LMFDB - Embedded newform 8048.2.a.p.1.2

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 9x^{8} + 14x^{7} + 27x^{6} - 27x^{5} - 34x^{4} + 14x^{3} + 17x^{2} + x - 1 $ x^10 - 2x^9 - 9x^8 + 14x^7 + 27x^6 - 27x^5 - 34x^4 + 14x^3 + 17x^2 + x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 503)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.95007$ of defining polynomial
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)

$f(q)$	$=$	$q-0.950069 q^{3} -2.28693 q^{5} -2.71022 q^{7} -2.09737 q^{9} +O(q^{10})$	$q-0.950069 q^{3} -2.28693 q^{5} -2.71022 q^{7} -2.09737 q^{9} -1.36880 q^{11} -2.93079 q^{13} +2.17275 q^{15} -2.61287 q^{17} +7.79104 q^{19} +2.57490 q^{21} +2.61064 q^{23} +0.230071 q^{25} +4.84285 q^{27} -0.314480 q^{29} +7.95126 q^{31} +1.30046 q^{33} +6.19810 q^{35} -4.17299 q^{37} +2.78446 q^{39} +6.16482 q^{41} -0.457851 q^{43} +4.79655 q^{45} -7.67118 q^{47} +0.345296 q^{49} +2.48241 q^{51} +7.26306 q^{53} +3.13037 q^{55} -7.40203 q^{57} -0.217166 q^{59} +7.26694 q^{61} +5.68433 q^{63} +6.70253 q^{65} -10.1022 q^{67} -2.48029 q^{69} +16.5088 q^{71} +2.86838 q^{73} -0.218584 q^{75} +3.70976 q^{77} +16.0222 q^{79} +1.69106 q^{81} -12.5312 q^{83} +5.97547 q^{85} +0.298777 q^{87} -0.0952234 q^{89} +7.94310 q^{91} -7.55424 q^{93} -17.8176 q^{95} -9.27640 q^{97} +2.87089 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 8 q^{3} - q^{5} + 5 q^{7} - 2 q^{9}+O(q^{10}) $ 10 * q + 8 * q^3 - q^5 + 5 * q^7 - 2 * q^9	$ 10 q + 8 q^{3} - q^{5} + 5 q^{7} - 2 q^{9} + 3 q^{11} - 18 q^{13} + 2 q^{15} - 11 q^{17} + q^{21} + 2 q^{23} - 27 q^{25} + 2 q^{27} - 9 q^{29} + 22 q^{31} - 10 q^{33} + 6 q^{35} - 35 q^{37} - 8 q^{39} - 4 q^{41} + 20 q^{43} + 2 q^{45} - 7 q^{47} - 27 q^{49} - 9 q^{51} - 24 q^{53} + 11 q^{55} - 23 q^{57} - 17 q^{59} - 4 q^{61} - 10 q^{63} - 16 q^{65} + 6 q^{67} - 2 q^{69} + q^{71} - 31 q^{73} - 30 q^{75} + 3 q^{77} + 10 q^{79} - 6 q^{81} - 22 q^{83} - 6 q^{85} - 25 q^{87} + q^{89} - 10 q^{91} - 6 q^{93} - 39 q^{95} - 57 q^{97} - 35 q^{99}+O(q^{100}) $ 10 * q + 8 * q^3 - q^5 + 5 * q^7 - 2 * q^9 + 3 * q^11 - 18 * q^13 + 2 * q^15 - 11 * q^17 + q^21 + 2 * q^23 - 27 * q^25 + 2 * q^27 - 9 * q^29 + 22 * q^31 - 10 * q^33 + 6 * q^35 - 35 * q^37 - 8 * q^39 - 4 * q^41 + 20 * q^43 + 2 * q^45 - 7 * q^47 - 27 * q^49 - 9 * q^51 - 24 * q^53 + 11 * q^55 - 23 * q^57 - 17 * q^59 - 4 * q^61 - 10 * q^63 - 16 * q^65 + 6 * q^67 - 2 * q^69 + q^71 - 31 * q^73 - 30 * q^75 + 3 * q^77 + 10 * q^79 - 6 * q^81 - 22 * q^83 - 6 * q^85 - 25 * q^87 + q^89 - 10 * q^91 - 6 * q^93 - 39 * q^95 - 57 * q^97 - 35 * 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
$3$	−0.950069	−0.548523	−0.274261	−	0.961655i	$-0.588433\pi$
$3$	−0.950069	−0.548523	−0.274261	+	0.961655i	$0.588433\pi$
$4$	0	0
$5$	−2.28693	−1.02275	−0.511374	−	0.859358i	$-0.670863\pi$
$5$	−2.28693	−1.02275	−0.511374	+	0.859358i	$0.670863\pi$
$6$	0	0
$7$	−2.71022	−1.02437	−0.512184	−	0.858876i	$-0.671163\pi$
$7$	−2.71022	−1.02437	−0.512184	+	0.858876i	$0.671163\pi$
$8$	0	0
$9$	−2.09737	−0.699123
$10$	0	0
$11$	−1.36880	−0.412710	−0.206355	−	0.978477i	$-0.566160\pi$
$11$	−1.36880	−0.412710	−0.206355	+	0.978477i	$0.566160\pi$
$12$	0	0
$13$	−2.93079	−0.812856	−0.406428	−	0.913683i	$-0.633226\pi$
$13$	−2.93079	−0.812856	−0.406428	+	0.913683i	$0.633226\pi$
$14$	0	0
$15$	2.17275	0.561001
$16$	0	0
$17$	−2.61287	−0.633715	−0.316857	−	0.948473i	$-0.602628\pi$
$17$	−2.61287	−0.633715	−0.316857	+	0.948473i	$0.602628\pi$
$18$	0	0
$19$	7.79104	1.78739	0.893694	−	0.448677i	$-0.148105\pi$
$19$	7.79104	1.78739	0.893694	+	0.448677i	$0.148105\pi$
$20$	0	0
$21$	2.57490	0.561888
$22$	0	0
$23$	2.61064	0.544356	0.272178	−	0.962247i	$-0.412256\pi$
$23$	2.61064	0.544356	0.272178	+	0.962247i	$0.412256\pi$
$24$	0	0
$25$	0.230071	0.0460143
$26$	0	0
$27$	4.84285	0.932007
$28$	0	0
$29$	−0.314480	−0.0583974	−0.0291987	−	0.999574i	$-0.509296\pi$
$29$	−0.314480	−0.0583974	−0.0291987	+	0.999574i	$0.509296\pi$
$30$	0	0
$31$	7.95126	1.42809	0.714044	−	0.700101i	$-0.246862\pi$
$31$	7.95126	1.42809	0.714044	+	0.700101i	$0.246862\pi$
$32$	0	0
$33$	1.30046	0.226381
$34$	0	0
$35$	6.19810	1.04767
$36$	0	0
$37$	−4.17299	−0.686035	−0.343017	−	0.939329i	$-0.611449\pi$
$37$	−4.17299	−0.686035	−0.343017	+	0.939329i	$0.611449\pi$
$38$	0	0
$39$	2.78446	0.445870
$40$	0	0
$41$	6.16482	0.962783	0.481391	−	0.876506i	$-0.340132\pi$
$41$	6.16482	0.962783	0.481391	+	0.876506i	$0.340132\pi$
$42$	0	0
$43$	−0.457851	−0.0698216	−0.0349108	−	0.999390i	$-0.511115\pi$
$43$	−0.457851	−0.0698216	−0.0349108	+	0.999390i	$0.511115\pi$
$44$	0	0
$45$	4.79655	0.715027
$46$	0	0
$47$	−7.67118	−1.11896	−0.559478	−	0.828845i	$-0.688998\pi$
$47$	−7.67118	−1.11896	−0.559478	+	0.828845i	$0.688998\pi$
$48$	0	0
$49$	0.345296	0.0493279
$50$	0	0
$51$	2.48241	0.347607
$52$	0	0
$53$	7.26306	0.997658	0.498829	−	0.866700i	$-0.333763\pi$
$53$	7.26306	0.997658	0.498829	+	0.866700i	$0.333763\pi$
$54$	0	0
$55$	3.13037	0.422099
$56$	0	0
$57$	−7.40203	−0.980423
$58$	0	0
$59$	−0.217166	−0.0282726	−0.0141363	−	0.999900i	$-0.504500\pi$
$59$	−0.217166	−0.0282726	−0.0141363	+	0.999900i	$0.504500\pi$
$60$	0	0
$61$	7.26694	0.930436	0.465218	−	0.885196i	$-0.345976\pi$
$61$	7.26694	0.930436	0.465218	+	0.885196i	$0.345976\pi$
$62$	0	0
$63$	5.68433	0.716159
$64$	0	0
$65$	6.70253	0.831347
$66$	0	0
$67$	−10.1022	−1.23418	−0.617091	−	0.786892i	$-0.711689\pi$
$67$	−10.1022	−1.23418	−0.617091	+	0.786892i	$0.711689\pi$
$68$	0	0
$69$	−2.48029	−0.298592
$70$	0	0
$71$	16.5088	1.95924	0.979619	−	0.200863i	$-0.0643746\pi$
$71$	16.5088	1.95924	0.979619	+	0.200863i	$0.0643746\pi$
$72$	0	0
$73$	2.86838	0.335718	0.167859	−	0.985811i	$-0.446315\pi$
$73$	2.86838	0.335718	0.167859	+	0.985811i	$0.446315\pi$
$74$	0	0
$75$	−0.218584	−0.0252399
$76$	0	0
$77$	3.70976	0.422767
$78$	0	0
$79$	16.0222	1.80264	0.901320	−	0.433153i	$-0.142599\pi$
$79$	16.0222	1.80264	0.901320	+	0.433153i	$0.142599\pi$
$80$	0	0
$81$	1.69106	0.187896
$82$	0	0
$83$	−12.5312	−1.37548	−0.687741	−	0.725956i	$-0.741398\pi$
$83$	−12.5312	−1.37548	−0.687741	+	0.725956i	$0.741398\pi$
$84$	0	0
$85$	5.97547	0.648131
$86$	0	0
$87$	0.298777	0.0320323
$88$	0	0
$89$	−0.0952234	−0.0100937	−0.00504683	−	0.999987i	$-0.501606\pi$
$89$	−0.0952234	−0.0100937	−0.00504683	+	0.999987i	$0.501606\pi$
$90$	0	0
$91$	7.94310	0.832663
$92$	0	0
$93$	−7.55424	−0.783339
$94$	0	0
$95$	−17.8176	−1.82805
$96$	0	0
$97$	−9.27640	−0.941875	−0.470938	−	0.882167i	$-0.656084\pi$
$97$	−9.27640	−0.941875	−0.470938	+	0.882167i	$0.656084\pi$
$98$	0	0
$99$	2.87089	0.288535
$100$	0	0
$101$	8.77727	0.873371	0.436686	−	0.899614i	$-0.356152\pi$
$101$	8.77727	0.873371	0.436686	+	0.899614i	$0.356152\pi$
$102$	0	0
$103$	−1.52475	−0.150238	−0.0751189	−	0.997175i	$-0.523934\pi$
$103$	−1.52475	−0.150238	−0.0751189	+	0.997175i	$0.523934\pi$
$104$	0	0
$105$	−5.88862	−0.574670
$106$	0	0
$107$	17.6358	1.70492	0.852460	−	0.522792i	$-0.175110\pi$
$107$	17.6358	1.70492	0.852460	+	0.522792i	$0.175110\pi$
$108$	0	0
$109$	−4.91427	−0.470702	−0.235351	−	0.971910i	$-0.575624\pi$
$109$	−4.91427	−0.470702	−0.235351	+	0.971910i	$0.575624\pi$
$110$	0	0
$111$	3.96462	0.376305
$112$	0	0
$113$	0.102880	0.00967811	0.00483905	−	0.999988i	$-0.498460\pi$
$113$	0.102880	0.00967811	0.00483905	+	0.999988i	$0.498460\pi$
$114$	0	0
$115$	−5.97037	−0.556740
$116$	0	0
$117$	6.14696	0.568286
$118$	0	0
$119$	7.08146	0.649157
$120$	0	0
$121$	−9.12637	−0.829670
$122$	0	0
$123$	−5.85700	−0.528108
$124$	0	0
$125$	10.9085	0.975687
$126$	0	0
$127$	−14.7682	−1.31046	−0.655232	−	0.755428i	$-0.727429\pi$
$127$	−14.7682	−1.31046	−0.655232	+	0.755428i	$0.727429\pi$
$128$	0	0
$129$	0.434990	0.0382987
$130$	0	0
$131$	−3.01646	−0.263549	−0.131775	−	0.991280i	$-0.542068\pi$
$131$	−3.01646	−0.263549	−0.131775	+	0.991280i	$0.542068\pi$
$132$	0	0
$133$	−21.1154	−1.83094
$134$	0	0
$135$	−11.0753	−0.953209
$136$	0	0
$137$	0.370190	0.0316275	0.0158137	−	0.999875i	$-0.494966\pi$
$137$	0.370190	0.0316275	0.0158137	+	0.999875i	$0.494966\pi$
$138$	0	0
$139$	−13.0944	−1.11065	−0.555324	−	0.831634i	$-0.687406\pi$
$139$	−13.0944	−1.11065	−0.555324	+	0.831634i	$0.687406\pi$
$140$	0	0
$141$	7.28815	0.613773
$142$	0	0
$143$	4.01168	0.335474
$144$	0	0
$145$	0.719195	0.0597259
$146$	0	0
$147$	−0.328055	−0.0270575
$148$	0	0
$149$	−5.61923	−0.460345	−0.230173	−	0.973150i	$-0.573929\pi$
$149$	−5.61923	−0.460345	−0.230173	+	0.973150i	$0.573929\pi$
$150$	0	0
$151$	6.99662	0.569377	0.284689	−	0.958620i	$-0.408110\pi$
$151$	6.99662	0.569377	0.284689	+	0.958620i	$0.408110\pi$
$152$	0	0
$153$	5.48016	0.443045
$154$	0	0
$155$	−18.1840	−1.46058
$156$	0	0
$157$	10.5849	0.844764	0.422382	−	0.906418i	$-0.361194\pi$
$157$	10.5849	0.844764	0.422382	+	0.906418i	$0.361194\pi$
$158$	0	0
$159$	−6.90041	−0.547238
$160$	0	0
$161$	−7.07541	−0.557621
$162$	0	0
$163$	4.74396	0.371576	0.185788	−	0.982590i	$-0.440516\pi$
$163$	4.74396	0.371576	0.185788	+	0.982590i	$0.440516\pi$
$164$	0	0
$165$	−2.97406	−0.231531
$166$	0	0
$167$	−15.4650	−1.19672	−0.598360	−	0.801227i	$-0.704181\pi$
$167$	−15.4650	−1.19672	−0.598360	+	0.801227i	$0.704181\pi$
$168$	0	0
$169$	−4.41045	−0.339265
$170$	0	0
$171$	−16.3407	−1.24960
$172$	0	0
$173$	−18.0030	−1.36874	−0.684372	−	0.729133i	$-0.739924\pi$
$173$	−18.0030	−1.36874	−0.684372	+	0.729133i	$0.739924\pi$
$174$	0	0
$175$	−0.623544	−0.0471355
$176$	0	0
$177$	0.206322	0.0155081
$178$	0	0
$179$	−22.6390	−1.69212	−0.846058	−	0.533092i	$-0.821030\pi$
$179$	−22.6390	−1.69212	−0.846058	+	0.533092i	$0.821030\pi$
$180$	0	0
$181$	2.97984	0.221490	0.110745	−	0.993849i	$-0.464676\pi$
$181$	2.97984	0.221490	0.110745	+	0.993849i	$0.464676\pi$
$182$	0	0
$183$	−6.90409	−0.510365
$184$	0	0
$185$	9.54335	0.701641
$186$	0	0
$187$	3.57651	0.261541
$188$	0	0
$189$	−13.1252	−0.954718
$190$	0	0
$191$	−4.09564	−0.296350	−0.148175	−	0.988961i	$-0.547340\pi$
$191$	−4.09564	−0.296350	−0.148175	+	0.988961i	$0.547340\pi$
$192$	0	0
$193$	−21.1708	−1.52391	−0.761955	−	0.647630i	$-0.775760\pi$
$193$	−21.1708	−1.52391	−0.761955	+	0.647630i	$0.775760\pi$
$194$	0	0
$195$	−6.36787	−0.456013
$196$	0	0
$197$	18.5139	1.31906	0.659531	−	0.751677i	$-0.270755\pi$
$197$	18.5139	1.31906	0.659531	+	0.751677i	$0.270755\pi$
$198$	0	0
$199$	1.56711	0.111089	0.0555447	−	0.998456i	$-0.482310\pi$
$199$	1.56711	0.111089	0.0555447	+	0.998456i	$0.482310\pi$
$200$	0	0
$201$	9.59779	0.676976
$202$	0	0
$203$	0.852309	0.0598204
$204$	0	0
$205$	−14.0985	−0.984685
$206$	0	0
$207$	−5.47548	−0.380572
$208$	0	0
$209$	−10.6644	−0.737673
$210$	0	0
$211$	22.8488	1.57298	0.786489	−	0.617604i	$-0.211897\pi$
$211$	22.8488	1.57298	0.786489	+	0.617604i	$0.211897\pi$
$212$	0	0
$213$	−15.6845	−1.07469
$214$	0	0
$215$	1.04708	0.0714100
$216$	0	0
$217$	−21.5497	−1.46289
$218$	0	0
$219$	−2.72516	−0.184149
$220$	0	0
$221$	7.65779	0.515119
$222$	0	0
$223$	−4.75557	−0.318457	−0.159228	−	0.987242i	$-0.550901\pi$
$223$	−4.75557	−0.318457	−0.159228	+	0.987242i	$0.550901\pi$
$224$	0	0
$225$	−0.482545	−0.0321696
$226$	0	0
$227$	27.4709	1.82331	0.911655	−	0.410956i	$-0.134805\pi$
$227$	27.4709	1.82331	0.911655	+	0.410956i	$0.134805\pi$
$228$	0	0
$229$	−4.52436	−0.298978	−0.149489	−	0.988763i	$-0.547763\pi$
$229$	−4.52436	−0.298978	−0.149489	+	0.988763i	$0.547763\pi$
$230$	0	0
$231$	−3.52453	−0.231897
$232$	0	0
$233$	−10.5567	−0.691594	−0.345797	−	0.938309i	$-0.612391\pi$
$233$	−10.5567	−0.691594	−0.345797	+	0.938309i	$0.612391\pi$
$234$	0	0
$235$	17.5435	1.14441
$236$	0	0
$237$	−15.2222	−0.988789
$238$	0	0
$239$	28.3072	1.83104	0.915522	−	0.402269i	$-0.131778\pi$
$239$	28.3072	1.83104	0.915522	+	0.402269i	$0.131778\pi$
$240$	0	0
$241$	−10.6922	−0.688748	−0.344374	−	0.938833i	$-0.611909\pi$
$241$	−10.6922	−0.688748	−0.344374	+	0.938833i	$0.611909\pi$
$242$	0	0
$243$	−16.1352	−1.03507
$244$	0	0
$245$	−0.789669	−0.0504501
$246$	0	0
$247$	−22.8339	−1.45289
$248$	0	0
$249$	11.9055	0.754483
$250$	0	0
$251$	3.42359	0.216095	0.108048	−	0.994146i	$-0.465540\pi$
$251$	3.42359	0.216095	0.108048	+	0.994146i	$0.465540\pi$
$252$	0	0
$253$	−3.57346	−0.224661
$254$	0	0
$255$	−5.67711	−0.355514
$256$	0	0
$257$	−11.2524	−0.701906	−0.350953	−	0.936393i	$-0.614142\pi$
$257$	−11.2524	−0.701906	−0.350953	+	0.936393i	$0.614142\pi$
$258$	0	0
$259$	11.3097	0.702751
$260$	0	0
$261$	0.659580	0.0408270
$262$	0	0
$263$	−0.429180	−0.0264643	−0.0132322	−	0.999912i	$-0.504212\pi$
$263$	−0.429180	−0.0264643	−0.0132322	+	0.999912i	$0.504212\pi$
$264$	0	0
$265$	−16.6102	−1.02035
$266$	0	0
$267$	0.0904688	0.00553660
$268$	0	0
$269$	15.2320	0.928712	0.464356	−	0.885649i	$-0.346286\pi$
$269$	15.2320	0.928712	0.464356	+	0.885649i	$0.346286\pi$
$270$	0	0
$271$	−17.7850	−1.08036	−0.540182	−	0.841548i	$-0.681645\pi$
$271$	−17.7850	−1.08036	−0.540182	+	0.841548i	$0.681645\pi$
$272$	0	0
$273$	−7.54649	−0.456734
$274$	0	0
$275$	−0.314923	−0.0189906
$276$	0	0
$277$	15.1882	0.912570	0.456285	−	0.889834i	$-0.349180\pi$
$277$	15.1882	0.912570	0.456285	+	0.889834i	$0.349180\pi$
$278$	0	0
$279$	−16.6767	−0.998409
$280$	0	0
$281$	20.0885	1.19838	0.599191	−	0.800606i	$-0.295489\pi$
$281$	20.0885	1.19838	0.599191	+	0.800606i	$0.295489\pi$
$282$	0	0
$283$	1.63497	0.0971889	0.0485945	−	0.998819i	$-0.484526\pi$
$283$	1.63497	0.0971889	0.0485945	+	0.998819i	$0.484526\pi$
$284$	0	0
$285$	16.9280	1.00273
$286$	0	0
$287$	−16.7080	−0.986243
$288$	0	0
$289$	−10.1729	−0.598405
$290$	0	0
$291$	8.81321	0.516640
$292$	0	0
$293$	−10.1892	−0.595262	−0.297631	−	0.954681i	$-0.596196\pi$
$293$	−10.1892	−0.595262	−0.297631	+	0.954681i	$0.596196\pi$
$294$	0	0
$295$	0.496644	0.0289157
$296$	0	0
$297$	−6.62892	−0.384649
$298$	0	0
$299$	−7.65125	−0.442483
$300$	0	0
$301$	1.24088	0.0715230
$302$	0	0
$303$	−8.33901	−0.479064
$304$	0	0
$305$	−16.6190	−0.951602
$306$	0	0
$307$	16.3682	0.934184	0.467092	−	0.884209i	$-0.345302\pi$
$307$	16.3682	0.934184	0.467092	+	0.884209i	$0.345302\pi$
$308$	0	0
$309$	1.44861	0.0824088
$310$	0	0
$311$	−6.98922	−0.396322	−0.198161	−	0.980169i	$-0.563497\pi$
$311$	−6.98922	−0.396322	−0.198161	+	0.980169i	$0.563497\pi$
$312$	0	0
$313$	2.22444	0.125733	0.0628665	−	0.998022i	$-0.479976\pi$
$313$	2.22444	0.125733	0.0628665	+	0.998022i	$0.479976\pi$
$314$	0	0
$315$	−12.9997	−0.732450
$316$	0	0
$317$	3.49687	0.196404	0.0982019	−	0.995167i	$-0.468691\pi$
$317$	3.49687	0.196404	0.0982019	+	0.995167i	$0.468691\pi$
$318$	0	0
$319$	0.430461	0.0241012
$320$	0	0
$321$	−16.7553	−0.935187
$322$	0	0
$323$	−20.3570	−1.13269
$324$	0	0
$325$	−0.674292	−0.0374030
$326$	0	0
$327$	4.66889	0.258190
$328$	0	0
$329$	20.7906	1.14622
$330$	0	0
$331$	13.8654	0.762112	0.381056	−	0.924552i	$-0.375561\pi$
$331$	13.8654	0.762112	0.381056	+	0.924552i	$0.375561\pi$
$332$	0	0
$333$	8.75229	0.479623
$334$	0	0
$335$	23.1031	1.26226
$336$	0	0
$337$	−33.0259	−1.79903	−0.899517	−	0.436885i	$-0.856082\pi$
$337$	−33.0259	−1.79903	−0.899517	+	0.436885i	$0.856082\pi$
$338$	0	0
$339$	−0.0977428	−0.00530866
$340$	0	0
$341$	−10.8837	−0.589387
$342$	0	0
$343$	18.0357	0.973837
$344$	0	0
$345$	5.67226	0.305384
$346$	0	0
$347$	0.417829	0.0224302	0.0112151	−	0.999937i	$-0.496430\pi$
$347$	0.417829	0.0224302	0.0112151	+	0.999937i	$0.496430\pi$
$348$	0	0
$349$	−8.68607	−0.464955	−0.232477	−	0.972602i	$-0.574683\pi$
$349$	−8.68607	−0.464955	−0.232477	+	0.972602i	$0.574683\pi$
$350$	0	0
$351$	−14.1934	−0.757588
$352$	0	0
$353$	−9.09582	−0.484122	−0.242061	−	0.970261i	$-0.577823\pi$
$353$	−9.09582	−0.484122	−0.242061	+	0.970261i	$0.577823\pi$
$354$	0	0
$355$	−37.7546	−2.00381
$356$	0	0
$357$	−6.72788	−0.356077
$358$	0	0
$359$	−5.51998	−0.291333	−0.145667	−	0.989334i	$-0.546533\pi$
$359$	−5.51998	−0.291333	−0.145667	+	0.989334i	$0.546533\pi$
$360$	0	0
$361$	41.7004	2.19476
$362$	0	0
$363$	8.67068	0.455093
$364$	0	0
$365$	−6.55979	−0.343355
$366$	0	0
$367$	−16.3639	−0.854188	−0.427094	−	0.904207i	$-0.640463\pi$
$367$	−16.3639	−0.854188	−0.427094	+	0.904207i	$0.640463\pi$
$368$	0	0
$369$	−12.9299	−0.673104
$370$	0	0
$371$	−19.6845	−1.02197
$372$	0	0
$373$	−23.5721	−1.22051	−0.610257	−	0.792203i	$-0.708934\pi$
$373$	−23.5721	−1.22051	−0.610257	+	0.792203i	$0.708934\pi$
$374$	0	0
$375$	−10.3638	−0.535186
$376$	0	0
$377$	0.921675	0.0474687
$378$	0	0
$379$	22.8307	1.17273	0.586367	−	0.810045i	$-0.300558\pi$
$379$	22.8307	1.17273	0.586367	+	0.810045i	$0.300558\pi$
$380$	0	0
$381$	14.0308	0.718819
$382$	0	0
$383$	−21.8257	−1.11524	−0.557622	−	0.830095i	$-0.688286\pi$
$383$	−21.8257	−1.11524	−0.557622	+	0.830095i	$0.688286\pi$
$384$	0	0
$385$	−8.48399	−0.432384
$386$	0	0
$387$	0.960283	0.0488139
$388$	0	0
$389$	29.7462	1.50819	0.754097	−	0.656763i	$-0.228075\pi$
$389$	29.7462	1.50819	0.754097	+	0.656763i	$0.228075\pi$
$390$	0	0
$391$	−6.82127	−0.344967
$392$	0	0
$393$	2.86584	0.144563
$394$	0	0
$395$	−36.6418	−1.84365
$396$	0	0
$397$	28.5561	1.43319	0.716595	−	0.697489i	$-0.245700\pi$
$397$	28.5561	1.43319	0.716595	+	0.697489i	$0.245700\pi$
$398$	0	0
$399$	20.0611	1.00431
$400$	0	0
$401$	13.9970	0.698975	0.349487	−	0.936941i	$-0.386356\pi$
$401$	13.9970	0.698975	0.349487	+	0.936941i	$0.386356\pi$
$402$	0	0
$403$	−23.3035	−1.16083
$404$	0	0
$405$	−3.86735	−0.192170
$406$	0	0
$407$	5.71200	0.283133
$408$	0	0
$409$	−18.2429	−0.902055	−0.451028	−	0.892510i	$-0.648942\pi$
$409$	−18.2429	−0.902055	−0.451028	+	0.892510i	$0.648942\pi$
$410$	0	0
$411$	−0.351706	−0.0173484
$412$	0	0
$413$	0.588567	0.0289615
$414$	0	0
$415$	28.6581	1.40677
$416$	0	0
$417$	12.4405	0.609216
$418$	0	0
$419$	−9.78645	−0.478099	−0.239050	−	0.971007i	$-0.576836\pi$
$419$	−9.78645	−0.478099	−0.239050	+	0.971007i	$0.576836\pi$
$420$	0	0
$421$	2.26639	0.110457	0.0552286	−	0.998474i	$-0.482411\pi$
$421$	2.26639	0.110457	0.0552286	+	0.998474i	$0.482411\pi$
$422$	0	0
$423$	16.0893	0.782288
$424$	0	0
$425$	−0.601147	−0.0291599
$426$	0	0
$427$	−19.6950	−0.953109
$428$	0	0
$429$	−3.81138	−0.184015
$430$	0	0
$431$	−4.29790	−0.207023	−0.103511	−	0.994628i	$-0.533008\pi$
$431$	−4.29790	−0.207023	−0.103511	+	0.994628i	$0.533008\pi$
$432$	0	0
$433$	−16.6114	−0.798292	−0.399146	−	0.916887i	$-0.630693\pi$
$433$	−16.6114	−0.798292	−0.399146	+	0.916887i	$0.630693\pi$
$434$	0	0
$435$	−0.683284	−0.0327610
$436$	0	0
$437$	20.3396	0.972976
$438$	0	0
$439$	21.4863	1.02548	0.512742	−	0.858542i	$-0.328630\pi$
$439$	21.4863	1.02548	0.512742	+	0.858542i	$0.328630\pi$
$440$	0	0
$441$	−0.724212	−0.0344863
$442$	0	0
$443$	19.6047	0.931447	0.465724	−	0.884930i	$-0.345794\pi$
$443$	19.6047	0.931447	0.465724	+	0.884930i	$0.345794\pi$
$444$	0	0
$445$	0.217770	0.0103233
$446$	0	0
$447$	5.33866	0.252510
$448$	0	0
$449$	−35.1455	−1.65862	−0.829310	−	0.558789i	$-0.811266\pi$
$449$	−35.1455	−1.65862	−0.829310	+	0.558789i	$0.811266\pi$
$450$	0	0
$451$	−8.43843	−0.397350
$452$	0	0
$453$	−6.64728	−0.312316
$454$	0	0
$455$	−18.1653	−0.851605
$456$	0	0
$457$	−11.8432	−0.554002	−0.277001	−	0.960870i	$-0.589341\pi$
$457$	−11.8432	−0.554002	−0.277001	+	0.960870i	$0.589341\pi$
$458$	0	0
$459$	−12.6538	−0.590627
$460$	0	0
$461$	17.1286	0.797756	0.398878	−	0.917004i	$-0.369400\pi$
$461$	17.1286	0.797756	0.398878	+	0.917004i	$0.369400\pi$
$462$	0	0
$463$	6.92218	0.321701	0.160850	−	0.986979i	$-0.448576\pi$
$463$	6.92218	0.321701	0.160850	+	0.986979i	$0.448576\pi$
$464$	0	0
$465$	17.2761	0.801158
$466$	0	0
$467$	32.6395	1.51038	0.755188	−	0.655508i	$-0.227545\pi$
$467$	32.6395	1.51038	0.755188	+	0.655508i	$0.227545\pi$
$468$	0	0
$469$	27.3792	1.26425
$470$	0	0
$471$	−10.0563	−0.463372
$472$	0	0
$473$	0.626709	0.0288161
$474$	0	0
$475$	1.79250	0.0822454
$476$	0	0
$477$	−15.2333	−0.697486
$478$	0	0
$479$	−8.78005	−0.401171	−0.200585	−	0.979676i	$-0.564284\pi$
$479$	−8.78005	−0.401171	−0.200585	+	0.979676i	$0.564284\pi$
$480$	0	0
$481$	12.2302	0.557647
$482$	0	0
$483$	6.72213	0.305868
$484$	0	0
$485$	21.2145	0.963301
$486$	0	0
$487$	23.1617	1.04956	0.524779	−	0.851238i	$-0.324148\pi$
$487$	23.1617	1.04956	0.524779	+	0.851238i	$0.324148\pi$
$488$	0	0
$489$	−4.50709	−0.203818
$490$	0	0
$491$	−8.15363	−0.367968	−0.183984	−	0.982929i	$-0.558900\pi$
$491$	−8.15363	−0.367968	−0.183984	+	0.982929i	$0.558900\pi$
$492$	0	0
$493$	0.821696	0.0370073
$494$	0	0
$495$	−6.56554	−0.295099
$496$	0	0
$497$	−44.7426	−2.00698
$498$	0	0
$499$	−30.6993	−1.37429	−0.687144	−	0.726521i	$-0.741136\pi$
$499$	−30.6993	−1.37429	−0.687144	+	0.726521i	$0.741136\pi$
$500$	0	0
$501$	14.6929	0.656428
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−20.0731	−0.893239
$506$	0	0
$507$	4.19023	0.186095
$508$	0	0
$509$	−6.39080	−0.283267	−0.141634	−	0.989919i	$-0.545235\pi$
$509$	−6.39080	−0.283267	−0.141634	+	0.989919i	$0.545235\pi$
$510$	0	0
$511$	−7.77394	−0.343899
$512$	0	0
$513$	37.7309	1.66586
$514$	0	0
$515$	3.48700	0.153655
$516$	0	0
$517$	10.5003	0.461805
$518$	0	0
$519$	17.1041	0.750787
$520$	0	0
$521$	−11.2745	−0.493944	−0.246972	−	0.969023i	$-0.579436\pi$
$521$	−11.2745	−0.493944	−0.246972	+	0.969023i	$0.579436\pi$
$522$	0	0
$523$	−18.6154	−0.813993	−0.406996	−	0.913430i	$-0.633424\pi$
$523$	−18.6154	−0.813993	−0.406996	+	0.913430i	$0.633424\pi$
$524$	0	0
$525$	0.592410	0.0258549
$526$	0	0
$527$	−20.7756	−0.905001
$528$	0	0
$529$	−16.1846	−0.703676
$530$	0	0
$531$	0.455477	0.0197660
$532$	0	0
$533$	−18.0678	−0.782604
$534$	0	0
$535$	−40.3320	−1.74370
$536$	0	0
$537$	21.5086	0.928163
$538$	0	0
$539$	−0.472642	−0.0203581
$540$	0	0
$541$	−36.6446	−1.57547	−0.787736	−	0.616013i	$-0.788747\pi$
$541$	−36.6446	−1.57547	−0.787736	+	0.616013i	$0.788747\pi$
$542$	0	0
$543$	−2.83105	−0.121492
$544$	0	0
$545$	11.2386	0.481409
$546$	0	0
$547$	−19.4186	−0.830280	−0.415140	−	0.909758i	$-0.636267\pi$
$547$	−19.4186	−0.830280	−0.415140	+	0.909758i	$0.636267\pi$
$548$	0	0
$549$	−15.2415	−0.650490
$550$	0	0
$551$	−2.45013	−0.104379
$552$	0	0
$553$	−43.4238	−1.84657
$554$	0	0
$555$	−9.06684	−0.384866
$556$	0	0
$557$	−35.6561	−1.51080	−0.755398	−	0.655266i	$-0.772556\pi$
$557$	−35.6561	−1.51080	−0.755398	+	0.655266i	$0.772556\pi$
$558$	0	0
$559$	1.34187	0.0567549
$560$	0	0
$561$	−3.39793	−0.143461
$562$	0	0
$563$	34.0879	1.43663	0.718316	−	0.695717i	$-0.244913\pi$
$563$	34.0879	1.43663	0.718316	+	0.695717i	$0.244913\pi$
$564$	0	0
$565$	−0.235279	−0.00989827
$566$	0	0
$567$	−4.58316	−0.192475
$568$	0	0
$569$	13.0323	0.546342	0.273171	−	0.961965i	$-0.411927\pi$
$569$	13.0323	0.546342	0.273171	+	0.961965i	$0.411927\pi$
$570$	0	0
$571$	−28.5710	−1.19566	−0.597829	−	0.801624i	$-0.703970\pi$
$571$	−28.5710	−1.19566	−0.597829	+	0.801624i	$0.703970\pi$
$572$	0	0
$573$	3.89114	0.162555
$574$	0	0
$575$	0.600634	0.0250482
$576$	0	0
$577$	40.1508	1.67150	0.835750	−	0.549110i	$-0.185033\pi$
$577$	40.1508	1.67150	0.835750	+	0.549110i	$0.185033\pi$
$578$	0	0
$579$	20.1137	0.835899
$580$	0	0
$581$	33.9624	1.40900
$582$	0	0
$583$	−9.94172	−0.411744
$584$	0	0
$585$	−14.0577	−0.581214
$586$	0	0
$587$	1.26390	0.0521666	0.0260833	−	0.999660i	$-0.491696\pi$
$587$	1.26390	0.0521666	0.0260833	+	0.999660i	$0.491696\pi$
$588$	0	0
$589$	61.9486	2.55255
$590$	0	0
$591$	−17.5895	−0.723535
$592$	0	0
$593$	0.473003	0.0194239	0.00971196	−	0.999953i	$-0.496909\pi$
$593$	0.473003	0.0194239	0.00971196	+	0.999953i	$0.496909\pi$
$594$	0	0
$595$	−16.1948	−0.663924
$596$	0	0
$597$	−1.48886	−0.0609350
$598$	0	0
$599$	−28.8689	−1.17955	−0.589775	−	0.807568i	$-0.700783\pi$
$599$	−28.8689	−1.17955	−0.589775	+	0.807568i	$0.700783\pi$
$600$	0	0
$601$	8.88531	0.362439	0.181220	−	0.983443i	$-0.441996\pi$
$601$	8.88531	0.362439	0.181220	+	0.983443i	$0.441996\pi$
$602$	0	0
$603$	21.1881	0.862845
$604$	0	0
$605$	20.8714	0.848544
$606$	0	0
$607$	−28.6612	−1.16332	−0.581662	−	0.813431i	$-0.697597\pi$
$607$	−28.6612	−1.16332	−0.581662	+	0.813431i	$0.697597\pi$
$608$	0	0
$609$	−0.809753	−0.0328128
$610$	0	0
$611$	22.4826	0.909550
$612$	0	0
$613$	46.9961	1.89815	0.949077	−	0.315043i	$-0.102019\pi$
$613$	46.9961	1.89815	0.949077	+	0.315043i	$0.102019\pi$
$614$	0	0
$615$	13.3946	0.540122
$616$	0	0
$617$	−24.3030	−0.978401	−0.489201	−	0.872171i	$-0.662711\pi$
$617$	−24.3030	−0.978401	−0.489201	+	0.872171i	$0.662711\pi$
$618$	0	0
$619$	−39.4552	−1.58584	−0.792920	−	0.609326i	$-0.791440\pi$
$619$	−39.4552	−1.58584	−0.792920	+	0.609326i	$0.791440\pi$
$620$	0	0
$621$	12.6429	0.507344
$622$	0	0
$623$	0.258077	0.0103396
$624$	0	0
$625$	−26.0974	−1.04390
$626$	0	0
$627$	10.1319	0.404630
$628$	0	0
$629$	10.9035	0.434750
$630$	0	0
$631$	−34.7457	−1.38320	−0.691602	−	0.722279i	$-0.743095\pi$
$631$	−34.7457	−1.38320	−0.691602	+	0.722279i	$0.743095\pi$
$632$	0	0
$633$	−21.7080	−0.862814
$634$	0	0
$635$	33.7738	1.34027
$636$	0	0
$637$	−1.01199	−0.0400965
$638$	0	0
$639$	−34.6251	−1.36975
$640$	0	0
$641$	−42.3173	−1.67143	−0.835716	−	0.549163i	$-0.814947\pi$
$641$	−42.3173	−1.67143	−0.835716	+	0.549163i	$0.814947\pi$
$642$	0	0
$643$	31.5672	1.24489	0.622444	−	0.782664i	$-0.286140\pi$
$643$	31.5672	1.24489	0.622444	+	0.782664i	$0.286140\pi$
$644$	0	0
$645$	−0.994794	−0.0391700
$646$	0	0
$647$	45.6664	1.79533	0.897667	−	0.440675i	$-0.145261\pi$
$647$	45.6664	1.79533	0.897667	+	0.440675i	$0.145261\pi$
$648$	0	0
$649$	0.297258	0.0116684
$650$	0	0
$651$	20.4737	0.802426
$652$	0	0
$653$	7.15705	0.280077	0.140039	−	0.990146i	$-0.455277\pi$
$653$	7.15705	0.280077	0.140039	+	0.990146i	$0.455277\pi$
$654$	0	0
$655$	6.89844	0.269545
$656$	0	0
$657$	−6.01605	−0.234708
$658$	0	0
$659$	−12.1603	−0.473696	−0.236848	−	0.971547i	$-0.576114\pi$
$659$	−12.1603	−0.473696	−0.236848	+	0.971547i	$0.576114\pi$
$660$	0	0
$661$	−15.2194	−0.591968	−0.295984	−	0.955193i	$-0.595647\pi$
$661$	−15.2194	−0.591968	−0.295984	+	0.955193i	$0.595647\pi$
$662$	0	0
$663$	−7.27543	−0.282554
$664$	0	0
$665$	48.2896	1.87259
$666$	0	0
$667$	−0.820994	−0.0317890
$668$	0	0
$669$	4.51812	0.174681
$670$	0	0
$671$	−9.94702	−0.384001
$672$	0	0
$673$	34.9359	1.34668	0.673340	−	0.739333i	$-0.264859\pi$
$673$	34.9359	1.34668	0.673340	+	0.739333i	$0.264859\pi$
$674$	0	0
$675$	1.11420	0.0428856
$676$	0	0
$677$	−17.6078	−0.676721	−0.338361	−	0.941016i	$-0.609872\pi$
$677$	−17.6078	−0.676721	−0.338361	+	0.941016i	$0.609872\pi$
$678$	0	0
$679$	25.1411	0.964826
$680$	0	0
$681$	−26.0993	−1.00013
$682$	0	0
$683$	43.5075	1.66477	0.832385	−	0.554198i	$-0.186975\pi$
$683$	43.5075	1.66477	0.832385	+	0.554198i	$0.186975\pi$
$684$	0	0
$685$	−0.846600	−0.0323469
$686$	0	0
$687$	4.29846	0.163996
$688$	0	0
$689$	−21.2865	−0.810953
$690$	0	0
$691$	2.05256	0.0780829	0.0390415	−	0.999238i	$-0.487570\pi$
$691$	2.05256	0.0780829	0.0390415	+	0.999238i	$0.487570\pi$
$692$	0	0
$693$	−7.78074	−0.295566
$694$	0	0
$695$	29.9459	1.13591
$696$	0	0
$697$	−16.1079	−0.610130
$698$	0	0
$699$	10.0296	0.379355
$700$	0	0
$701$	−9.44349	−0.356676	−0.178338	−	0.983969i	$-0.557072\pi$
$701$	−9.44349	−0.356676	−0.178338	+	0.983969i	$0.557072\pi$
$702$	0	0
$703$	−32.5119	−1.22621
$704$	0	0
$705$	−16.6675	−0.627735
$706$	0	0
$707$	−23.7883	−0.894653
$708$	0	0
$709$	15.9123	0.597598	0.298799	−	0.954316i	$-0.403414\pi$
$709$	15.9123	0.597598	0.298799	+	0.954316i	$0.403414\pi$
$710$	0	0
$711$	−33.6045	−1.26027
$712$	0	0
$713$	20.7579	0.777389
$714$	0	0
$715$	−9.17446	−0.343105
$716$	0	0
$717$	−26.8938	−1.00437
$718$	0	0
$719$	−24.9061	−0.928842	−0.464421	−	0.885615i	$-0.653737\pi$
$719$	−24.9061	−0.928842	−0.464421	+	0.885615i	$0.653737\pi$
$720$	0	0
$721$	4.13240	0.153899
$722$	0	0
$723$	10.1584	0.377794
$724$	0	0
$725$	−0.0723528	−0.00268712
$726$	0	0
$727$	−11.5573	−0.428636	−0.214318	−	0.976764i	$-0.568753\pi$
$727$	−11.5573	−0.428636	−0.214318	+	0.976764i	$0.568753\pi$
$728$	0	0
$729$	10.2563	0.379865
$730$	0	0
$731$	1.19631	0.0442470
$732$	0	0
$733$	11.9918	0.442926	0.221463	−	0.975169i	$-0.428917\pi$
$733$	11.9918	0.442926	0.221463	+	0.975169i	$0.428917\pi$
$734$	0	0
$735$	0.750240	0.0276730
$736$	0	0
$737$	13.8280	0.509359
$738$	0	0
$739$	−48.3255	−1.77768	−0.888840	−	0.458217i	$-0.848488\pi$
$739$	−48.3255	−1.77768	−0.888840	+	0.458217i	$0.848488\pi$
$740$	0	0
$741$	21.6938	0.796942
$742$	0	0
$743$	−40.3146	−1.47900	−0.739499	−	0.673158i	$-0.764937\pi$
$743$	−40.3146	−1.47900	−0.739499	+	0.673158i	$0.764937\pi$
$744$	0	0
$745$	12.8508	0.470817
$746$	0	0
$747$	26.2826	0.961632
$748$	0	0
$749$	−47.7970	−1.74646
$750$	0	0
$751$	48.6091	1.77377	0.886885	−	0.461990i	$-0.152864\pi$
$751$	48.6091	1.77377	0.886885	+	0.461990i	$0.152864\pi$
$752$	0	0
$753$	−3.25265	−0.118533
$754$	0	0
$755$	−16.0008	−0.582330
$756$	0	0
$757$	−44.8319	−1.62944	−0.814721	−	0.579853i	$-0.803110\pi$
$757$	−44.8319	−1.62944	−0.814721	+	0.579853i	$0.803110\pi$
$758$	0	0
$759$	3.39503	0.123232
$760$	0	0
$761$	−31.4812	−1.14119	−0.570596	−	0.821231i	$-0.693288\pi$
$761$	−31.4812	−1.14119	−0.570596	+	0.821231i	$0.693288\pi$
$762$	0	0
$763$	13.3188	0.482171
$764$	0	0
$765$	−12.5328	−0.453123
$766$	0	0
$767$	0.636468	0.0229815
$768$	0	0
$769$	30.6134	1.10395	0.551974	−	0.833861i	$-0.313875\pi$
$769$	30.6134	1.10395	0.551974	+	0.833861i	$0.313875\pi$
$770$	0	0
$771$	10.6906	0.385011
$772$	0	0
$773$	25.8605	0.930136	0.465068	−	0.885275i	$-0.346030\pi$
$773$	25.8605	0.930136	0.465068	+	0.885275i	$0.346030\pi$
$774$	0	0
$775$	1.82936	0.0657125
$776$	0	0
$777$	−10.7450	−0.385475
$778$	0	0
$779$	48.0304	1.72087
$780$	0	0
$781$	−22.5974	−0.808598
$782$	0	0
$783$	−1.52298	−0.0544268
$784$	0	0
$785$	−24.2069	−0.863981
$786$	0	0
$787$	−31.7233	−1.13081	−0.565406	−	0.824813i	$-0.691281\pi$
$787$	−31.7233	−1.13081	−0.565406	+	0.824813i	$0.691281\pi$
$788$	0	0
$789$	0.407750	0.0145163
$790$	0	0
$791$	−0.278827	−0.00991394
$792$	0	0
$793$	−21.2979	−0.756311
$794$	0	0
$795$	15.7808	0.559687
$796$	0	0
$797$	−55.1753	−1.95441	−0.977205	−	0.212296i	$-0.931906\pi$
$797$	−55.1753	−1.95441	−0.977205	+	0.212296i	$0.931906\pi$
$798$	0	0
$799$	20.0438	0.709099
$800$	0	0
$801$	0.199719	0.00705671
$802$	0	0
$803$	−3.92625	−0.138554
$804$	0	0
$805$	16.1810	0.570306
$806$	0	0
$807$	−14.4715	−0.509420
$808$	0	0
$809$	−6.04923	−0.212680	−0.106340	−	0.994330i	$-0.533913\pi$
$809$	−6.04923	−0.212680	−0.106340	+	0.994330i	$0.533913\pi$
$810$	0	0
$811$	3.97693	0.139649	0.0698244	−	0.997559i	$-0.477756\pi$
$811$	3.97693	0.139649	0.0698244	+	0.997559i	$0.477756\pi$
$812$	0	0
$813$	16.8970	0.592604
$814$	0	0
$815$	−10.8491	−0.380029
$816$	0	0
$817$	−3.56714	−0.124798
$818$	0	0
$819$	−16.6596	−0.582134
$820$	0	0
$821$	16.5562	0.577815	0.288907	−	0.957357i	$-0.406708\pi$
$821$	16.5562	0.577815	0.288907	+	0.957357i	$0.406708\pi$
$822$	0	0
$823$	29.9588	1.04430	0.522150	−	0.852854i	$-0.325130\pi$
$823$	29.9588	1.04430	0.522150	+	0.852854i	$0.325130\pi$
$824$	0	0
$825$	0.299198	0.0104168
$826$	0	0
$827$	39.4061	1.37029	0.685143	−	0.728409i	$-0.259740\pi$
$827$	39.4061	1.37029	0.685143	+	0.728409i	$0.259740\pi$
$828$	0	0
$829$	6.63817	0.230553	0.115277	−	0.993333i	$-0.463225\pi$
$829$	6.63817	0.230553	0.115277	+	0.993333i	$0.463225\pi$
$830$	0	0
$831$	−14.4298	−0.500565
$832$	0	0
$833$	−0.902214	−0.0312599
$834$	0	0
$835$	35.3675	1.22394
$836$	0	0
$837$	38.5068	1.33099
$838$	0	0
$839$	55.4839	1.91552	0.957759	−	0.287574i	$-0.0928486\pi$
$839$	55.4839	1.91552	0.957759	+	0.287574i	$0.0928486\pi$
$840$	0	0
$841$	−28.9011	−0.996590
$842$	0	0
$843$	−19.0855	−0.657340
$844$	0	0
$845$	10.0864	0.346983
$846$	0	0
$847$	24.7345	0.849887
$848$	0	0
$849$	−1.55334	−0.0533103
$850$	0	0
$851$	−10.8942	−0.373447
$852$	0	0
$853$	−23.3264	−0.798680	−0.399340	−	0.916803i	$-0.630761\pi$
$853$	−23.3264	−0.798680	−0.399340	+	0.916803i	$0.630761\pi$
$854$	0	0
$855$	37.3701	1.27803
$856$	0	0
$857$	20.5822	0.703075	0.351537	−	0.936174i	$-0.385659\pi$
$857$	20.5822	0.703075	0.351537	+	0.936174i	$0.385659\pi$
$858$	0	0
$859$	45.7526	1.56106	0.780529	−	0.625120i	$-0.214950\pi$
$859$	45.7526	1.56106	0.780529	+	0.625120i	$0.214950\pi$
$860$	0	0
$861$	15.8738	0.540977
$862$	0	0
$863$	−10.0848	−0.343290	−0.171645	−	0.985159i	$-0.554908\pi$
$863$	−10.0848	−0.343290	−0.171645	+	0.985159i	$0.554908\pi$
$864$	0	0
$865$	41.1717	1.39988
$866$	0	0
$867$	9.66495	0.328239
$868$	0	0
$869$	−21.9313	−0.743968
$870$	0	0
$871$	29.6075	1.00321
$872$	0	0
$873$	19.4560	0.658487
$874$	0	0
$875$	−29.5645	−0.999462
$876$	0	0
$877$	3.30564	0.111624	0.0558118	−	0.998441i	$-0.482225\pi$
$877$	3.30564	0.111624	0.0558118	+	0.998441i	$0.482225\pi$
$878$	0	0
$879$	9.68048	0.326515
$880$	0	0
$881$	8.30907	0.279940	0.139970	−	0.990156i	$-0.455299\pi$
$881$	8.30907	0.279940	0.139970	+	0.990156i	$0.455299\pi$
$882$	0	0
$883$	−2.65895	−0.0894807	−0.0447403	−	0.998999i	$-0.514246\pi$
$883$	−2.65895	−0.0894807	−0.0447403	+	0.998999i	$0.514246\pi$
$884$	0	0
$885$	−0.471846	−0.0158609
$886$	0	0
$887$	21.5846	0.724740	0.362370	−	0.932034i	$-0.381968\pi$
$887$	21.5846	0.724740	0.362370	+	0.932034i	$0.381968\pi$
$888$	0	0
$889$	40.0250	1.34240
$890$	0	0
$891$	−2.31474	−0.0775466
$892$	0	0
$893$	−59.7665	−2.00001
$894$	0	0
$895$	51.7738	1.73061
$896$	0	0
$897$	7.26921	0.242712
$898$	0	0
$899$	−2.50051	−0.0833967
$900$	0	0
$901$	−18.9775	−0.632231
$902$	0	0
$903$	−1.17892	−0.0392320
$904$	0	0
$905$	−6.81470	−0.226528
$906$	0	0
$907$	5.95616	0.197771	0.0988855	−	0.995099i	$-0.468472\pi$
$907$	5.95616	0.197771	0.0988855	+	0.995099i	$0.468472\pi$
$908$	0	0
$909$	−18.4092	−0.610594
$910$	0	0
$911$	−14.9730	−0.496079	−0.248040	−	0.968750i	$-0.579786\pi$
$911$	−14.9730	−0.496079	−0.248040	+	0.968750i	$0.579786\pi$
$912$	0	0
$913$	17.1528	0.567676
$914$	0	0
$915$	15.7892	0.521975
$916$	0	0
$917$	8.17527	0.269971
$918$	0	0
$919$	21.4164	0.706463	0.353232	−	0.935536i	$-0.385083\pi$
$919$	21.4164	0.706463	0.353232	+	0.935536i	$0.385083\pi$
$920$	0	0
$921$	−15.5509	−0.512421
$922$	0	0
$923$	−48.3840	−1.59258
$924$	0	0
$925$	−0.960085	−0.0315674
$926$	0	0
$927$	3.19796	0.105035
$928$	0	0
$929$	−51.4153	−1.68688	−0.843441	−	0.537221i	$-0.819474\pi$
$929$	−51.4153	−1.68688	−0.843441	+	0.537221i	$0.819474\pi$
$930$	0	0
$931$	2.69021	0.0881682
$932$	0	0
$933$	6.64024	0.217392
$934$	0	0
$935$	−8.17925	−0.267490
$936$	0	0
$937$	−41.9350	−1.36996	−0.684979	−	0.728563i	$-0.740188\pi$
$937$	−41.9350	−1.36996	−0.684979	+	0.728563i	$0.740188\pi$
$938$	0	0
$939$	−2.11337	−0.0689674
$940$	0	0
$941$	−36.8467	−1.20117	−0.600585	−	0.799561i	$-0.705065\pi$
$941$	−36.8467	−1.20117	−0.600585	+	0.799561i	$0.705065\pi$
$942$	0	0
$943$	16.0941	0.524097
$944$	0	0
$945$	30.0165	0.976436
$946$	0	0
$947$	−2.12177	−0.0689481	−0.0344741	−	0.999406i	$-0.510976\pi$
$947$	−2.12177	−0.0689481	−0.0344741	+	0.999406i	$0.510976\pi$
$948$	0	0
$949$	−8.40663	−0.272891
$950$	0	0
$951$	−3.32227	−0.107732
$952$	0	0
$953$	13.5643	0.439391	0.219695	−	0.975569i	$-0.429494\pi$
$953$	13.5643	0.439391	0.219695	+	0.975569i	$0.429494\pi$
$954$	0	0
$955$	9.36645	0.303091
$956$	0	0
$957$	−0.408968	−0.0132201
$958$	0	0
$959$	−1.00330	−0.0323981
$960$	0	0
$961$	32.2225	1.03944
$962$	0	0
$963$	−36.9889	−1.19195
$964$	0	0
$965$	48.4163	1.55858
$966$	0	0
$967$	−5.95719	−0.191570	−0.0957851	−	0.995402i	$-0.530536\pi$
$967$	−5.95719	−0.191570	−0.0957851	+	0.995402i	$0.530536\pi$
$968$	0	0
$969$	19.3406	0.621308
$970$	0	0
$971$	−12.1772	−0.390785	−0.195392	−	0.980725i	$-0.562598\pi$
$971$	−12.1772	−0.390785	−0.195392	+	0.980725i	$0.562598\pi$
$972$	0	0
$973$	35.4886	1.13771
$974$	0	0
$975$	0.640624	0.0205164
$976$	0	0
$977$	7.69439	0.246165	0.123083	−	0.992396i	$-0.460722\pi$
$977$	7.69439	0.246165	0.123083	+	0.992396i	$0.460722\pi$
$978$	0	0
$979$	0.130342	0.00416576
$980$	0	0
$981$	10.3070	0.329078
$982$	0	0
$983$	12.2144	0.389578	0.194789	−	0.980845i	$-0.437598\pi$
$983$	12.2144	0.389578	0.194789	+	0.980845i	$0.437598\pi$
$984$	0	0
$985$	−42.3401	−1.34907
$986$	0	0
$987$	−19.7525	−0.628729
$988$	0	0
$989$	−1.19529	−0.0380079
$990$	0	0
$991$	−2.35228	−0.0747227	−0.0373613	−	0.999302i	$-0.511895\pi$
$991$	−2.35228	−0.0747227	−0.0373613	+	0.999302i	$0.511895\pi$
$992$	0	0
$993$	−13.1731	−0.418036
$994$	0	0
$995$	−3.58387	−0.113616
$996$	0	0
$997$	19.2415	0.609384	0.304692	−	0.952451i	$-0.401446\pi$
$997$	19.2415	0.609384	0.304692	+	0.952451i	$0.401446\pi$
$998$	0	0
$999$	−20.2092	−0.639389

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.p.1.2		10
4.3	odd	2		503.2.a.e.1.6	✓	10
12.11	even	2		4527.2.a.k.1.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.a.e.1.6	✓	10	4.3	odd	2
4527.2.a.k.1.5		10	12.11	even	2
8048.2.a.p.1.2		10	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)

\(f(q)\)	\(=\)	\(q-0.950069 q^{3} -2.28693 q^{5} -2.71022 q^{7} -2.09737 q^{9} +O(q^{10})\)	\(q-0.950069 q^{3} -2.28693 q^{5} -2.71022 q^{7} -2.09737 q^{9} -1.36880 q^{11} -2.93079 q^{13} +2.17275 q^{15} -2.61287 q^{17} +7.79104 q^{19} +2.57490 q^{21} +2.61064 q^{23} +0.230071 q^{25} +4.84285 q^{27} -0.314480 q^{29} +7.95126 q^{31} +1.30046 q^{33} +6.19810 q^{35} -4.17299 q^{37} +2.78446 q^{39} +6.16482 q^{41} -0.457851 q^{43} +4.79655 q^{45} -7.67118 q^{47} +0.345296 q^{49} +2.48241 q^{51} +7.26306 q^{53} +3.13037 q^{55} -7.40203 q^{57} -0.217166 q^{59} +7.26694 q^{61} +5.68433 q^{63} +6.70253 q^{65} -10.1022 q^{67} -2.48029 q^{69} +16.5088 q^{71} +2.86838 q^{73} -0.218584 q^{75} +3.70976 q^{77} +16.0222 q^{79} +1.69106 q^{81} -12.5312 q^{83} +5.97547 q^{85} +0.298777 q^{87} -0.0952234 q^{89} +7.94310 q^{91} -7.55424 q^{93} -17.8176 q^{95} -9.27640 q^{97} +2.87089 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 8 q^{3} - q^{5} + 5 q^{7} - 2 q^{9}+O(q^{10}) \) 10 * q + 8 * q^3 - q^5 + 5 * q^7 - 2 * q^9	\( 10 q + 8 q^{3} - q^{5} + 5 q^{7} - 2 q^{9} + 3 q^{11} - 18 q^{13} + 2 q^{15} - 11 q^{17} + q^{21} + 2 q^{23} - 27 q^{25} + 2 q^{27} - 9 q^{29} + 22 q^{31} - 10 q^{33} + 6 q^{35} - 35 q^{37} - 8 q^{39} - 4 q^{41} + 20 q^{43} + 2 q^{45} - 7 q^{47} - 27 q^{49} - 9 q^{51} - 24 q^{53} + 11 q^{55} - 23 q^{57} - 17 q^{59} - 4 q^{61} - 10 q^{63} - 16 q^{65} + 6 q^{67} - 2 q^{69} + q^{71} - 31 q^{73} - 30 q^{75} + 3 q^{77} + 10 q^{79} - 6 q^{81} - 22 q^{83} - 6 q^{85} - 25 q^{87} + q^{89} - 10 q^{91} - 6 q^{93} - 39 q^{95} - 57 q^{97} - 35 q^{99}+O(q^{100}) \) 10 * q + 8 * q^3 - q^5 + 5 * q^7 - 2 * q^9 + 3 * q^11 - 18 * q^13 + 2 * q^15 - 11 * q^17 + q^21 + 2 * q^23 - 27 * q^25 + 2 * q^27 - 9 * q^29 + 22 * q^31 - 10 * q^33 + 6 * q^35 - 35 * q^37 - 8 * q^39 - 4 * q^41 + 20 * q^43 + 2 * q^45 - 7 * q^47 - 27 * q^49 - 9 * q^51 - 24 * q^53 + 11 * q^55 - 23 * q^57 - 17 * q^59 - 4 * q^61 - 10 * q^63 - 16 * q^65 + 6 * q^67 - 2 * q^69 + q^71 - 31 * q^73 - 30 * q^75 + 3 * q^77 + 10 * q^79 - 6 * q^81 - 22 * q^83 - 6 * q^85 - 25 * q^87 + q^89 - 10 * q^91 - 6 * q^93 - 39 * q^95 - 57 * q^97 - 35 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more