2.19934	0.0761112
$836$	0	0
$837$	0	0
$838$	0	0
$839$	38.9424	1.34444	0.672220	−	0.740351i	$-0.265341\pi$
$839$	38.9424	1.34444	0.672220	+	0.740351i	$0.265341\pi$
$840$	0	0
$841$	8.09967	0.279299
$842$	0	0
$843$	0	0
$844$	0	0
$845$	31.6531	1.08890
$846$	0	0
$847$	27.8279	0.956178
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.90033	0.0994221
$852$	0	0
$853$	15.5309	0.531768	0.265884	−	0.964005i	$-0.414336\pi$
$853$	15.5309	0.531768	0.265884	+	0.964005i	$0.414336\pi$
$854$	0	0
$855$	3.94027	0.134754
$856$	0	0
$857$	−44.1993	−1.50982	−0.754910	−	0.655828i	$-0.772320\pi$
$857$	−44.1993	−1.50982	−0.754910	+	0.655828i	$0.772320\pi$
$858$	0	0
$859$	12.4743	0.425616	0.212808	−	0.977094i	$-0.431739\pi$
$859$	12.4743	0.425616	0.212808	+	0.977094i	$0.431739\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	27.9430	0.951190	0.475595	−	0.879664i	$-0.342233\pi$
$863$	27.9430	0.951190	0.475595	+	0.879664i	$0.342233\pi$
$864$	0	0
$865$	11.4502	0.389317
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−9.78523	−0.331941
$870$	0	0
$871$	−52.0766	−1.76455
$872$	0	0
$873$	−4.35050	−0.147242
$874$	0	0
$875$	51.9244	1.75537
$876$	0	0
$877$	−38.3353	−1.29449	−0.647245	−	0.762282i	$-0.724079\pi$
$877$	−38.3353	−1.29449	−0.647245	+	0.762282i	$0.724079\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−20.8248	−0.701604	−0.350802	−	0.936450i	$-0.614091\pi$
$881$	−20.8248	−0.701604	−0.350802	+	0.936450i	$0.614091\pi$
$882$	0	0
$883$	51.3746	1.72889	0.864446	−	0.502725i	$-0.167669\pi$
$883$	51.3746	1.72889	0.864446	+	0.502725i	$0.167669\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−9.55505	−0.320827	−0.160414	−	0.987050i	$-0.551283\pi$
$887$	−9.55505	−0.320827	−0.160414	+	0.987050i	$0.551283\pi$
$888$	0	0
$889$	−83.2990	−2.79376
$890$	0	0
$891$	−20.4743	−0.685913
$892$	0	0
$893$	4.77753	0.159874
$894$	0	0
$895$	−11.9517	−0.399502
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−16.0000	−0.533630
$900$	0	0
$901$	44.4262	1.48005
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4.54983	−0.151242
$906$	0	0
$907$	21.6495	0.718860	0.359430	−	0.933172i	$-0.382971\pi$
$907$	21.6495	0.718860	0.359430	+	0.933172i	$0.382971\pi$
$908$	0	0
$909$	−41.5692	−1.37876
$910$	0	0
$911$	−44.4262	−1.47191	−0.735954	−	0.677032i	$-0.763266\pi$
$911$	−44.4262	−1.47191	−0.735954	+	0.677032i	$0.763266\pi$
$912$	0	0
$913$	−29.8007	−0.986258
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−10.8685	−0.358909
$918$	0	0
$919$	34.7561	1.14650	0.573249	−	0.819381i	$-0.305683\pi$
$919$	34.7561	1.14650	0.573249	+	0.819381i	$0.305683\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−90.1993	−2.96895
$924$	0	0
$925$	2.74194	0.0901543
$926$	0	0
$927$	−7.88054	−0.258831
$928$	0	0
$929$	23.0997	0.757876	0.378938	−	0.925422i	$-0.376289\pi$
$929$	23.0997	0.757876	0.378938	+	0.925422i	$0.376289\pi$
$930$	0	0
$931$	−15.8248	−0.518635
$932$	0	0
$933$	0	0
$934$	0	0
$935$	12.7732	0.417727
$936$	0	0
$937$	−37.9244	−1.23894	−0.619468	−	0.785022i	$-0.712652\pi$
$937$	−37.9244	−1.23894	−0.619468	+	0.785022i	$0.712652\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	31.1769	1.01634	0.508169	−	0.861257i	$-0.330322\pi$
$941$	31.1769	1.01634	0.508169	+	0.861257i	$0.330322\pi$
$942$	0	0
$943$	−36.5457	−1.19009
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−30.1993	−0.981347	−0.490673	−	0.871344i	$-0.663249\pi$
$947$	−30.1993	−0.981347	−0.490673	+	0.871344i	$0.663249\pi$
$948$	0	0
$949$	−26.0383	−0.845239
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−45.2990	−1.46738	−0.733689	−	0.679485i	$-0.762203\pi$
$953$	−45.2990	−1.46738	−0.733689	+	0.679485i	$0.762203\pi$
$954$	0	0
$955$	−13.1752	−0.426341
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−96.8639	−3.12790
$960$	0	0
$961$	−24.0997	−0.777409
$962$	0	0
$963$	−49.6495	−1.59993
$964$	0	0
$965$	−25.0860	−0.807546
$966$	0	0
$967$	−20.6695	−0.664687	−0.332344	−	0.943158i	$-0.607839\pi$
$967$	−20.6695	−0.664687	−0.332344	+	0.943158i	$0.607839\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	44.0000	1.41203	0.706014	−	0.708198i	$-0.250492\pi$
$971$	44.0000	1.41203	0.706014	+	0.708198i	$0.250492\pi$
$972$	0	0
$973$	−49.0887	−1.57371
$974$	0	0
$975$	0	0
$976$	0	0
$977$	32.1993	1.03015	0.515074	−	0.857146i	$-0.327764\pi$
$977$	32.1993	1.03015	0.515074	+	0.857146i	$0.327764\pi$
$978$	0	0
$979$	−22.7492	−0.727067
$980$	0	0
$981$	10.3923	0.331801
$982$	0	0
$983$	−29.6175	−0.944651	−0.472326	−	0.881424i	$-0.656585\pi$
$983$	−29.6175	−0.944651	−0.472326	+	0.881424i	$0.656585\pi$
$984$	0	0
$985$	16.0000	0.509802
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−35.5934	−1.13180
$990$	0	0
$991$	−40.1249	−1.27461	−0.637305	−	0.770612i	$-0.719951\pi$
$991$	−40.1249	−1.27461	−0.637305	+	0.770612i	$0.719951\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	24.4743	0.775886
$996$	0	0
$997$	−11.5906	−0.367079	−0.183540	−	0.983012i	$-0.558756\pi$
$997$	−11.5906	−0.367079	−0.183540	+	0.983012i	$0.558756\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4864.2.a.bk.1.3		4
4.3	odd	2		4864.2.a.bj.1.3		4
8.3	odd	2	inner	4864.2.a.bk.1.2		4
8.5	even	2		4864.2.a.bj.1.2		4
16.3	odd	4		1216.2.c.i.609.3	✓	8
16.5	even	4		1216.2.c.i.609.6	yes	8
16.11	odd	4		1216.2.c.i.609.5	yes	8
16.13	even	4		1216.2.c.i.609.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.2.c.i.609.3	✓	8	16.3	odd	4
1216.2.c.i.609.4	yes	8	16.13	even	4
1216.2.c.i.609.5	yes	8	16.11	odd	4
1216.2.c.i.609.6	yes	8	16.5	even	4
4864.2.a.bj.1.2		4	8.5	even	2
4864.2.a.bj.1.3		4	4.3	odd	2
4864.2.a.bk.1.2		4	8.3	odd	2	inner
4864.2.a.bk.1.3		4	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:	\( N \)	\(=\)	\( 4864 = 2^{8} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4864.a (trivial)

\(f(q)\)	\(=\)	\(q+1.31342 q^{5} -4.77753 q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+1.31342 q^{5} -4.77753 q^{7} -3.00000 q^{9} -2.27492 q^{11} +6.09095 q^{13} -4.27492 q^{17} -1.00000 q^{19} -3.46410 q^{23} -3.27492 q^{25} +6.09095 q^{29} -2.62685 q^{31} -6.27492 q^{35} -0.837253 q^{37} +10.5498 q^{41} +10.2749 q^{43} -3.94027 q^{45} -4.77753 q^{47} +15.8248 q^{49} -10.3923 q^{53} -2.98793 q^{55} +8.54983 q^{59} -1.31342 q^{61} +14.3326 q^{63} +8.00000 q^{65} -8.54983 q^{67} -14.8087 q^{71} -4.27492 q^{73} +10.8685 q^{77} +4.30136 q^{79} +9.00000 q^{81} +13.0997 q^{83} -5.61478 q^{85} +10.0000 q^{89} -29.0997 q^{91} -1.31342 q^{95} +1.45017 q^{97} +6.82475 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9}+O(q^{10}) \) 4 * q - 12 * q^9	\( 4 q - 12 q^{9} + 6 q^{11} - 2 q^{17} - 4 q^{19} + 2 q^{25} - 10 q^{35} + 12 q^{41} + 26 q^{43} + 18 q^{49} + 4 q^{59} + 32 q^{65} - 4 q^{67} - 2 q^{73} + 36 q^{81} - 8 q^{83} + 40 q^{89} - 56 q^{91} + 36 q^{97} - 18 q^{99}+O(q^{100}) \) 4 * q - 12 * q^9 + 6 * q^11 - 2 * q^17 - 4 * q^19 + 2 * q^25 - 10 * q^35 + 12 * q^41 + 26 * q^43 + 18 * q^49 + 4 * q^59 + 32 * q^65 - 4 * q^67 - 2 * q^73 + 36 * q^81 - 8 * q^83 + 40 * q^89 - 56 * q^91 + 36 * q^97 - 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more