LMFDB - Embedded newform 4864.2.a.bh.1.2

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:	$1$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2432)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.34292$ 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)

$f(q)$	$=$	$q+1.14637 q^{3} +3.34292 q^{5} -3.34292 q^{7} -1.68585 q^{9} +O(q^{10})$	$q+1.14637 q^{3} +3.34292 q^{5} -3.34292 q^{7} -1.68585 q^{9} +0.489289 q^{11} -2.29273 q^{13} +3.83221 q^{15} +0.196558 q^{17} -1.00000 q^{19} -3.83221 q^{21} -4.00000 q^{23} +6.17513 q^{25} -5.37169 q^{27} -6.12494 q^{29} +8.22533 q^{31} +0.560904 q^{33} -11.1751 q^{35} -3.83221 q^{37} -2.62831 q^{39} +4.12494 q^{41} -7.17513 q^{43} -5.63565 q^{45} -11.0073 q^{47} +4.17513 q^{49} +0.225327 q^{51} +4.81079 q^{53} +1.63565 q^{55} -1.14637 q^{57} +5.31415 q^{59} +7.73604 q^{61} +5.63565 q^{63} -7.66442 q^{65} -15.4966 q^{67} -4.58546 q^{69} -11.2713 q^{71} +15.4679 q^{73} +7.07896 q^{75} -1.63565 q^{77} -10.5181 q^{79} -1.10038 q^{81} -0.728692 q^{83} +0.657077 q^{85} -7.02142 q^{87} -8.51806 q^{89} +7.66442 q^{91} +9.42923 q^{93} -3.34292 q^{95} -3.31415 q^{97} -0.824865 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} + 4 q^{5} - 4 q^{7} + 7 q^{9}+O(q^{10}) $ 3 * q + 2 * q^3 + 4 * q^5 - 4 * q^7 + 7 * q^9	$ 3 q + 2 q^{3} + 4 q^{5} - 4 q^{7} + 7 q^{9} - 6 q^{11} - 4 q^{13} - 2 q^{15} - 4 q^{17} - 3 q^{19} + 2 q^{21} - 12 q^{23} - q^{25} + 8 q^{27} - 2 q^{29} + 2 q^{31} + 6 q^{33} - 14 q^{35} + 2 q^{37} - 32 q^{39} - 4 q^{41} - 2 q^{43} - 8 q^{45} - 7 q^{49} - 22 q^{51} - 14 q^{53} - 4 q^{55} - 2 q^{57} + 28 q^{59} + 8 q^{61} + 8 q^{63} + 4 q^{65} - 6 q^{67} - 8 q^{69} - 16 q^{71} + 24 q^{73} + 4 q^{77} - 6 q^{79} + 3 q^{81} - 20 q^{83} + 8 q^{85} - 36 q^{87} - 4 q^{91} - 32 q^{93} - 4 q^{95} - 22 q^{97} - 22 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 + 4 * q^5 - 4 * q^7 + 7 * q^9 - 6 * q^11 - 4 * q^13 - 2 * q^15 - 4 * q^17 - 3 * q^19 + 2 * q^21 - 12 * q^23 - q^25 + 8 * q^27 - 2 * q^29 + 2 * q^31 + 6 * q^33 - 14 * q^35 + 2 * q^37 - 32 * q^39 - 4 * q^41 - 2 * q^43 - 8 * q^45 - 7 * q^49 - 22 * q^51 - 14 * q^53 - 4 * q^55 - 2 * q^57 + 28 * q^59 + 8 * q^61 + 8 * q^63 + 4 * q^65 - 6 * q^67 - 8 * q^69 - 16 * q^71 + 24 * q^73 + 4 * q^77 - 6 * q^79 + 3 * q^81 - 20 * q^83 + 8 * q^85 - 36 * q^87 - 4 * q^91 - 32 * q^93 - 4 * q^95 - 22 * 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
$3$	1.14637	0.661854	0.330927	−	0.943656i	$-0.392639\pi$
$3$	1.14637	0.661854	0.330927	+	0.943656i	$0.392639\pi$
$4$	0	0
$5$	3.34292	1.49500	0.747500	−	0.664261i	$-0.231254\pi$
$5$	3.34292	1.49500	0.747500	+	0.664261i	$0.231254\pi$
$6$	0	0
$7$	−3.34292	−1.26351	−0.631753	−	0.775170i	$-0.717664\pi$
$7$	−3.34292	−1.26351	−0.631753	+	0.775170i	$0.717664\pi$
$8$	0	0
$9$	−1.68585	−0.561949
$10$	0	0
$11$	0.489289	0.147526	0.0737630	−	0.997276i	$-0.476499\pi$
$11$	0.489289	0.147526	0.0737630	+	0.997276i	$0.476499\pi$
$12$	0	0
$13$	−2.29273	−0.635889	−0.317945	−	0.948109i	$-0.602993\pi$
$13$	−2.29273	−0.635889	−0.317945	+	0.948109i	$0.602993\pi$
$14$	0	0
$15$	3.83221	0.989473
$16$	0	0
$17$	0.196558	0.0476722	0.0238361	−	0.999716i	$-0.492412\pi$
$17$	0.196558	0.0476722	0.0238361	+	0.999716i	$0.492412\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−3.83221	−0.836257
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	6.17513	1.23503
$26$	0	0
$27$	−5.37169	−1.03378
$28$	0	0
$29$	−6.12494	−1.13737	−0.568687	−	0.822554i	$-0.692548\pi$
$29$	−6.12494	−1.13737	−0.568687	+	0.822554i	$0.692548\pi$
$30$	0	0
$31$	8.22533	1.47731	0.738656	−	0.674082i	$-0.235461\pi$
$31$	8.22533	1.47731	0.738656	+	0.674082i	$0.235461\pi$
$32$	0	0
$33$	0.560904	0.0976408
$34$	0	0
$35$	−11.1751	−1.88894
$36$	0	0
$37$	−3.83221	−0.630012	−0.315006	−	0.949090i	$-0.602006\pi$
$37$	−3.83221	−0.630012	−0.315006	+	0.949090i	$0.602006\pi$
$38$	0	0
$39$	−2.62831	−0.420866
$40$	0	0
$41$	4.12494	0.644208	0.322104	−	0.946704i	$-0.395610\pi$
$41$	4.12494	0.644208	0.322104	+	0.946704i	$0.395610\pi$
$42$	0	0
$43$	−7.17513	−1.09420	−0.547099	−	0.837068i	$-0.684268\pi$
$43$	−7.17513	−1.09420	−0.547099	+	0.837068i	$0.684268\pi$
$44$	0	0
$45$	−5.63565	−0.840114
$46$	0	0
$47$	−11.0073	−1.60559	−0.802793	−	0.596258i	$-0.796654\pi$
$47$	−11.0073	−1.60559	−0.802793	+	0.596258i	$0.796654\pi$
$48$	0	0
$49$	4.17513	0.596448
$50$	0	0
$51$	0.225327	0.0315521
$52$	0	0
$53$	4.81079	0.660813	0.330406	−	0.943839i	$-0.392814\pi$
$53$	4.81079	0.660813	0.330406	+	0.943839i	$0.392814\pi$
$54$	0	0
$55$	1.63565	0.220552
$56$	0	0
$57$	−1.14637	−0.151840
$58$	0	0
$59$	5.31415	0.691844	0.345922	−	0.938263i	$-0.387566\pi$
$59$	5.31415	0.691844	0.345922	+	0.938263i	$0.387566\pi$
$60$	0	0
$61$	7.73604	0.990498	0.495249	−	0.868751i	$-0.335077\pi$
$61$	7.73604	0.990498	0.495249	+	0.868751i	$0.335077\pi$
$62$	0	0
$63$	5.63565	0.710026
$64$	0	0
$65$	−7.66442	−0.950655
$66$	0	0
$67$	−15.4966	−1.89322	−0.946608	−	0.322388i	$-0.895514\pi$
$67$	−15.4966	−1.89322	−0.946608	+	0.322388i	$0.895514\pi$
$68$	0	0
$69$	−4.58546	−0.552025
$70$	0	0
$71$	−11.2713	−1.33766	−0.668829	−	0.743416i	$-0.733204\pi$
$71$	−11.2713	−1.33766	−0.668829	+	0.743416i	$0.733204\pi$
$72$	0	0
$73$	15.4679	1.81038	0.905188	−	0.425011i	$-0.139730\pi$
$73$	15.4679	1.81038	0.905188	+	0.425011i	$0.139730\pi$
$74$	0	0
$75$	7.07896	0.817408
$76$	0	0
$77$	−1.63565	−0.186400
$78$	0	0
$79$	−10.5181	−1.18337	−0.591687	−	0.806168i	$-0.701538\pi$
$79$	−10.5181	−1.18337	−0.591687	+	0.806168i	$0.701538\pi$
$80$	0	0
$81$	−1.10038	−0.122265
$82$	0	0
$83$	−0.728692	−0.0799843	−0.0399922	−	0.999200i	$-0.512733\pi$
$83$	−0.728692	−0.0799843	−0.0399922	+	0.999200i	$0.512733\pi$
$84$	0	0
$85$	0.657077	0.0712700
$86$	0	0
$87$	−7.02142	−0.752776
$88$	0	0
$89$	−8.51806	−0.902912	−0.451456	−	0.892293i	$-0.649095\pi$
$89$	−8.51806	−0.902912	−0.451456	+	0.892293i	$0.649095\pi$
$90$	0	0
$91$	7.66442	0.803450
$92$	0	0
$93$	9.42923	0.977766
$94$	0	0
$95$	−3.34292	−0.342977
$96$	0	0
$97$	−3.31415	−0.336501	−0.168251	−	0.985744i	$-0.553812\pi$
$97$	−3.31415	−0.336501	−0.168251	+	0.985744i	$0.553812\pi$
$98$	0	0
$99$	−0.824865	−0.0829021
$100$	0	0
$101$	−10.6858	−1.06328	−0.531641	−	0.846970i	$-0.678424\pi$
$101$	−10.6858	−1.06328	−0.531641	+	0.846970i	$0.678424\pi$
$102$	0	0
$103$	−12.0575	−1.18806	−0.594032	−	0.804441i	$-0.702465\pi$
$103$	−12.0575	−1.18806	−0.594032	+	0.804441i	$0.702465\pi$
$104$	0	0
$105$	−12.8108	−1.25020
$106$	0	0
$107$	8.39312	0.811393	0.405697	−	0.914008i	$-0.367029\pi$
$107$	8.39312	0.811393	0.405697	+	0.914008i	$0.367029\pi$
$108$	0	0
$109$	−8.22533	−0.787843	−0.393922	−	0.919144i	$-0.628882\pi$
$109$	−8.22533	−0.787843	−0.393922	+	0.919144i	$0.628882\pi$
$110$	0	0
$111$	−4.39312	−0.416976
$112$	0	0
$113$	13.4966	1.26966	0.634828	−	0.772653i	$-0.281071\pi$
$113$	13.4966	1.26966	0.634828	+	0.772653i	$0.281071\pi$
$114$	0	0
$115$	−13.3717	−1.24692
$116$	0	0
$117$	3.86519	0.357337
$118$	0	0
$119$	−0.657077	−0.0602341
$120$	0	0
$121$	−10.7606	−0.978236
$122$	0	0
$123$	4.72869	0.426372
$124$	0	0
$125$	3.92839	0.351365
$126$	0	0
$127$	−1.53948	−0.136607	−0.0683034	−	0.997665i	$-0.521759\pi$
$127$	−1.53948	−0.136607	−0.0683034	+	0.997665i	$0.521759\pi$
$128$	0	0
$129$	−8.22533	−0.724200
$130$	0	0
$131$	−3.56825	−0.311759	−0.155880	−	0.987776i	$-0.549821\pi$
$131$	−3.56825	−0.311759	−0.155880	+	0.987776i	$0.549821\pi$
$132$	0	0
$133$	3.34292	0.289868
$134$	0	0
$135$	−17.9572	−1.54551
$136$	0	0
$137$	10.1537	0.867490	0.433745	−	0.901036i	$-0.357192\pi$
$137$	10.1537	0.867490	0.433745	+	0.901036i	$0.357192\pi$
$138$	0	0
$139$	−3.90383	−0.331118	−0.165559	−	0.986200i	$-0.552943\pi$
$139$	−3.90383	−0.331118	−0.165559	+	0.986200i	$0.552943\pi$
$140$	0	0
$141$	−12.6184	−1.06266
$142$	0	0
$143$	−1.12181	−0.0938102
$144$	0	0
$145$	−20.4752	−1.70037
$146$	0	0
$147$	4.78623	0.394762
$148$	0	0
$149$	−17.3576	−1.42199	−0.710996	−	0.703196i	$-0.751755\pi$
$149$	−17.3576	−1.42199	−0.710996	+	0.703196i	$0.751755\pi$
$150$	0	0
$151$	−4.05754	−0.330198	−0.165099	−	0.986277i	$-0.552794\pi$
$151$	−4.05754	−0.330198	−0.165099	+	0.986277i	$0.552794\pi$
$152$	0	0
$153$	−0.331366	−0.0267893
$154$	0	0
$155$	27.4966	2.20858
$156$	0	0
$157$	−2.35027	−0.187572	−0.0937860	−	0.995592i	$-0.529897\pi$
$157$	−2.35027	−0.187572	−0.0937860	+	0.995592i	$0.529897\pi$
$158$	0	0
$159$	5.51492	0.437362
$160$	0	0
$161$	13.3717	1.05384
$162$	0	0
$163$	13.7648	1.07814	0.539071	−	0.842260i	$-0.318775\pi$
$163$	13.7648	1.07814	0.539071	+	0.842260i	$0.318775\pi$
$164$	0	0
$165$	1.87506	0.145973
$166$	0	0
$167$	0.417674	0.0323206	0.0161603	−	0.999869i	$-0.494856\pi$
$167$	0.417674	0.0323206	0.0161603	+	0.999869i	$0.494856\pi$
$168$	0	0
$169$	−7.74338	−0.595645
$170$	0	0
$171$	1.68585	0.128920
$172$	0	0
$173$	−3.60688	−0.274226	−0.137113	−	0.990555i	$-0.543782\pi$
$173$	−3.60688	−0.274226	−0.137113	+	0.990555i	$0.543782\pi$
$174$	0	0
$175$	−20.6430	−1.56046
$176$	0	0
$177$	6.09196	0.457900
$178$	0	0
$179$	−6.51806	−0.487183	−0.243591	−	0.969878i	$-0.578326\pi$
$179$	−6.51806	−0.487183	−0.243591	+	0.969878i	$0.578326\pi$
$180$	0	0
$181$	1.31415	0.0976803	0.0488401	−	0.998807i	$-0.484448\pi$
$181$	1.31415	0.0976803	0.0488401	+	0.998807i	$0.484448\pi$
$182$	0	0
$183$	8.86833	0.655566
$184$	0	0
$185$	−12.8108	−0.941868
$186$	0	0
$187$	0.0961734	0.00703289
$188$	0	0
$189$	17.9572	1.30619
$190$	0	0
$191$	−9.24254	−0.668767	−0.334383	−	0.942437i	$-0.608528\pi$
$191$	−9.24254	−0.668767	−0.334383	+	0.942437i	$0.608528\pi$
$192$	0	0
$193$	8.85363	0.637299	0.318649	−	0.947873i	$-0.396771\pi$
$193$	8.85363	0.637299	0.318649	+	0.947873i	$0.396771\pi$
$194$	0	0
$195$	−8.78623	−0.629195
$196$	0	0
$197$	27.3288	1.94710	0.973550	−	0.228475i	$-0.0733738\pi$
$197$	27.3288	1.94710	0.973550	+	0.228475i	$0.0733738\pi$
$198$	0	0
$199$	−18.3643	−1.30181	−0.650907	−	0.759157i	$-0.725611\pi$
$199$	−18.3643	−1.30181	−0.650907	+	0.759157i	$0.725611\pi$
$200$	0	0
$201$	−17.7648	−1.25303
$202$	0	0
$203$	20.4752	1.43708
$204$	0	0
$205$	13.7894	0.963091
$206$	0	0
$207$	6.74338	0.468698
$208$	0	0
$209$	−0.489289	−0.0338448
$210$	0	0
$211$	18.3503	1.26328	0.631642	−	0.775260i	$-0.282381\pi$
$211$	18.3503	1.26328	0.631642	+	0.775260i	$0.282381\pi$
$212$	0	0
$213$	−12.9210	−0.885335
$214$	0	0
$215$	−23.9859	−1.63583
$216$	0	0
$217$	−27.4966	−1.86659
$218$	0	0
$219$	17.7318	1.19821
$220$	0	0
$221$	−0.450654	−0.0303142
$222$	0	0
$223$	18.3748	1.23047	0.615235	−	0.788344i	$-0.289061\pi$
$223$	18.3748	1.23047	0.615235	+	0.788344i	$0.289061\pi$
$224$	0	0
$225$	−10.4103	−0.694022
$226$	0	0
$227$	18.9933	1.26063	0.630314	−	0.776340i	$-0.282926\pi$
$227$	18.9933	1.26063	0.630314	+	0.776340i	$0.282926\pi$
$228$	0	0
$229$	−9.05019	−0.598054	−0.299027	−	0.954245i	$-0.596662\pi$
$229$	−9.05019	−0.598054	−0.299027	+	0.954245i	$0.596662\pi$
$230$	0	0
$231$	−1.87506	−0.123370
$232$	0	0
$233$	9.76060	0.639438	0.319719	−	0.947512i	$-0.396411\pi$
$233$	9.76060	0.639438	0.319719	+	0.947512i	$0.396411\pi$
$234$	0	0
$235$	−36.7967	−2.40035
$236$	0	0
$237$	−12.0575	−0.783221
$238$	0	0
$239$	12.9645	0.838604	0.419302	−	0.907847i	$-0.362275\pi$
$239$	12.9645	0.838604	0.419302	+	0.907847i	$0.362275\pi$
$240$	0	0
$241$	−20.3257	−1.30929	−0.654647	−	0.755935i	$-0.727183\pi$
$241$	−20.3257	−1.30929	−0.654647	+	0.755935i	$0.727183\pi$
$242$	0	0
$243$	14.8536	0.952861
$244$	0	0
$245$	13.9572	0.891690
$246$	0	0
$247$	2.29273	0.145883
$248$	0	0
$249$	−0.835347	−0.0529380
$250$	0	0
$251$	−6.05333	−0.382083	−0.191041	−	0.981582i	$-0.561186\pi$
$251$	−6.05333	−0.382083	−0.191041	+	0.981582i	$0.561186\pi$
$252$	0	0
$253$	−1.95715	−0.123045
$254$	0	0
$255$	0.753250	0.0471704
$256$	0	0
$257$	14.0575	0.876885	0.438443	−	0.898759i	$-0.355530\pi$
$257$	14.0575	0.876885	0.438443	+	0.898759i	$0.355530\pi$
$258$	0	0
$259$	12.8108	0.796024
$260$	0	0
$261$	10.3257	0.639145
$262$	0	0
$263$	−8.26396	−0.509578	−0.254789	−	0.966997i	$-0.582006\pi$
$263$	−8.26396	−0.509578	−0.254789	+	0.966997i	$0.582006\pi$
$264$	0	0
$265$	16.0821	0.987915
$266$	0	0
$267$	−9.76481	−0.597597
$268$	0	0
$269$	−3.18921	−0.194450	−0.0972248	−	0.995262i	$-0.530997\pi$
$269$	−3.18921	−0.194450	−0.0972248	+	0.995262i	$0.530997\pi$
$270$	0	0
$271$	−9.89962	−0.601359	−0.300679	−	0.953725i	$-0.597213\pi$
$271$	−9.89962	−0.601359	−0.300679	+	0.953725i	$0.597213\pi$
$272$	0	0
$273$	8.78623	0.531767
$274$	0	0
$275$	3.02142	0.182199
$276$	0	0
$277$	−12.7722	−0.767404	−0.383702	−	0.923457i	$-0.625351\pi$
$277$	−12.7722	−0.767404	−0.383702	+	0.923457i	$0.625351\pi$
$278$	0	0
$279$	−13.8666	−0.830174
$280$	0	0
$281$	29.5296	1.76159	0.880795	−	0.473499i	$-0.157009\pi$
$281$	29.5296	1.76159	0.880795	+	0.473499i	$0.157009\pi$
$282$	0	0
$283$	−25.3246	−1.50539	−0.752697	−	0.658367i	$-0.771247\pi$
$283$	−25.3246	−1.50539	−0.752697	+	0.658367i	$0.771247\pi$
$284$	0	0
$285$	−3.83221	−0.227001
$286$	0	0
$287$	−13.7894	−0.813961
$288$	0	0
$289$	−16.9614	−0.997727
$290$	0	0
$291$	−3.79923	−0.222715
$292$	0	0
$293$	16.8353	0.983531	0.491766	−	0.870728i	$-0.336352\pi$
$293$	16.8353	0.983531	0.491766	+	0.870728i	$0.336352\pi$
$294$	0	0
$295$	17.7648	1.03431
$296$	0	0
$297$	−2.62831	−0.152510
$298$	0	0
$299$	9.17092	0.530368
$300$	0	0
$301$	23.9859	1.38253
$302$	0	0
$303$	−12.2499	−0.703738
$304$	0	0
$305$	25.8610	1.48080
$306$	0	0
$307$	17.7894	1.01529	0.507646	−	0.861566i	$-0.330516\pi$
$307$	17.7894	1.01529	0.507646	+	0.861566i	$0.330516\pi$
$308$	0	0
$309$	−13.8223	−0.786326
$310$	0	0
$311$	−20.3215	−1.15233	−0.576163	−	0.817335i	$-0.695451\pi$
$311$	−20.3215	−1.15233	−0.576163	+	0.817335i	$0.695451\pi$
$312$	0	0
$313$	34.8929	1.97226	0.986131	−	0.165967i	$-0.0530746\pi$
$313$	34.8929	1.97226	0.986131	+	0.165967i	$0.0530746\pi$
$314$	0	0
$315$	18.8396	1.06149
$316$	0	0
$317$	−12.7287	−0.714915	−0.357457	−	0.933929i	$-0.616356\pi$
$317$	−12.7287	−0.714915	−0.357457	+	0.933929i	$0.616356\pi$
$318$	0	0
$319$	−2.99686	−0.167792
$320$	0	0
$321$	9.62158	0.537024
$322$	0	0
$323$	−0.196558	−0.0109368
$324$	0	0
$325$	−14.1579	−0.785340
$326$	0	0
$327$	−9.42923	−0.521438
$328$	0	0
$329$	36.7967	2.02867
$330$	0	0
$331$	−18.4605	−1.01468	−0.507341	−	0.861745i	$-0.669372\pi$
$331$	−18.4605	−1.01468	−0.507341	+	0.861745i	$0.669372\pi$
$332$	0	0
$333$	6.46052	0.354034
$334$	0	0
$335$	−51.8041	−2.83036
$336$	0	0
$337$	28.5510	1.55527	0.777637	−	0.628713i	$-0.216418\pi$
$337$	28.5510	1.55527	0.777637	+	0.628713i	$0.216418\pi$
$338$	0	0
$339$	15.4721	0.840328
$340$	0	0
$341$	4.02456	0.217942
$342$	0	0
$343$	9.44331	0.509891
$344$	0	0
$345$	−15.3288	−0.825277
$346$	0	0
$347$	30.3116	1.62721	0.813607	−	0.581415i	$-0.197501\pi$
$347$	30.3116	1.62721	0.813607	+	0.581415i	$0.197501\pi$
$348$	0	0
$349$	19.6503	1.05186	0.525929	−	0.850528i	$-0.323718\pi$
$349$	19.6503	1.05186	0.525929	+	0.850528i	$0.323718\pi$
$350$	0	0
$351$	12.3158	0.657371
$352$	0	0
$353$	1.80765	0.0962117	0.0481058	−	0.998842i	$-0.484682\pi$
$353$	1.80765	0.0962117	0.0481058	+	0.998842i	$0.484682\pi$
$354$	0	0
$355$	−37.6791	−1.99980
$356$	0	0
$357$	−0.753250	−0.0398662
$358$	0	0
$359$	23.4496	1.23762	0.618811	−	0.785540i	$-0.287615\pi$
$359$	23.4496	1.23762	0.618811	+	0.785540i	$0.287615\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−12.3356	−0.647450
$364$	0	0
$365$	51.7079	2.70651
$366$	0	0
$367$	16.2499	0.848237	0.424119	−	0.905607i	$-0.360584\pi$
$367$	16.2499	0.848237	0.424119	+	0.905607i	$0.360584\pi$
$368$	0	0
$369$	−6.95402	−0.362012
$370$	0	0
$371$	−16.0821	−0.834941
$372$	0	0
$373$	16.8353	0.871701	0.435851	−	0.900019i	$-0.356448\pi$
$373$	16.8353	0.871701	0.435851	+	0.900019i	$0.356448\pi$
$374$	0	0
$375$	4.50337	0.232553
$376$	0	0
$377$	14.0428	0.723243
$378$	0	0
$379$	2.68585	0.137963	0.0689813	−	0.997618i	$-0.478025\pi$
$379$	2.68585	0.137963	0.0689813	+	0.997618i	$0.478025\pi$
$380$	0	0
$381$	−1.76481	−0.0904138
$382$	0	0
$383$	−0.110250	−0.00563350	−0.00281675	−	0.999996i	$-0.500897\pi$
$383$	−0.110250	−0.00563350	−0.00281675	+	0.999996i	$0.500897\pi$
$384$	0	0
$385$	−5.46787	−0.278668
$386$	0	0
$387$	12.0962	0.614883
$388$	0	0
$389$	9.24254	0.468615	0.234308	−	0.972162i	$-0.424718\pi$
$389$	9.24254	0.468615	0.234308	+	0.972162i	$0.424718\pi$
$390$	0	0
$391$	−0.786230	−0.0397614
$392$	0	0
$393$	−4.09052	−0.206339
$394$	0	0
$395$	−35.1611	−1.76914
$396$	0	0
$397$	−32.9070	−1.65155	−0.825777	−	0.563997i	$-0.809263\pi$
$397$	−32.9070	−1.65155	−0.825777	+	0.563997i	$0.809263\pi$
$398$	0	0
$399$	3.83221	0.191851
$400$	0	0
$401$	−23.9817	−1.19759	−0.598795	−	0.800902i	$-0.704353\pi$
$401$	−23.9817	−1.19759	−0.598795	+	0.800902i	$0.704353\pi$
$402$	0	0
$403$	−18.8585	−0.939407
$404$	0	0
$405$	−3.67850	−0.182786
$406$	0	0
$407$	−1.87506	−0.0929431
$408$	0	0
$409$	0.182481	0.00902311	0.00451156	−	0.999990i	$-0.498564\pi$
$409$	0.182481	0.00902311	0.00451156	+	0.999990i	$0.498564\pi$
$410$	0	0
$411$	11.6399	0.574152
$412$	0	0
$413$	−17.7648	−0.874149
$414$	0	0
$415$	−2.43596	−0.119577
$416$	0	0
$417$	−4.47521	−0.219152
$418$	0	0
$419$	−13.6497	−0.666833	−0.333416	−	0.942780i	$-0.608202\pi$
$419$	−13.6497	−0.666833	−0.333416	+	0.942780i	$0.608202\pi$
$420$	0	0
$421$	14.4324	0.703390	0.351695	−	0.936115i	$-0.385605\pi$
$421$	14.4324	0.703390	0.351695	+	0.936115i	$0.385605\pi$
$422$	0	0
$423$	18.5567	0.902257
$424$	0	0
$425$	1.21377	0.0588765
$426$	0	0
$427$	−25.8610	−1.25150
$428$	0	0
$429$	−1.28600	−0.0620887
$430$	0	0
$431$	−17.1218	−0.824728	−0.412364	−	0.911019i	$-0.635297\pi$
$431$	−17.1218	−0.824728	−0.412364	+	0.911019i	$0.635297\pi$
$432$	0	0
$433$	−28.3748	−1.36361	−0.681804	−	0.731535i	$-0.738804\pi$
$433$	−28.3748	−1.36361	−0.681804	+	0.731535i	$0.738804\pi$
$434$	0	0
$435$	−23.4721	−1.12540
$436$	0	0
$437$	4.00000	0.191346
$438$	0	0
$439$	15.5823	0.743704	0.371852	−	0.928292i	$-0.378723\pi$
$439$	15.5823	0.743704	0.371852	+	0.928292i	$0.378723\pi$
$440$	0	0
$441$	−7.03863	−0.335173
$442$	0	0
$443$	−24.7476	−1.17579	−0.587897	−	0.808936i	$-0.700044\pi$
$443$	−24.7476	−1.17579	−0.587897	+	0.808936i	$0.700044\pi$
$444$	0	0
$445$	−28.4752	−1.34985
$446$	0	0
$447$	−19.8982	−0.941151
$448$	0	0
$449$	−37.7220	−1.78021	−0.890105	−	0.455756i	$-0.849369\pi$
$449$	−37.7220	−1.78021	−0.890105	+	0.455756i	$0.849369\pi$
$450$	0	0
$451$	2.01829	0.0950374
$452$	0	0
$453$	−4.65142	−0.218543
$454$	0	0
$455$	25.6216	1.20116
$456$	0	0
$457$	−21.6258	−1.01161	−0.505806	−	0.862647i	$-0.668805\pi$
$457$	−21.6258	−1.01161	−0.505806	+	0.862647i	$0.668805\pi$
$458$	0	0
$459$	−1.05585	−0.0492827
$460$	0	0
$461$	11.5353	0.537251	0.268626	−	0.963245i	$-0.413431\pi$
$461$	11.5353	0.537251	0.268626	+	0.963245i	$0.413431\pi$
$462$	0	0
$463$	−40.4282	−1.87886	−0.939428	−	0.342747i	$-0.888643\pi$
$463$	−40.4282	−1.87886	−0.939428	+	0.342747i	$0.888643\pi$
$464$	0	0
$465$	31.5212	1.46176
$466$	0	0
$467$	27.7606	1.28461	0.642304	−	0.766450i	$-0.277979\pi$
$467$	27.7606	1.28461	0.642304	+	0.766450i	$0.277979\pi$
$468$	0	0
$469$	51.8041	2.39209
$470$	0	0
$471$	−2.69427	−0.124145
$472$	0	0
$473$	−3.51071	−0.161423
$474$	0	0
$475$	−6.17513	−0.283335
$476$	0	0
$477$	−8.11025	−0.371343
$478$	0	0
$479$	10.4935	0.479460	0.239730	−	0.970840i	$-0.422941\pi$
$479$	10.4935	0.479460	0.239730	+	0.970840i	$0.422941\pi$
$480$	0	0
$481$	8.78623	0.400618
$482$	0	0
$483$	15.3288	0.697487
$484$	0	0
$485$	−11.0790	−0.503070
$486$	0	0
$487$	−18.2646	−0.827647	−0.413824	−	0.910357i	$-0.635807\pi$
$487$	−18.2646	−0.827647	−0.413824	+	0.910357i	$0.635807\pi$
$488$	0	0
$489$	15.7795	0.713574
$490$	0	0
$491$	41.6791	1.88095	0.940476	−	0.339860i	$-0.110380\pi$
$491$	41.6791	1.88095	0.940476	+	0.339860i	$0.110380\pi$
$492$	0	0
$493$	−1.20390	−0.0542211
$494$	0	0
$495$	−2.75746	−0.123939
$496$	0	0
$497$	37.6791	1.69014
$498$	0	0
$499$	13.0748	0.585306	0.292653	−	0.956219i	$-0.405462\pi$
$499$	13.0748	0.585306	0.292653	+	0.956219i	$0.405462\pi$
$500$	0	0
$501$	0.478807	0.0213915
$502$	0	0
$503$	18.4935	0.824584	0.412292	−	0.911052i	$-0.364728\pi$
$503$	18.4935	0.824584	0.412292	+	0.911052i	$0.364728\pi$
$504$	0	0
$505$	−35.7220	−1.58961
$506$	0	0
$507$	−8.87675	−0.394230
$508$	0	0
$509$	11.8652	0.525915	0.262958	−	0.964807i	$-0.415302\pi$
$509$	11.8652	0.525915	0.262958	+	0.964807i	$0.415302\pi$
$510$	0	0
$511$	−51.7079	−2.28742
$512$	0	0
$513$	5.37169	0.237166
$514$	0	0
$515$	−40.3074	−1.77616
$516$	0	0
$517$	−5.38577	−0.236866
$518$	0	0
$519$	−4.13481	−0.181498
$520$	0	0
$521$	−24.3832	−1.06825	−0.534125	−	0.845406i	$-0.679359\pi$
$521$	−24.3832	−1.06825	−0.534125	+	0.845406i	$0.679359\pi$
$522$	0	0
$523$	−38.5510	−1.68572	−0.842860	−	0.538134i	$-0.819130\pi$
$523$	−38.5510	−1.68572	−0.842860	+	0.538134i	$0.819130\pi$
$524$	0	0
$525$	−23.6644	−1.03280
$526$	0	0
$527$	1.61675	0.0704268
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−8.95885	−0.388781
$532$	0	0
$533$	−9.45738	−0.409645
$534$	0	0
$535$	28.0575	1.21303
$536$	0	0
$537$	−7.47208	−0.322444
$538$	0	0
$539$	2.04285	0.0879916
$540$	0	0
$541$	−0.857845	−0.0368817	−0.0184408	−	0.999830i	$-0.505870\pi$
$541$	−0.857845	−0.0368817	−0.0184408	+	0.999830i	$0.505870\pi$
$542$	0	0
$543$	1.50650	0.0646501
$544$	0	0
$545$	−27.4966	−1.17783
$546$	0	0
$547$	−10.8291	−0.463018	−0.231509	−	0.972833i	$-0.574366\pi$
$547$	−10.8291	−0.463018	−0.231509	+	0.972833i	$0.574366\pi$
$548$	0	0
$549$	−13.0418	−0.556609
$550$	0	0
$551$	6.12494	0.260931
$552$	0	0
$553$	35.1611	1.49520
$554$	0	0
$555$	−14.6858	−0.623379
$556$	0	0
$557$	36.7722	1.55809	0.779043	−	0.626970i	$-0.215705\pi$
$557$	36.7722	1.55809	0.779043	+	0.626970i	$0.215705\pi$
$558$	0	0
$559$	16.4507	0.695789
$560$	0	0
$561$	0.110250	0.00465475
$562$	0	0
$563$	29.8469	1.25790	0.628949	−	0.777447i	$-0.283486\pi$
$563$	29.8469	1.25790	0.628949	+	0.777447i	$0.283486\pi$
$564$	0	0
$565$	45.1182	1.89814
$566$	0	0
$567$	3.67850	0.154482
$568$	0	0
$569$	−36.2646	−1.52029	−0.760145	−	0.649753i	$-0.774872\pi$
$569$	−36.2646	−1.52029	−0.760145	+	0.649753i	$0.774872\pi$
$570$	0	0
$571$	−11.3570	−0.475276	−0.237638	−	0.971354i	$-0.576373\pi$
$571$	−11.3570	−0.475276	−0.237638	+	0.971354i	$0.576373\pi$
$572$	0	0
$573$	−10.5953	−0.442626
$574$	0	0
$575$	−24.7005	−1.03008
$576$	0	0
$577$	18.2113	0.758144	0.379072	−	0.925367i	$-0.376243\pi$
$577$	18.2113	0.758144	0.379072	+	0.925367i	$0.376243\pi$
$578$	0	0
$579$	10.1495	0.421799
$580$	0	0
$581$	2.43596	0.101061
$582$	0	0
$583$	2.35386	0.0974871
$584$	0	0
$585$	12.9210	0.534219
$586$	0	0
$587$	−26.4464	−1.09156	−0.545781	−	0.837928i	$-0.683767\pi$
$587$	−26.4464	−1.09156	−0.545781	+	0.837928i	$0.683767\pi$
$588$	0	0
$589$	−8.22533	−0.338919
$590$	0	0
$591$	31.3288	1.28870
$592$	0	0
$593$	−26.8929	−1.10436	−0.552179	−	0.833725i	$-0.686204\pi$
$593$	−26.8929	−1.10436	−0.552179	+	0.833725i	$0.686204\pi$
$594$	0	0
$595$	−2.19656	−0.0900501
$596$	0	0
$597$	−21.0523	−0.861611
$598$	0	0
$599$	−34.5181	−1.41037	−0.705185	−	0.709024i	$-0.749136\pi$
$599$	−34.5181	−1.41037	−0.705185	+	0.709024i	$0.749136\pi$
$600$	0	0
$601$	−27.8715	−1.13690	−0.568450	−	0.822718i	$-0.692457\pi$
$601$	−27.8715	−1.13690	−0.568450	+	0.822718i	$0.692457\pi$
$602$	0	0
$603$	26.1249	1.06389
$604$	0	0
$605$	−35.9718	−1.46246
$606$	0	0
$607$	−27.5787	−1.11939	−0.559693	−	0.828700i	$-0.689081\pi$
$607$	−27.5787	−1.11939	−0.559693	+	0.828700i	$0.689081\pi$
$608$	0	0
$609$	23.4721	0.951137
$610$	0	0
$611$	25.2369	1.02098
$612$	0	0
$613$	−2.67177	−0.107912	−0.0539559	−	0.998543i	$-0.517183\pi$
$613$	−2.67177	−0.107912	−0.0539559	+	0.998543i	$0.517183\pi$
$614$	0	0
$615$	15.8077	0.637426
$616$	0	0
$617$	15.0832	0.607226	0.303613	−	0.952795i	$-0.401807\pi$
$617$	15.0832	0.607226	0.303613	+	0.952795i	$0.401807\pi$
$618$	0	0
$619$	−31.7220	−1.27501	−0.637507	−	0.770445i	$-0.720034\pi$
$619$	−31.7220	−1.27501	−0.637507	+	0.770445i	$0.720034\pi$
$620$	0	0
$621$	21.4868	0.862234
$622$	0	0
$623$	28.4752	1.14084
$624$	0	0
$625$	−17.7434	−0.709735
$626$	0	0
$627$	−0.560904	−0.0224003
$628$	0	0
$629$	−0.753250	−0.0300341
$630$	0	0
$631$	47.1800	1.87820	0.939102	−	0.343638i	$-0.111659\pi$
$631$	47.1800	1.87820	0.939102	+	0.343638i	$0.111659\pi$
$632$	0	0
$633$	21.0361	0.836111
$634$	0	0
$635$	−5.14637	−0.204227
$636$	0	0
$637$	−9.57246	−0.379275
$638$	0	0
$639$	19.0017	0.751695
$640$	0	0
$641$	34.1067	1.34713	0.673566	−	0.739127i	$-0.264762\pi$
$641$	34.1067	1.34713	0.673566	+	0.739127i	$0.264762\pi$
$642$	0	0
$643$	21.6686	0.854528	0.427264	−	0.904127i	$-0.359478\pi$
$643$	21.6686	0.854528	0.427264	+	0.904127i	$0.359478\pi$
$644$	0	0
$645$	−27.4966	−1.08268
$646$	0	0
$647$	−35.8427	−1.40912	−0.704561	−	0.709644i	$-0.748856\pi$
$647$	−35.8427	−1.40912	−0.704561	+	0.709644i	$0.748856\pi$
$648$	0	0
$649$	2.60015	0.102065
$650$	0	0
$651$	−31.5212	−1.23541
$652$	0	0
$653$	−21.7507	−0.851172	−0.425586	−	0.904918i	$-0.639932\pi$
$653$	−21.7507	−0.851172	−0.425586	+	0.904918i	$0.639932\pi$
$654$	0	0
$655$	−11.9284	−0.466081
$656$	0	0
$657$	−26.0764	−1.01734
$658$	0	0
$659$	10.3012	0.401276	0.200638	−	0.979665i	$-0.435699\pi$
$659$	10.3012	0.401276	0.200638	+	0.979665i	$0.435699\pi$
$660$	0	0
$661$	−1.95715	−0.0761245	−0.0380622	−	0.999275i	$-0.512119\pi$
$661$	−1.95715	−0.0761245	−0.0380622	+	0.999275i	$0.512119\pi$
$662$	0	0
$663$	−0.516614	−0.0200636
$664$	0	0
$665$	11.1751	0.433353
$666$	0	0
$667$	24.4998	0.948635
$668$	0	0
$669$	21.0643	0.814392
$670$	0	0
$671$	3.78516	0.146124
$672$	0	0
$673$	−0.0428457	−0.00165158	−0.000825790	−	1.00000i	$-0.500263\pi$
$673$	−0.0428457	−0.00165158	−0.000825790		1.00000i	$0.500263\pi$
$674$	0	0
$675$	−33.1709	−1.27675
$676$	0	0
$677$	−42.1543	−1.62012	−0.810061	−	0.586345i	$-0.800566\pi$
$677$	−42.1543	−1.62012	−0.810061	+	0.586345i	$0.800566\pi$
$678$	0	0
$679$	11.0790	0.425172
$680$	0	0
$681$	21.7732	0.834352
$682$	0	0
$683$	−28.8353	−1.10335	−0.551677	−	0.834058i	$-0.686012\pi$
$683$	−28.8353	−1.10335	−0.551677	+	0.834058i	$0.686012\pi$
$684$	0	0
$685$	33.9431	1.29690
$686$	0	0
$687$	−10.3748	−0.395824
$688$	0	0
$689$	−11.0298	−0.420204
$690$	0	0
$691$	21.4103	0.814487	0.407244	−	0.913320i	$-0.366490\pi$
$691$	21.4103	0.814487	0.407244	+	0.913320i	$0.366490\pi$
$692$	0	0
$693$	2.75746	0.104747
$694$	0	0
$695$	−13.0502	−0.495022
$696$	0	0
$697$	0.810789	0.0307108
$698$	0	0
$699$	11.1892	0.423215
$700$	0	0
$701$	3.07054	0.115973	0.0579863	−	0.998317i	$-0.481532\pi$
$701$	3.07054	0.115973	0.0579863	+	0.998317i	$0.481532\pi$
$702$	0	0
$703$	3.83221	0.144535
$704$	0	0
$705$	−42.1825	−1.58868
$706$	0	0
$707$	35.7220	1.34346
$708$	0	0
$709$	−42.8500	−1.60927	−0.804634	−	0.593772i	$-0.797638\pi$
$709$	−42.8500	−1.60927	−0.804634	+	0.593772i	$0.797638\pi$
$710$	0	0
$711$	17.7318	0.664995
$712$	0	0
$713$	−32.9013	−1.23216
$714$	0	0
$715$	−3.75011	−0.140246
$716$	0	0
$717$	14.8621	0.555034
$718$	0	0
$719$	34.5285	1.28770	0.643849	−	0.765153i	$-0.277337\pi$
$719$	34.5285	1.28770	0.643849	+	0.765153i	$0.277337\pi$
$720$	0	0
$721$	40.3074	1.50113
$722$	0	0
$723$	−23.3007	−0.866562
$724$	0	0
$725$	−37.8223	−1.40469
$726$	0	0
$727$	44.0350	1.63317	0.816585	−	0.577226i	$-0.195865\pi$
$727$	44.0350	1.63317	0.816585	+	0.577226i	$0.195865\pi$
$728$	0	0
$729$	20.3288	0.752920
$730$	0	0
$731$	−1.41033	−0.0521628
$732$	0	0
$733$	35.4721	1.31019	0.655096	−	0.755546i	$-0.272628\pi$
$733$	35.4721	1.31019	0.655096	+	0.755546i	$0.272628\pi$
$734$	0	0
$735$	16.0000	0.590169
$736$	0	0
$737$	−7.58233	−0.279299
$738$	0	0
$739$	34.4956	1.26894	0.634470	−	0.772948i	$-0.281218\pi$
$739$	34.4956	1.26894	0.634470	+	0.772948i	$0.281218\pi$
$740$	0	0
$741$	2.62831	0.0965533
$742$	0	0
$743$	−2.40298	−0.0881568	−0.0440784	−	0.999028i	$-0.514035\pi$
$743$	−2.40298	−0.0881568	−0.0440784	+	0.999028i	$0.514035\pi$
$744$	0	0
$745$	−58.0252	−2.12588
$746$	0	0
$747$	1.22846	0.0449471
$748$	0	0
$749$	−28.0575	−1.02520
$750$	0	0
$751$	46.3221	1.69032	0.845159	−	0.534515i	$-0.179506\pi$
$751$	46.3221	1.69032	0.845159	+	0.534515i	$0.179506\pi$
$752$	0	0
$753$	−6.93933	−0.252883
$754$	0	0
$755$	−13.5640	−0.493646
$756$	0	0
$757$	28.5798	1.03875	0.519375	−	0.854546i	$-0.326165\pi$
$757$	28.5798	1.03875	0.519375	+	0.854546i	$0.326165\pi$
$758$	0	0
$759$	−2.24361	−0.0814380
$760$	0	0
$761$	21.5401	0.780828	0.390414	−	0.920639i	$-0.372332\pi$
$761$	21.5401	0.780828	0.390414	+	0.920639i	$0.372332\pi$
$762$	0	0
$763$	27.4966	0.995445
$764$	0	0
$765$	−1.10773	−0.0400501
$766$	0	0
$767$	−12.1839	−0.439936
$768$	0	0
$769$	16.6472	0.600314	0.300157	−	0.953890i	$-0.402961\pi$
$769$	16.6472	0.600314	0.300157	+	0.953890i	$0.402961\pi$
$770$	0	0
$771$	16.1151	0.580370
$772$	0	0
$773$	38.8009	1.39557	0.697786	−	0.716306i	$-0.254169\pi$
$773$	38.8009	1.39557	0.697786	+	0.716306i	$0.254169\pi$
$774$	0	0
$775$	50.7925	1.82452
$776$	0	0
$777$	14.6858	0.526852
$778$	0	0
$779$	−4.12494	−0.147791
$780$	0	0
$781$	−5.51492	−0.197339
$782$	0	0
$783$	32.9013	1.17580
$784$	0	0
$785$	−7.85677	−0.280420
$786$	0	0
$787$	23.8652	0.850702	0.425351	−	0.905028i	$-0.360151\pi$
$787$	23.8652	0.850702	0.425351	+	0.905028i	$0.360151\pi$
$788$	0	0
$789$	−9.47352	−0.337266
$790$	0	0
$791$	−45.1182	−1.60422
$792$	0	0
$793$	−17.7367	−0.629847
$794$	0	0
$795$	18.4360	0.653856
$796$	0	0
$797$	2.01829	0.0714914	0.0357457	−	0.999361i	$-0.488619\pi$
$797$	2.01829	0.0714914	0.0357457	+	0.999361i	$0.488619\pi$
$798$	0	0
$799$	−2.16358	−0.0765419
$800$	0	0
$801$	14.3601	0.507390
$802$	0	0
$803$	7.56825	0.267078
$804$	0	0
$805$	44.7005	1.57549
$806$	0	0
$807$	−3.65600	−0.128697
$808$	0	0
$809$	−52.0533	−1.83010	−0.915049	−	0.403343i	$-0.867848\pi$
$809$	−52.0533	−1.83010	−0.915049	+	0.403343i	$0.867848\pi$
$810$	0	0
$811$	−34.1579	−1.19945	−0.599723	−	0.800207i	$-0.704723\pi$
$811$	−34.1579	−1.19945	−0.599723	+	0.800207i	$0.704723\pi$
$812$	0	0
$813$	−11.3486	−0.398012
$814$	0	0
$815$	46.0147	1.61182
$816$	0	0
$817$	7.17513	0.251026
$818$	0	0
$819$	−12.9210	−0.451498
$820$	0	0
$821$	−30.2787	−1.05673	−0.528366	−	0.849017i	$-0.677195\pi$
$821$	−30.2787	−1.05673	−0.528366	+	0.849017i	$0.677195\pi$
$822$	0	0
$823$	39.9284	1.39182	0.695908	−	0.718131i	$-0.255002\pi$
$823$	39.9284	1.39182	0.695908	+	0.718131i	$0.255002\pi$
$824$	0	0
$825$	3.46365	0.120589
$826$	0	0
$827$	12.0330	0.418428	0.209214	−	0.977870i	$-0.432910\pi$
$827$	12.0330	0.418428	0.209214	+	0.977870i	$0.432910\pi$
$828$	0	0
$829$	−9.84208	−0.341829	−0.170915	−	0.985286i	$-0.554672\pi$
$829$	−9.84208	−0.341829	−0.170915	+	0.985286i	$0.554672\pi$
$830$	0	0
$831$	−14.6416	−0.507910
$832$	0	0
$833$	0.820654	0.0284340
$834$	0	0
$835$	1.39625	0.0483192
$836$	0	0
$837$	−44.1839	−1.52722
$838$	0	0
$839$	−19.2713	−0.665319	−0.332660	−	0.943047i	$-0.607946\pi$
$839$	−19.2713	−0.665319	−0.332660	+	0.943047i	$0.607946\pi$
$840$	0	0
$841$	8.51492	0.293618
$842$	0	0
$843$	33.8517	1.16592
$844$	0	0
$845$	−25.8855	−0.890490
$846$	0	0
$847$	35.9718	1.23601
$848$	0	0
$849$	−29.0313	−0.996351
$850$	0	0
$851$	15.3288	0.525466
$852$	0	0
$853$	−27.4721	−0.940626	−0.470313	−	0.882500i	$-0.655859\pi$
$853$	−27.4721	−0.940626	−0.470313	+	0.882500i	$0.655859\pi$
$854$	0	0
$855$	5.63565	0.192735
$856$	0	0
$857$	0.292731	0.00999950	0.00499975	−	0.999988i	$-0.498409\pi$
$857$	0.292731	0.00999950	0.00499975	+	0.999988i	$0.498409\pi$
$858$	0	0
$859$	23.9614	0.817551	0.408776	−	0.912635i	$-0.365956\pi$
$859$	23.9614	0.817551	0.408776	+	0.912635i	$0.365956\pi$
$860$	0	0
$861$	−15.8077	−0.538723
$862$	0	0
$863$	6.12494	0.208495	0.104248	−	0.994551i	$-0.466757\pi$
$863$	6.12494	0.208495	0.104248	+	0.994551i	$0.466757\pi$
$864$	0	0
$865$	−12.0575	−0.409969
$866$	0	0
$867$	−19.4439	−0.660350
$868$	0	0
$869$	−5.14637	−0.174578
$870$	0	0
$871$	35.5296	1.20388
$872$	0	0
$873$	5.58715	0.189096
$874$	0	0
$875$	−13.1323	−0.443952
$876$	0	0
$877$	−22.3503	−0.754715	−0.377357	−	0.926068i	$-0.623167\pi$
$877$	−22.3503	−0.754715	−0.377357	+	0.926068i	$0.623167\pi$
$878$	0	0
$879$	19.2995	0.650955
$880$	0	0
$881$	5.70306	0.192141	0.0960705	−	0.995375i	$-0.469373\pi$
$881$	5.70306	0.192141	0.0960705	+	0.995375i	$0.469373\pi$
$882$	0	0
$883$	8.29694	0.279214	0.139607	−	0.990207i	$-0.455416\pi$
$883$	8.29694	0.279214	0.139607	+	0.990207i	$0.455416\pi$
$884$	0	0
$885$	20.3650	0.684561
$886$	0	0
$887$	−25.4783	−0.855479	−0.427740	−	0.903902i	$-0.640690\pi$
$887$	−25.4783	−0.855479	−0.427740	+	0.903902i	$0.640690\pi$
$888$	0	0
$889$	5.14637	0.172604
$890$	0	0
$891$	−0.538405	−0.0180373
$892$	0	0
$893$	11.0073	0.368347
$894$	0	0
$895$	−21.7894	−0.728338
$896$	0	0
$897$	10.5132	0.351027
$898$	0	0
$899$	−50.3797	−1.68026
$900$	0	0
$901$	0.945597	0.0315024
$902$	0	0
$903$	27.4966	0.915031
$904$	0	0
$905$	4.39312	0.146032
$906$	0	0
$907$	39.9964	1.32806	0.664029	−	0.747706i	$-0.268845\pi$
$907$	39.9964	1.32806	0.664029	+	0.747706i	$0.268845\pi$
$908$	0	0
$909$	18.0147	0.597510
$910$	0	0
$911$	19.3864	0.642300	0.321150	−	0.947028i	$-0.395931\pi$
$911$	19.3864	0.642300	0.321150	+	0.947028i	$0.395931\pi$
$912$	0	0
$913$	−0.356541	−0.0117998
$914$	0	0
$915$	29.6461	0.980071
$916$	0	0
$917$	11.9284	0.393910
$918$	0	0
$919$	−38.1213	−1.25751	−0.628754	−	0.777605i	$-0.716435\pi$
$919$	−38.1213	−1.25751	−0.628754	+	0.777605i	$0.716435\pi$
$920$	0	0
$921$	20.3931	0.671976
$922$	0	0
$923$	25.8421	0.850602
$924$	0	0
$925$	−23.6644	−0.778081
$926$	0	0
$927$	20.3272	0.667631
$928$	0	0
$929$	−16.8500	−0.552832	−0.276416	−	0.961038i	$-0.589147\pi$
$929$	−16.8500	−0.552832	−0.276416	+	0.961038i	$0.589147\pi$
$930$	0	0
$931$	−4.17513	−0.136835
$932$	0	0
$933$	−23.2959	−0.762672
$934$	0	0
$935$	0.321500	0.0105142
$936$	0	0
$937$	−39.4973	−1.29032	−0.645159	−	0.764048i	$-0.723209\pi$
$937$	−39.4973	−1.29032	−0.645159	+	0.764048i	$0.723209\pi$
$938$	0	0
$939$	40.0000	1.30535
$940$	0	0
$941$	55.9718	1.82463	0.912315	−	0.409489i	$-0.134293\pi$
$941$	55.9718	1.82463	0.912315	+	0.409489i	$0.134293\pi$
$942$	0	0
$943$	−16.4998	−0.537306
$944$	0	0
$945$	60.0294	1.95276
$946$	0	0
$947$	13.5725	0.441046	0.220523	−	0.975382i	$-0.429224\pi$
$947$	13.5725	0.441046	0.220523	+	0.975382i	$0.429224\pi$
$948$	0	0
$949$	−35.4637	−1.15120
$950$	0	0
$951$	−14.5917	−0.473169
$952$	0	0
$953$	26.0246	0.843018	0.421509	−	0.906824i	$-0.361501\pi$
$953$	26.0246	0.843018	0.421509	+	0.906824i	$0.361501\pi$
$954$	0	0
$955$	−30.8971	−0.999807
$956$	0	0
$957$	−3.43550	−0.111054
$958$	0	0
$959$	−33.9431	−1.09608
$960$	0	0
$961$	36.6560	1.18245
$962$	0	0
$963$	−14.1495	−0.455961
$964$	0	0
$965$	29.5970	0.952762
$966$	0	0
$967$	13.0277	0.418942	0.209471	−	0.977815i	$-0.432826\pi$
$967$	13.0277	0.418942	0.209471	+	0.977815i	$0.432826\pi$
$968$	0	0
$969$	−0.225327	−0.00723854
$970$	0	0
$971$	−60.6148	−1.94522	−0.972612	−	0.232437i	$-0.925330\pi$
$971$	−60.6148	−1.94522	−0.972612	+	0.232437i	$0.925330\pi$
$972$	0	0
$973$	13.0502	0.418370
$974$	0	0
$975$	−16.2302	−0.519781
$976$	0	0
$977$	−62.1396	−1.98802	−0.994012	−	0.109275i	$-0.965147\pi$
$977$	−62.1396	−1.98802	−0.994012	+	0.109275i	$0.965147\pi$
$978$	0	0
$979$	−4.16779	−0.133203
$980$	0	0
$981$	13.8666	0.442728
$982$	0	0
$983$	−6.37842	−0.203440	−0.101720	−	0.994813i	$-0.532435\pi$
$983$	−6.37842	−0.203440	−0.101720	+	0.994813i	$0.532435\pi$
$984$	0	0
$985$	91.3582	2.91092
$986$	0	0
$987$	42.1825	1.34268
$988$	0	0
$989$	28.7005	0.912624
$990$	0	0
$991$	14.1249	0.448694	0.224347	−	0.974509i	$-0.427975\pi$
$991$	14.1249	0.448694	0.224347	+	0.974509i	$0.427975\pi$
$992$	0	0
$993$	−21.1625	−0.671572
$994$	0	0
$995$	−61.3906	−1.94621
$996$	0	0
$997$	−13.1568	−0.416682	−0.208341	−	0.978056i	$-0.566806\pi$
$997$	−13.1568	−0.416682	−0.208341	+	0.978056i	$0.566806\pi$
$998$	0	0
$999$	20.5855	0.651295

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4864.2.a.bh.1.2		3
4.3	odd	2		4864.2.a.bb.1.2		3
8.3	odd	2		4864.2.a.bg.1.2		3
8.5	even	2		4864.2.a.ba.1.2		3
16.3	odd	4		2432.2.c.e.1217.3	✓	6
16.5	even	4		2432.2.c.h.1217.3	yes	6
16.11	odd	4		2432.2.c.e.1217.4	yes	6
16.13	even	4		2432.2.c.h.1217.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.e.1217.3	✓	6	16.3	odd	4
2432.2.c.e.1217.4	yes	6	16.11	odd	4
2432.2.c.h.1217.3	yes	6	16.5	even	4
2432.2.c.h.1217.4	yes	6	16.13	even	4
4864.2.a.ba.1.2		3	8.5	even	2
4864.2.a.bb.1.2		3	4.3	odd	2
4864.2.a.bg.1.2		3	8.3	odd	2
4864.2.a.bh.1.2		3	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)

\(f(q)\)	\(=\)	\(q+1.14637 q^{3} +3.34292 q^{5} -3.34292 q^{7} -1.68585 q^{9} +O(q^{10})\)	\(q+1.14637 q^{3} +3.34292 q^{5} -3.34292 q^{7} -1.68585 q^{9} +0.489289 q^{11} -2.29273 q^{13} +3.83221 q^{15} +0.196558 q^{17} -1.00000 q^{19} -3.83221 q^{21} -4.00000 q^{23} +6.17513 q^{25} -5.37169 q^{27} -6.12494 q^{29} +8.22533 q^{31} +0.560904 q^{33} -11.1751 q^{35} -3.83221 q^{37} -2.62831 q^{39} +4.12494 q^{41} -7.17513 q^{43} -5.63565 q^{45} -11.0073 q^{47} +4.17513 q^{49} +0.225327 q^{51} +4.81079 q^{53} +1.63565 q^{55} -1.14637 q^{57} +5.31415 q^{59} +7.73604 q^{61} +5.63565 q^{63} -7.66442 q^{65} -15.4966 q^{67} -4.58546 q^{69} -11.2713 q^{71} +15.4679 q^{73} +7.07896 q^{75} -1.63565 q^{77} -10.5181 q^{79} -1.10038 q^{81} -0.728692 q^{83} +0.657077 q^{85} -7.02142 q^{87} -8.51806 q^{89} +7.66442 q^{91} +9.42923 q^{93} -3.34292 q^{95} -3.31415 q^{97} -0.824865 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} + 4 q^{5} - 4 q^{7} + 7 q^{9}+O(q^{10}) \) 3 * q + 2 * q^3 + 4 * q^5 - 4 * q^7 + 7 * q^9	\( 3 q + 2 q^{3} + 4 q^{5} - 4 q^{7} + 7 q^{9} - 6 q^{11} - 4 q^{13} - 2 q^{15} - 4 q^{17} - 3 q^{19} + 2 q^{21} - 12 q^{23} - q^{25} + 8 q^{27} - 2 q^{29} + 2 q^{31} + 6 q^{33} - 14 q^{35} + 2 q^{37} - 32 q^{39} - 4 q^{41} - 2 q^{43} - 8 q^{45} - 7 q^{49} - 22 q^{51} - 14 q^{53} - 4 q^{55} - 2 q^{57} + 28 q^{59} + 8 q^{61} + 8 q^{63} + 4 q^{65} - 6 q^{67} - 8 q^{69} - 16 q^{71} + 24 q^{73} + 4 q^{77} - 6 q^{79} + 3 q^{81} - 20 q^{83} + 8 q^{85} - 36 q^{87} - 4 q^{91} - 32 q^{93} - 4 q^{95} - 22 q^{97} - 22 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 + 4 * q^5 - 4 * q^7 + 7 * q^9 - 6 * q^11 - 4 * q^13 - 2 * q^15 - 4 * q^17 - 3 * q^19 + 2 * q^21 - 12 * q^23 - q^25 + 8 * q^27 - 2 * q^29 + 2 * q^31 + 6 * q^33 - 14 * q^35 + 2 * q^37 - 32 * q^39 - 4 * q^41 - 2 * q^43 - 8 * q^45 - 7 * q^49 - 22 * q^51 - 14 * q^53 - 4 * q^55 - 2 * q^57 + 28 * q^59 + 8 * q^61 + 8 * q^63 + 4 * q^65 - 6 * q^67 - 8 * q^69 - 16 * q^71 + 24 * q^73 + 4 * q^77 - 6 * q^79 + 3 * q^81 - 20 * q^83 + 8 * q^85 - 36 * q^87 - 4 * q^91 - 32 * q^93 - 4 * q^95 - 22 * q^97 - 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more