LMFDB - Embedded newform 3040.2.a.r.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3040,2,Mod(1,3040)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3040, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3040.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3040 = 2^{5} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3040.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-1,0,4,0,-5,0,1,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$24.2745222145$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.17428.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 4x + 6 $ x^4 - x^3 - 6x^2 + 4x + 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.36865$ of defining polynomial
Character	$\chi$	$=$	3040.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.36865 q^{3} +1.00000 q^{5} -3.61050 q^{7} +2.61050 q^{9} -2.20421 q^{11} +2.57286 q^{13} -2.36865 q^{15} +0.922589 q^{17} -1.00000 q^{19} +8.55201 q^{21} +7.23992 q^{23} +1.00000 q^{25} +0.922589 q^{27} +3.81471 q^{29} -4.48370 q^{31} +5.22100 q^{33} -3.61050 q^{35} -4.13397 q^{37} -6.09420 q^{39} -4.73730 q^{41} +5.42521 q^{43} +2.61050 q^{45} +0.279491 q^{47} +6.03571 q^{49} -2.18529 q^{51} +9.86914 q^{53} -2.20421 q^{55} +2.36865 q^{57} +1.33101 q^{59} -11.1299 q^{61} -9.42521 q^{63} +2.57286 q^{65} +13.7028 q^{67} -17.1488 q^{69} -11.1457 q^{71} -9.28931 q^{73} -2.36865 q^{75} +7.95830 q^{77} +4.85042 q^{79} -10.0168 q^{81} -9.42521 q^{83} +0.922589 q^{85} -9.03571 q^{87} -10.3667 q^{89} -9.28931 q^{91} +10.6203 q^{93} -1.00000 q^{95} +2.75815 q^{97} -5.75409 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} + 4 q^{5} - 5 q^{7} + q^{9} - 6 q^{11} - q^{13} - q^{15} - q^{17} - 4 q^{19} + 5 q^{21} - 5 q^{23} + 4 q^{25} - q^{27} + 3 q^{29} - 16 q^{31} + 2 q^{33} - 5 q^{35} - 8 q^{37} - 13 q^{39}+ \cdots + 10 q^{99}+O(q^{100}) $ 4 * q - q^3 + 4 * q^5 - 5 * q^7 + q^9 - 6 * q^11 - q^13 - q^15 - q^17 - 4 * q^19 + 5 * q^21 - 5 * q^23 + 4 * q^25 - q^27 + 3 * q^29 - 16 * q^31 + 2 * q^33 - 5 * q^35 - 8 * q^37 - 13 * q^39 - 2 * q^41 + q^45 + 2 * q^47 - 7 * q^49 - 21 * q^51 + 13 * q^53 - 6 * q^55 + q^57 - 5 * q^59 - 2 * q^61 - 16 * q^63 - q^65 + q^67 - 11 * q^69 - 22 * q^71 + 9 * q^73 - q^75 - 4 * q^77 - 24 * q^79 - 24 * q^81 - 16 * q^83 - q^85 - 5 * q^87 + 9 * q^91 - 14 * q^93 - 4 * q^95 + 12 * q^97 + 10 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.36865	−1.36754	−0.683770	−	0.729697i	$-0.739661\pi$
$3$	−2.36865	−1.36754	−0.683770	+	0.729697i	$0.739661\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.61050	−1.36464	−0.682320	−	0.731053i	$-0.739029\pi$
$7$	−3.61050	−1.36464	−0.682320	+	0.731053i	$0.739029\pi$
$8$	0	0
$9$	2.61050	0.870167
$10$	0	0
$11$	−2.20421	−0.664594	−0.332297	−	0.943175i	$-0.607824\pi$
$11$	−2.20421	−0.664594	−0.332297	+	0.943175i	$0.607824\pi$
$12$	0	0
$13$	2.57286	0.713583	0.356791	−	0.934184i	$-0.383871\pi$
$13$	2.57286	0.713583	0.356791	+	0.934184i	$0.383871\pi$
$14$	0	0
$15$	−2.36865	−0.611583
$16$	0	0
$17$	0.922589	0.223761	0.111880	−	0.993722i	$-0.464313\pi$
$17$	0.922589	0.223761	0.111880	+	0.993722i	$0.464313\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	8.55201	1.86620
$22$	0	0
$23$	7.23992	1.50963	0.754814	−	0.655939i	$-0.227727\pi$
$23$	7.23992	1.50963	0.754814	+	0.655939i	$0.227727\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0.922589	0.177552
$28$	0	0
$29$	3.81471	0.708374	0.354187	−	0.935175i	$-0.384758\pi$
$29$	3.81471	0.708374	0.354187	+	0.935175i	$0.384758\pi$
$30$	0	0
$31$	−4.48370	−0.805296	−0.402648	−	0.915355i	$-0.631910\pi$
$31$	−4.48370	−0.805296	−0.402648	+	0.915355i	$0.631910\pi$
$32$	0	0
$33$	5.22100	0.908859
$34$	0	0
$35$	−3.61050	−0.610286
$36$	0	0
$37$	−4.13397	−0.679621	−0.339810	−	0.940494i	$-0.610363\pi$
$37$	−4.13397	−0.679621	−0.339810	+	0.940494i	$0.610363\pi$
$38$	0	0
$39$	−6.09420	−0.975853
$40$	0	0
$41$	−4.73730	−0.739842	−0.369921	−	0.929063i	$-0.620615\pi$
$41$	−4.73730	−0.739842	−0.369921	+	0.929063i	$0.620615\pi$
$42$	0	0
$43$	5.42521	0.827337	0.413668	−	0.910428i	$-0.364247\pi$
$43$	5.42521	0.827337	0.413668	+	0.910428i	$0.364247\pi$
$44$	0	0
$45$	2.61050	0.389150
$46$	0	0
$47$	0.279491	0.0407680	0.0203840	−	0.999792i	$-0.493511\pi$
$47$	0.279491	0.0407680	0.0203840	+	0.999792i	$0.493511\pi$
$48$	0	0
$49$	6.03571	0.862244
$50$	0	0
$51$	−2.18529	−0.306002
$52$	0	0
$53$	9.86914	1.35563	0.677815	−	0.735232i	$-0.262927\pi$
$53$	9.86914	1.35563	0.677815	+	0.735232i	$0.262927\pi$
$54$	0	0
$55$	−2.20421	−0.297216
$56$	0	0
$57$	2.36865	0.313735
$58$	0	0
$59$	1.33101	0.173283	0.0866413	−	0.996240i	$-0.472387\pi$
$59$	1.33101	0.173283	0.0866413	+	0.996240i	$0.472387\pi$
$60$	0	0
$61$	−11.1299	−1.42504	−0.712519	−	0.701652i	$-0.752446\pi$
$61$	−11.1299	−1.42504	−0.712519	+	0.701652i	$0.752446\pi$
$62$	0	0
$63$	−9.42521	−1.18746
$64$	0	0
$65$	2.57286	0.319124
$66$	0	0
$67$	13.7028	1.67406	0.837030	−	0.547157i	$-0.184290\pi$
$67$	13.7028	1.67406	0.837030	+	0.547157i	$0.184290\pi$
$68$	0	0
$69$	−17.1488	−2.06448
$70$	0	0
$71$	−11.1457	−1.32275	−0.661377	−	0.750054i	$-0.730028\pi$
$71$	−11.1457	−1.32275	−0.661377	+	0.750054i	$0.730028\pi$
$72$	0	0
$73$	−9.28931	−1.08723	−0.543616	−	0.839334i	$-0.682945\pi$
$73$	−9.28931	−1.08723	−0.543616	+	0.839334i	$0.682945\pi$
$74$	0	0
$75$	−2.36865	−0.273508
$76$	0	0
$77$	7.95830	0.906932
$78$	0	0
$79$	4.85042	0.545715	0.272857	−	0.962054i	$-0.412031\pi$
$79$	4.85042	0.545715	0.272857	+	0.962054i	$0.412031\pi$
$80$	0	0
$81$	−10.0168	−1.11298
$82$	0	0
$83$	−9.42521	−1.03455	−0.517276	−	0.855819i	$-0.673054\pi$
$83$	−9.42521	−1.03455	−0.517276	+	0.855819i	$0.673054\pi$
$84$	0	0
$85$	0.922589	0.100069
$86$	0	0
$87$	−9.03571	−0.968730
$88$	0	0
$89$	−10.3667	−1.09887	−0.549435	−	0.835537i	$-0.685157\pi$
$89$	−10.3667	−1.09887	−0.549435	+	0.835537i	$0.685157\pi$
$90$	0	0
$91$	−9.28931	−0.973784
$92$	0	0
$93$	10.6203	1.10128
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	2.75815	0.280048	0.140024	−	0.990148i	$-0.455282\pi$
$97$	2.75815	0.280048	0.140024	+	0.990148i	$0.455282\pi$
$98$	0	0
$99$	−5.75409	−0.578308
$100$	0	0
$101$	18.5292	1.84373	0.921863	−	0.387515i	$-0.126666\pi$
$101$	18.5292	1.84373	0.921863	+	0.387515i	$0.126666\pi$
$102$	0	0
$103$	−5.44606	−0.536616	−0.268308	−	0.963333i	$-0.586465\pi$
$103$	−5.44606	−0.536616	−0.268308	+	0.963333i	$0.586465\pi$
$104$	0	0
$105$	8.55201	0.834591
$106$	0	0
$107$	−3.26077	−0.315230	−0.157615	−	0.987501i	$-0.550381\pi$
$107$	−3.26077	−0.315230	−0.157615	+	0.987501i	$0.550381\pi$
$108$	0	0
$109$	−15.6182	−1.49595	−0.747975	−	0.663726i	$-0.768974\pi$
$109$	−15.6182	−1.49595	−0.747975	+	0.663726i	$0.768974\pi$
$110$	0	0
$111$	9.79193	0.929409
$112$	0	0
$113$	−9.53329	−0.896816	−0.448408	−	0.893829i	$-0.648009\pi$
$113$	−9.53329	−0.896816	−0.448408	+	0.893829i	$0.648009\pi$
$114$	0	0
$115$	7.23992	0.675126
$116$	0	0
$117$	6.71645	0.620936
$118$	0	0
$119$	−3.33101	−0.305353
$120$	0	0
$121$	−6.14146	−0.558315
$122$	0	0
$123$	11.2210	1.01176
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	1.92976	0.171239	0.0856193	−	0.996328i	$-0.472713\pi$
$127$	1.92976	0.171239	0.0856193	+	0.996328i	$0.472713\pi$
$128$	0	0
$129$	−12.8504	−1.13142
$130$	0	0
$131$	14.8087	1.29384	0.646922	−	0.762556i	$-0.276056\pi$
$131$	14.8087	1.29384	0.646922	+	0.762556i	$0.276056\pi$
$132$	0	0
$133$	3.61050	0.313070
$134$	0	0
$135$	0.922589	0.0794038
$136$	0	0
$137$	17.5572	1.50002	0.750008	−	0.661428i	$-0.230049\pi$
$137$	17.5572	1.50002	0.750008	+	0.661428i	$0.230049\pi$
$138$	0	0
$139$	−4.23779	−0.359445	−0.179722	−	0.983717i	$-0.557520\pi$
$139$	−4.23779	−0.359445	−0.179722	+	0.983717i	$0.557520\pi$
$140$	0	0
$141$	−0.662017	−0.0557519
$142$	0	0
$143$	−5.67112	−0.474243
$144$	0	0
$145$	3.81471	0.316794
$146$	0	0
$147$	−14.2965	−1.17915
$148$	0	0
$149$	6.06373	0.496760	0.248380	−	0.968663i	$-0.420102\pi$
$149$	6.06373	0.496760	0.248380	+	0.968663i	$0.420102\pi$
$150$	0	0
$151$	−6.85428	−0.557794	−0.278897	−	0.960321i	$-0.589969\pi$
$151$	−6.85428	−0.557794	−0.278897	+	0.960321i	$0.589969\pi$
$152$	0	0
$153$	2.40842	0.194709
$154$	0	0
$155$	−4.48370	−0.360140
$156$	0	0
$157$	8.85042	0.706340	0.353170	−	0.935559i	$-0.385104\pi$
$157$	8.85042	0.706340	0.353170	+	0.935559i	$0.385104\pi$
$158$	0	0
$159$	−23.3765	−1.85388
$160$	0	0
$161$	−26.1397	−2.06010
$162$	0	0
$163$	−16.6840	−1.30680	−0.653398	−	0.757015i	$-0.726657\pi$
$163$	−16.6840	−1.30680	−0.653398	+	0.757015i	$0.726657\pi$
$164$	0	0
$165$	5.22100	0.406454
$166$	0	0
$167$	−10.5875	−0.819287	−0.409643	−	0.912246i	$-0.634347\pi$
$167$	−10.5875	−0.819287	−0.409643	+	0.912246i	$0.634347\pi$
$168$	0	0
$169$	−6.38040	−0.490800
$170$	0	0
$171$	−2.61050	−0.199630
$172$	0	0
$173$	−18.7879	−1.42842	−0.714208	−	0.699934i	$-0.753213\pi$
$173$	−18.7879	−1.42842	−0.714208	+	0.699934i	$0.753213\pi$
$174$	0	0
$175$	−3.61050	−0.272928
$176$	0	0
$177$	−3.15269	−0.236971
$178$	0	0
$179$	8.91136	0.666066	0.333033	−	0.942915i	$-0.391928\pi$
$179$	8.91136	0.666066	0.333033	+	0.942915i	$0.391928\pi$
$180$	0	0
$181$	6.96740	0.517883	0.258941	−	0.965893i	$-0.416626\pi$
$181$	6.96740	0.517883	0.258941	+	0.965893i	$0.416626\pi$
$182$	0	0
$183$	26.3629	1.94880
$184$	0	0
$185$	−4.13397	−0.303936
$186$	0	0
$187$	−2.03358	−0.148710
$188$	0	0
$189$	−3.33101	−0.242295
$190$	0	0
$191$	−23.9278	−1.73136	−0.865678	−	0.500600i	$-0.833113\pi$
$191$	−23.9278	−1.73136	−0.865678	+	0.500600i	$0.833113\pi$
$192$	0	0
$193$	−9.89961	−0.712589	−0.356295	−	0.934374i	$-0.615960\pi$
$193$	−9.89961	−0.712589	−0.356295	+	0.934374i	$0.615960\pi$
$194$	0	0
$195$	−6.09420	−0.436415
$196$	0	0
$197$	−12.3015	−0.876447	−0.438224	−	0.898866i	$-0.644392\pi$
$197$	−12.3015	−0.876447	−0.438224	+	0.898866i	$0.644392\pi$
$198$	0	0
$199$	−9.07741	−0.643481	−0.321740	−	0.946828i	$-0.604268\pi$
$199$	−9.07741	−0.643481	−0.321740	+	0.946828i	$0.604268\pi$
$200$	0	0
$201$	−32.4571	−2.28934
$202$	0	0
$203$	−13.7730	−0.966676
$204$	0	0
$205$	−4.73730	−0.330867
$206$	0	0
$207$	18.8998	1.31363
$208$	0	0
$209$	2.20421	0.152468
$210$	0	0
$211$	−5.40629	−0.372184	−0.186092	−	0.982532i	$-0.559582\pi$
$211$	−5.40629	−0.372184	−0.186092	+	0.982532i	$0.559582\pi$
$212$	0	0
$213$	26.4003	1.80892
$214$	0	0
$215$	5.42521	0.369996
$216$	0	0
$217$	16.1884	1.09894
$218$	0	0
$219$	22.0031	1.48683
$220$	0	0
$221$	2.37369	0.159672
$222$	0	0
$223$	2.85546	0.191216	0.0956079	−	0.995419i	$-0.469520\pi$
$223$	2.85546	0.191216	0.0956079	+	0.995419i	$0.469520\pi$
$224$	0	0
$225$	2.61050	0.174033
$226$	0	0
$227$	−3.73013	−0.247577	−0.123789	−	0.992309i	$-0.539504\pi$
$227$	−3.73013	−0.247577	−0.123789	+	0.992309i	$0.539504\pi$
$228$	0	0
$229$	−24.3744	−1.61071	−0.805353	−	0.592796i	$-0.798024\pi$
$229$	−24.3744	−1.61071	−0.805353	+	0.592796i	$0.798024\pi$
$230$	0	0
$231$	−18.8504	−1.24027
$232$	0	0
$233$	−20.4319	−1.33854	−0.669270	−	0.743020i	$-0.733393\pi$
$233$	−20.4319	−1.33854	−0.669270	+	0.743020i	$0.733393\pi$
$234$	0	0
$235$	0.279491	0.0182320
$236$	0	0
$237$	−11.4889	−0.746287
$238$	0	0
$239$	−16.2189	−1.04911	−0.524556	−	0.851376i	$-0.675769\pi$
$239$	−16.2189	−1.04911	−0.524556	+	0.851376i	$0.675769\pi$
$240$	0	0
$241$	9.43290	0.607626	0.303813	−	0.952732i	$-0.401740\pi$
$241$	9.43290	0.607626	0.303813	+	0.952732i	$0.401740\pi$
$242$	0	0
$243$	20.9585	1.34449
$244$	0	0
$245$	6.03571	0.385607
$246$	0	0
$247$	−2.57286	−0.163707
$248$	0	0
$249$	22.3250	1.41479
$250$	0	0
$251$	1.37582	0.0868411	0.0434205	−	0.999057i	$-0.486174\pi$
$251$	1.37582	0.0868411	0.0434205	+	0.999057i	$0.486174\pi$
$252$	0	0
$253$	−15.9583	−1.00329
$254$	0	0
$255$	−2.18529	−0.136848
$256$	0	0
$257$	10.3080	0.642997	0.321499	−	0.946910i	$-0.395813\pi$
$257$	10.3080	0.642997	0.321499	+	0.946910i	$0.395813\pi$
$258$	0	0
$259$	14.9257	0.927438
$260$	0	0
$261$	9.95830	0.616403
$262$	0	0
$263$	−15.6371	−0.964225	−0.482113	−	0.876109i	$-0.660130\pi$
$263$	−15.6371	−0.964225	−0.482113	+	0.876109i	$0.660130\pi$
$264$	0	0
$265$	9.86914	0.606256
$266$	0	0
$267$	24.5551	1.50275
$268$	0	0
$269$	−16.7944	−1.02397	−0.511986	−	0.858994i	$-0.671090\pi$
$269$	−16.7944	−1.02397	−0.511986	+	0.858994i	$0.671090\pi$
$270$	0	0
$271$	−20.9068	−1.27000	−0.634998	−	0.772514i	$-0.718999\pi$
$271$	−20.9068	−1.27000	−0.634998	+	0.772514i	$0.718999\pi$
$272$	0	0
$273$	22.0031	1.33169
$274$	0	0
$275$	−2.20421	−0.132919
$276$	0	0
$277$	20.4798	1.23051	0.615257	−	0.788327i	$-0.289052\pi$
$277$	20.4798	1.23051	0.615257	+	0.788327i	$0.289052\pi$
$278$	0	0
$279$	−11.7047	−0.700742
$280$	0	0
$281$	16.1467	0.963231	0.481616	−	0.876383i	$-0.340050\pi$
$281$	16.1467	0.963231	0.481616	+	0.876383i	$0.340050\pi$
$282$	0	0
$283$	5.19897	0.309047	0.154523	−	0.987989i	$-0.450616\pi$
$283$	5.19897	0.309047	0.154523	+	0.987989i	$0.450616\pi$
$284$	0	0
$285$	2.36865	0.140307
$286$	0	0
$287$	17.1040	1.00962
$288$	0	0
$289$	−16.1488	−0.949931
$290$	0	0
$291$	−6.53309	−0.382976
$292$	0	0
$293$	−7.79386	−0.455322	−0.227661	−	0.973740i	$-0.573108\pi$
$293$	−7.79386	−0.455322	−0.227661	+	0.973740i	$0.573108\pi$
$294$	0	0
$295$	1.33101	0.0774943
$296$	0	0
$297$	−2.03358	−0.118000
$298$	0	0
$299$	18.6273	1.07724
$300$	0	0
$301$	−19.5877	−1.12902
$302$	0	0
$303$	−43.8892	−2.52137
$304$	0	0
$305$	−11.1299	−0.637297
$306$	0	0
$307$	−6.27298	−0.358018	−0.179009	−	0.983847i	$-0.557289\pi$
$307$	−6.27298	−0.358018	−0.179009	+	0.983847i	$0.557289\pi$
$308$	0	0
$309$	12.8998	0.733844
$310$	0	0
$311$	−29.2634	−1.65938	−0.829688	−	0.558227i	$-0.811482\pi$
$311$	−29.2634	−1.65938	−0.829688	+	0.558227i	$0.811482\pi$
$312$	0	0
$313$	−20.4725	−1.15717	−0.578586	−	0.815621i	$-0.696395\pi$
$313$	−20.4725	−1.15717	−0.578586	+	0.815621i	$0.696395\pi$
$314$	0	0
$315$	−9.42521	−0.531050
$316$	0	0
$317$	−4.64814	−0.261066	−0.130533	−	0.991444i	$-0.541669\pi$
$317$	−4.64814	−0.261066	−0.130533	+	0.991444i	$0.541669\pi$
$318$	0	0
$319$	−8.40842	−0.470781
$320$	0	0
$321$	7.72362	0.431090
$322$	0	0
$323$	−0.922589	−0.0513342
$324$	0	0
$325$	2.57286	0.142717
$326$	0	0
$327$	36.9940	2.04577
$328$	0	0
$329$	−1.00910	−0.0556337
$330$	0	0
$331$	26.3891	1.45047	0.725237	−	0.688499i	$-0.241730\pi$
$331$	26.3891	1.45047	0.725237	+	0.688499i	$0.241730\pi$
$332$	0	0
$333$	−10.7917	−0.591383
$334$	0	0
$335$	13.7028	0.748662
$336$	0	0
$337$	17.1441	0.933899	0.466949	−	0.884284i	$-0.345353\pi$
$337$	17.1441	0.933899	0.466949	+	0.884284i	$0.345353\pi$
$338$	0	0
$339$	22.5810	1.22643
$340$	0	0
$341$	9.88302	0.535195
$342$	0	0
$343$	3.48157	0.187987
$344$	0	0
$345$	−17.1488	−0.923262
$346$	0	0
$347$	2.84273	0.152606	0.0763029	−	0.997085i	$-0.475688\pi$
$347$	2.84273	0.152606	0.0763029	+	0.997085i	$0.475688\pi$
$348$	0	0
$349$	−17.6487	−0.944711	−0.472355	−	0.881408i	$-0.656596\pi$
$349$	−17.6487	−0.944711	−0.472355	+	0.881408i	$0.656596\pi$
$350$	0	0
$351$	2.37369	0.126698
$352$	0	0
$353$	−1.88999	−0.100594	−0.0502970	−	0.998734i	$-0.516017\pi$
$353$	−1.88999	−0.100594	−0.0502970	+	0.998734i	$0.516017\pi$
$354$	0	0
$355$	−11.1457	−0.591553
$356$	0	0
$357$	7.88999	0.417583
$358$	0	0
$359$	28.8234	1.52124	0.760620	−	0.649198i	$-0.224895\pi$
$359$	28.8234	1.52124	0.760620	+	0.649198i	$0.224895\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	14.5470	0.763518
$364$	0	0
$365$	−9.28931	−0.486225
$366$	0	0
$367$	8.92716	0.465994	0.232997	−	0.972477i	$-0.425147\pi$
$367$	8.92716	0.465994	0.232997	+	0.972477i	$0.425147\pi$
$368$	0	0
$369$	−12.3667	−0.643786
$370$	0	0
$371$	−35.6325	−1.84995
$372$	0	0
$373$	−11.7939	−0.610663	−0.305331	−	0.952246i	$-0.598767\pi$
$373$	−11.7939	−0.610663	−0.305331	+	0.952246i	$0.598767\pi$
$374$	0	0
$375$	−2.36865	−0.122317
$376$	0	0
$377$	9.81471	0.505483
$378$	0	0
$379$	10.6354	0.546304	0.273152	−	0.961971i	$-0.411934\pi$
$379$	10.6354	0.546304	0.273152	+	0.961971i	$0.411934\pi$
$380$	0	0
$381$	−4.57093	−0.234176
$382$	0	0
$383$	−2.86358	−0.146322	−0.0731611	−	0.997320i	$-0.523309\pi$
$383$	−2.86358	−0.146322	−0.0731611	+	0.997320i	$0.523309\pi$
$384$	0	0
$385$	7.95830	0.405592
$386$	0	0
$387$	14.1625	0.719921
$388$	0	0
$389$	3.45536	0.175194	0.0875969	−	0.996156i	$-0.472081\pi$
$389$	3.45536	0.175194	0.0875969	+	0.996156i	$0.472081\pi$
$390$	0	0
$391$	6.67947	0.337795
$392$	0	0
$393$	−35.0767	−1.76938
$394$	0	0
$395$	4.85042	0.244051
$396$	0	0
$397$	−0.854282	−0.0428752	−0.0214376	−	0.999770i	$-0.506824\pi$
$397$	−0.854282	−0.0428752	−0.0214376	+	0.999770i	$0.506824\pi$
$398$	0	0
$399$	−8.55201	−0.428136
$400$	0	0
$401$	16.0336	0.800679	0.400339	−	0.916367i	$-0.368892\pi$
$401$	16.0336	0.800679	0.400339	+	0.916367i	$0.368892\pi$
$402$	0	0
$403$	−11.5359	−0.574646
$404$	0	0
$405$	−10.0168	−0.497738
$406$	0	0
$407$	9.11214	0.451672
$408$	0	0
$409$	13.8452	0.684600	0.342300	−	0.939591i	$-0.388794\pi$
$409$	13.8452	0.684600	0.342300	+	0.939591i	$0.388794\pi$
$410$	0	0
$411$	−41.5870	−2.05133
$412$	0	0
$413$	−4.80561	−0.236468
$414$	0	0
$415$	−9.42521	−0.462665
$416$	0	0
$417$	10.0378	0.491555
$418$	0	0
$419$	16.1783	0.790362	0.395181	−	0.918603i	$-0.370682\pi$
$419$	16.1783	0.790362	0.395181	+	0.918603i	$0.370682\pi$
$420$	0	0
$421$	−1.71593	−0.0836295	−0.0418147	−	0.999125i	$-0.513314\pi$
$421$	−1.71593	−0.0836295	−0.0418147	+	0.999125i	$0.513314\pi$
$422$	0	0
$423$	0.729612	0.0354750
$424$	0	0
$425$	0.922589	0.0447522
$426$	0	0
$427$	40.1845	1.94467
$428$	0	0
$429$	13.4329	0.648546
$430$	0	0
$431$	−30.8087	−1.48400	−0.742002	−	0.670398i	$-0.766123\pi$
$431$	−30.8087	−1.48400	−0.742002	+	0.670398i	$0.766123\pi$
$432$	0	0
$433$	30.3084	1.45653	0.728265	−	0.685296i	$-0.240327\pi$
$433$	30.3084	1.45653	0.728265	+	0.685296i	$0.240327\pi$
$434$	0	0
$435$	−9.03571	−0.433229
$436$	0	0
$437$	−7.23992	−0.346332
$438$	0	0
$439$	0.722955	0.0345048	0.0172524	−	0.999851i	$-0.494508\pi$
$439$	0.722955	0.0345048	0.0172524	+	0.999851i	$0.494508\pi$
$440$	0	0
$441$	15.7562	0.750296
$442$	0	0
$443$	2.32159	0.110302	0.0551510	−	0.998478i	$-0.482436\pi$
$443$	2.32159	0.110302	0.0551510	+	0.998478i	$0.482436\pi$
$444$	0	0
$445$	−10.3667	−0.491430
$446$	0	0
$447$	−14.3629	−0.679340
$448$	0	0
$449$	39.8039	1.87846	0.939230	−	0.343287i	$-0.111540\pi$
$449$	39.8039	1.87846	0.939230	+	0.343287i	$0.111540\pi$
$450$	0	0
$451$	10.4420	0.491695
$452$	0	0
$453$	16.2354	0.762805
$454$	0	0
$455$	−9.28931	−0.435489
$456$	0	0
$457$	−17.9657	−0.840398	−0.420199	−	0.907432i	$-0.638040\pi$
$457$	−17.9657	−0.840398	−0.420199	+	0.907432i	$0.638040\pi$
$458$	0	0
$459$	0.851171	0.0397293
$460$	0	0
$461$	−34.8840	−1.62471	−0.812355	−	0.583163i	$-0.801815\pi$
$461$	−34.8840	−1.62471	−0.812355	+	0.583163i	$0.801815\pi$
$462$	0	0
$463$	−18.4118	−0.855671	−0.427836	−	0.903857i	$-0.640724\pi$
$463$	−18.4118	−0.855671	−0.427836	+	0.903857i	$0.640724\pi$
$464$	0	0
$465$	10.6203	0.492505
$466$	0	0
$467$	−17.7541	−0.821561	−0.410781	−	0.911734i	$-0.634744\pi$
$467$	−17.7541	−0.821561	−0.410781	+	0.911734i	$0.634744\pi$
$468$	0	0
$469$	−49.4738	−2.28449
$470$	0	0
$471$	−20.9635	−0.965949
$472$	0	0
$473$	−11.9583	−0.549843
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	25.7634	1.17962
$478$	0	0
$479$	23.7545	1.08537	0.542685	−	0.839936i	$-0.317408\pi$
$479$	23.7545	1.08537	0.542685	+	0.839936i	$0.317408\pi$
$480$	0	0
$481$	−10.6361	−0.484965
$482$	0	0
$483$	61.9158	2.81727
$484$	0	0
$485$	2.75815	0.125241
$486$	0	0
$487$	28.9581	1.31222	0.656108	−	0.754667i	$-0.272201\pi$
$487$	28.9581	1.31222	0.656108	+	0.754667i	$0.272201\pi$
$488$	0	0
$489$	39.5187	1.78710
$490$	0	0
$491$	−2.40416	−0.108498	−0.0542491	−	0.998527i	$-0.517277\pi$
$491$	−2.40416	−0.108498	−0.0542491	+	0.998527i	$0.517277\pi$
$492$	0	0
$493$	3.51941	0.158506
$494$	0	0
$495$	−5.75409	−0.258627
$496$	0	0
$497$	40.2416	1.80508
$498$	0	0
$499$	−12.2756	−0.549533	−0.274766	−	0.961511i	$-0.588600\pi$
$499$	−12.2756	−0.549533	−0.274766	+	0.961511i	$0.588600\pi$
$500$	0	0
$501$	25.0781	1.12041
$502$	0	0
$503$	0.488277	0.0217712	0.0108856	−	0.999941i	$-0.496535\pi$
$503$	0.488277	0.0217712	0.0108856	+	0.999941i	$0.496535\pi$
$504$	0	0
$505$	18.5292	0.824540
$506$	0	0
$507$	15.1129	0.671188
$508$	0	0
$509$	−14.6246	−0.648223	−0.324111	−	0.946019i	$-0.605065\pi$
$509$	−14.6246	−0.648223	−0.324111	+	0.946019i	$0.605065\pi$
$510$	0	0
$511$	33.5390	1.48368
$512$	0	0
$513$	−0.922589	−0.0407333
$514$	0	0
$515$	−5.44606	−0.239982
$516$	0	0
$517$	−0.616058	−0.0270942
$518$	0	0
$519$	44.5019	1.95342
$520$	0	0
$521$	14.5450	0.637229	0.318615	−	0.947884i	$-0.396782\pi$
$521$	14.5450	0.637229	0.318615	+	0.947884i	$0.396782\pi$
$522$	0	0
$523$	22.0590	0.964573	0.482287	−	0.876014i	$-0.339806\pi$
$523$	22.0590	0.964573	0.482287	+	0.876014i	$0.339806\pi$
$524$	0	0
$525$	8.55201	0.373240
$526$	0	0
$527$	−4.13661	−0.180194
$528$	0	0
$529$	29.4164	1.27897
$530$	0	0
$531$	3.47460	0.150785
$532$	0	0
$533$	−12.1884	−0.527938
$534$	0	0
$535$	−3.26077	−0.140975
$536$	0	0
$537$	−21.1079	−0.910872
$538$	0	0
$539$	−13.3040	−0.573042
$540$	0	0
$541$	19.2388	0.827139	0.413570	−	0.910472i	$-0.364282\pi$
$541$	19.2388	0.827139	0.413570	+	0.910472i	$0.364282\pi$
$542$	0	0
$543$	−16.5033	−0.708226
$544$	0	0
$545$	−15.6182	−0.669010
$546$	0	0
$547$	−44.3833	−1.89769	−0.948846	−	0.315740i	$-0.897747\pi$
$547$	−44.3833	−1.89769	−0.948846	+	0.315740i	$0.897747\pi$
$548$	0	0
$549$	−29.0546	−1.24002
$550$	0	0
$551$	−3.81471	−0.162512
$552$	0	0
$553$	−17.5124	−0.744705
$554$	0	0
$555$	9.79193	0.415644
$556$	0	0
$557$	−16.1030	−0.682307	−0.341154	−	0.940008i	$-0.610818\pi$
$557$	−16.1030	−0.682307	−0.341154	+	0.940008i	$0.610818\pi$
$558$	0	0
$559$	13.9583	0.590373
$560$	0	0
$561$	4.81684	0.203367
$562$	0	0
$563$	−0.107683	−0.00453828	−0.00226914	−	0.999997i	$-0.500722\pi$
$563$	−0.107683	−0.00453828	−0.00226914	+	0.999997i	$0.500722\pi$
$564$	0	0
$565$	−9.53329	−0.401068
$566$	0	0
$567$	36.1656	1.51881
$568$	0	0
$569$	17.7842	0.745554	0.372777	−	0.927921i	$-0.378406\pi$
$569$	17.7842	0.745554	0.372777	+	0.927921i	$0.378406\pi$
$570$	0	0
$571$	26.5902	1.11276	0.556382	−	0.830927i	$-0.312189\pi$
$571$	26.5902	1.11276	0.556382	+	0.830927i	$0.312189\pi$
$572$	0	0
$573$	56.6766	2.36770
$574$	0	0
$575$	7.23992	0.301925
$576$	0	0
$577$	33.3565	1.38865	0.694324	−	0.719663i	$-0.255704\pi$
$577$	33.3565	1.38865	0.694324	+	0.719663i	$0.255704\pi$
$578$	0	0
$579$	23.4487	0.974495
$580$	0	0
$581$	34.0297	1.41179
$582$	0	0
$583$	−21.7537	−0.900944
$584$	0	0
$585$	6.71645	0.277691
$586$	0	0
$587$	29.0322	1.19829	0.599143	−	0.800642i	$-0.295508\pi$
$587$	29.0322	1.19829	0.599143	+	0.800642i	$0.295508\pi$
$588$	0	0
$589$	4.48370	0.184748
$590$	0	0
$591$	29.1380	1.19858
$592$	0	0
$593$	0.722955	0.0296882	0.0148441	−	0.999890i	$-0.495275\pi$
$593$	0.722955	0.0296882	0.0148441	+	0.999890i	$0.495275\pi$
$594$	0	0
$595$	−3.33101	−0.136558
$596$	0	0
$597$	21.5012	0.879986
$598$	0	0
$599$	20.7516	0.847889	0.423945	−	0.905688i	$-0.360645\pi$
$599$	20.7516	0.847889	0.423945	+	0.905688i	$0.360645\pi$
$600$	0	0
$601$	0.919478	0.0375063	0.0187532	−	0.999824i	$-0.494030\pi$
$601$	0.919478	0.0375063	0.0187532	+	0.999824i	$0.494030\pi$
$602$	0	0
$603$	35.7711	1.45671
$604$	0	0
$605$	−6.14146	−0.249686
$606$	0	0
$607$	3.23416	0.131271	0.0656353	−	0.997844i	$-0.479093\pi$
$607$	3.23416	0.131271	0.0656353	+	0.997844i	$0.479093\pi$
$608$	0	0
$609$	32.6234	1.32197
$610$	0	0
$611$	0.719092	0.0290913
$612$	0	0
$613$	−47.0392	−1.89990	−0.949948	−	0.312408i	$-0.898864\pi$
$613$	−47.0392	−1.89990	−0.949948	+	0.312408i	$0.898864\pi$
$614$	0	0
$615$	11.2210	0.452474
$616$	0	0
$617$	−32.1510	−1.29435	−0.647174	−	0.762342i	$-0.724049\pi$
$617$	−32.1510	−1.29435	−0.647174	+	0.762342i	$0.724049\pi$
$618$	0	0
$619$	25.7727	1.03589	0.517946	−	0.855413i	$-0.326697\pi$
$619$	25.7727	1.03589	0.517946	+	0.855413i	$0.326697\pi$
$620$	0	0
$621$	6.67947	0.268038
$622$	0	0
$623$	37.4290	1.49956
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−5.22100	−0.208507
$628$	0	0
$629$	−3.81396	−0.152072
$630$	0	0
$631$	−35.0133	−1.39386	−0.696929	−	0.717140i	$-0.745451\pi$
$631$	−35.0133	−1.39386	−0.696929	+	0.717140i	$0.745451\pi$
$632$	0	0
$633$	12.8056	0.508977
$634$	0	0
$635$	1.92976	0.0765802
$636$	0	0
$637$	15.5290	0.615283
$638$	0	0
$639$	−29.0959	−1.15102
$640$	0	0
$641$	28.0340	1.10728	0.553638	−	0.832758i	$-0.313239\pi$
$641$	28.0340	1.10728	0.553638	+	0.832758i	$0.313239\pi$
$642$	0	0
$643$	−19.1831	−0.756508	−0.378254	−	0.925702i	$-0.623475\pi$
$643$	−19.1831	−0.756508	−0.378254	+	0.925702i	$0.623475\pi$
$644$	0	0
$645$	−12.8504	−0.505985
$646$	0	0
$647$	18.0046	0.707833	0.353916	−	0.935277i	$-0.384850\pi$
$647$	18.0046	0.707833	0.353916	+	0.935277i	$0.384850\pi$
$648$	0	0
$649$	−2.93382	−0.115163
$650$	0	0
$651$	−38.3446	−1.50285
$652$	0	0
$653$	−22.8130	−0.892741	−0.446370	−	0.894848i	$-0.647284\pi$
$653$	−22.8130	−0.892741	−0.446370	+	0.894848i	$0.647284\pi$
$654$	0	0
$655$	14.8087	0.578624
$656$	0	0
$657$	−24.2497	−0.946072
$658$	0	0
$659$	−46.1401	−1.79736	−0.898682	−	0.438601i	$-0.855474\pi$
$659$	−46.1401	−1.79736	−0.898682	+	0.438601i	$0.855474\pi$
$660$	0	0
$661$	−8.44511	−0.328477	−0.164238	−	0.986421i	$-0.552517\pi$
$661$	−8.44511	−0.328477	−0.164238	+	0.986421i	$0.552517\pi$
$662$	0	0
$663$	−5.62244	−0.218358
$664$	0	0
$665$	3.61050	0.140009
$666$	0	0
$667$	27.6182	1.06938
$668$	0	0
$669$	−6.76359	−0.261495
$670$	0	0
$671$	24.5327	0.947073
$672$	0	0
$673$	10.8334	0.417598	0.208799	−	0.977959i	$-0.433045\pi$
$673$	10.8334	0.417598	0.208799	+	0.977959i	$0.433045\pi$
$674$	0	0
$675$	0.922589	0.0355105
$676$	0	0
$677$	4.74202	0.182251	0.0911254	−	0.995839i	$-0.470954\pi$
$677$	4.74202	0.182251	0.0911254	+	0.995839i	$0.470954\pi$
$678$	0	0
$679$	−9.95830	−0.382164
$680$	0	0
$681$	8.83536	0.338572
$682$	0	0
$683$	−8.84963	−0.338622	−0.169311	−	0.985563i	$-0.554154\pi$
$683$	−8.84963	−0.338622	−0.169311	+	0.985563i	$0.554154\pi$
$684$	0	0
$685$	17.5572	0.670828
$686$	0	0
$687$	57.7344	2.20271
$688$	0	0
$689$	25.3919	0.967355
$690$	0	0
$691$	−2.41571	−0.0918980	−0.0459490	−	0.998944i	$-0.514631\pi$
$691$	−2.41571	−0.0918980	−0.0459490	+	0.998944i	$0.514631\pi$
$692$	0	0
$693$	20.7751	0.789182
$694$	0	0
$695$	−4.23779	−0.160749
$696$	0	0
$697$	−4.37058	−0.165548
$698$	0	0
$699$	48.3960	1.83051
$700$	0	0
$701$	−3.62173	−0.136791	−0.0683955	−	0.997658i	$-0.521788\pi$
$701$	−3.62173	−0.136791	−0.0683955	+	0.997658i	$0.521788\pi$
$702$	0	0
$703$	4.13397	0.155916
$704$	0	0
$705$	−0.662017	−0.0249330
$706$	0	0
$707$	−66.8998	−2.51602
$708$	0	0
$709$	−36.9909	−1.38922	−0.694611	−	0.719385i	$-0.744424\pi$
$709$	−36.9909	−1.38922	−0.694611	+	0.719385i	$0.744424\pi$
$710$	0	0
$711$	12.6620	0.474863
$712$	0	0
$713$	−32.4616	−1.21570
$714$	0	0
$715$	−5.67112	−0.212088
$716$	0	0
$717$	38.4168	1.43470
$718$	0	0
$719$	−27.8175	−1.03742	−0.518709	−	0.854951i	$-0.673587\pi$
$719$	−27.8175	−1.03742	−0.518709	+	0.854951i	$0.673587\pi$
$720$	0	0
$721$	19.6630	0.732288
$722$	0	0
$723$	−22.3432	−0.830953
$724$	0	0
$725$	3.81471	0.141675
$726$	0	0
$727$	−37.5947	−1.39431	−0.697154	−	0.716921i	$-0.745551\pi$
$727$	−37.5947	−1.39431	−0.697154	+	0.716921i	$0.745551\pi$
$728$	0	0
$729$	−19.5930	−0.725665
$730$	0	0
$731$	5.00524	0.185125
$732$	0	0
$733$	−19.7201	−0.728378	−0.364189	−	0.931325i	$-0.618654\pi$
$733$	−19.7201	−0.728378	−0.364189	+	0.931325i	$0.618654\pi$
$734$	0	0
$735$	−14.2965	−0.527334
$736$	0	0
$737$	−30.2038	−1.11257
$738$	0	0
$739$	48.3629	1.77906	0.889528	−	0.456880i	$-0.151033\pi$
$739$	48.3629	1.77906	0.889528	+	0.456880i	$0.151033\pi$
$740$	0	0
$741$	6.09420	0.223876
$742$	0	0
$743$	41.0765	1.50695	0.753475	−	0.657476i	$-0.228376\pi$
$743$	41.0765	1.50695	0.753475	+	0.657476i	$0.228376\pi$
$744$	0	0
$745$	6.06373	0.222158
$746$	0	0
$747$	−24.6045	−0.900232
$748$	0	0
$749$	11.7730	0.430176
$750$	0	0
$751$	−0.0508527	−0.00185564	−0.000927821	−	1.00000i	$-0.500295\pi$
$751$	−0.0508527	−0.00185564	−0.000927821		1.00000i	$0.500295\pi$
$752$	0	0
$753$	−3.25884	−0.118759
$754$	0	0
$755$	−6.85428	−0.249453
$756$	0	0
$757$	27.7109	1.00717	0.503585	−	0.863946i	$-0.332014\pi$
$757$	27.7109	1.00717	0.503585	+	0.863946i	$0.332014\pi$
$758$	0	0
$759$	37.7996	1.37204
$760$	0	0
$761$	−13.4140	−0.486256	−0.243128	−	0.969994i	$-0.578174\pi$
$761$	−13.4140	−0.486256	−0.243128	+	0.969994i	$0.578174\pi$
$762$	0	0
$763$	56.3895	2.04144
$764$	0	0
$765$	2.40842	0.0870766
$766$	0	0
$767$	3.42450	0.123651
$768$	0	0
$769$	−36.5362	−1.31753	−0.658765	−	0.752349i	$-0.728921\pi$
$769$	−36.5362	−1.31753	−0.658765	+	0.752349i	$0.728921\pi$
$770$	0	0
$771$	−24.4161	−0.879325
$772$	0	0
$773$	46.0124	1.65495	0.827475	−	0.561503i	$-0.189777\pi$
$773$	46.0124	1.65495	0.827475	+	0.561503i	$0.189777\pi$
$774$	0	0
$775$	−4.48370	−0.161059
$776$	0	0
$777$	−35.3538	−1.26831
$778$	0	0
$779$	4.73730	0.169731
$780$	0	0
$781$	24.5675	0.879094
$782$	0	0
$783$	3.51941	0.125773
$784$	0	0
$785$	8.85042	0.315885
$786$	0	0
$787$	28.4971	1.01581	0.507907	−	0.861412i	$-0.330419\pi$
$787$	28.4971	1.01581	0.507907	+	0.861412i	$0.330419\pi$
$788$	0	0
$789$	37.0388	1.31862
$790$	0	0
$791$	34.4199	1.22383
$792$	0	0
$793$	−28.6357	−1.01688
$794$	0	0
$795$	−23.3765	−0.829080
$796$	0	0
$797$	−37.2978	−1.32116	−0.660578	−	0.750758i	$-0.729689\pi$
$797$	−37.2978	−1.32116	−0.660578	+	0.750758i	$0.729689\pi$
$798$	0	0
$799$	0.257856	0.00912228
$800$	0	0
$801$	−27.0623	−0.956200
$802$	0	0
$803$	20.4756	0.722568
$804$	0	0
$805$	−26.1397	−0.921304
$806$	0	0
$807$	39.7800	1.40032
$808$	0	0
$809$	−20.6609	−0.726398	−0.363199	−	0.931712i	$-0.618315\pi$
$809$	−20.6609	−0.726398	−0.363199	+	0.931712i	$0.618315\pi$
$810$	0	0
$811$	31.4210	1.10334	0.551671	−	0.834062i	$-0.313990\pi$
$811$	31.4210	1.10334	0.551671	+	0.834062i	$0.313990\pi$
$812$	0	0
$813$	49.5208	1.73677
$814$	0	0
$815$	−16.6840	−0.584417
$816$	0	0
$817$	−5.42521	−0.189804
$818$	0	0
$819$	−24.2497	−0.847354
$820$	0	0
$821$	40.0297	1.39705	0.698523	−	0.715587i	$-0.253841\pi$
$821$	40.0297	1.39705	0.698523	+	0.715587i	$0.253841\pi$
$822$	0	0
$823$	55.9610	1.95068	0.975338	−	0.220714i	$-0.0708389\pi$
$823$	55.9610	1.95068	0.975338	+	0.220714i	$0.0708389\pi$
$824$	0	0
$825$	5.22100	0.181772
$826$	0	0
$827$	−36.5868	−1.27225	−0.636123	−	0.771587i	$-0.719463\pi$
$827$	−36.5868	−1.27225	−0.636123	+	0.771587i	$0.719463\pi$
$828$	0	0
$829$	42.3474	1.47078	0.735392	−	0.677642i	$-0.236998\pi$
$829$	42.3474	1.47078	0.735392	+	0.677642i	$0.236998\pi$
$830$	0	0
$831$	−48.5096	−1.68278
$832$	0	0
$833$	5.56848	0.192936
$834$	0	0
$835$	−10.5875	−0.366396
$836$	0	0
$837$	−4.13661	−0.142982
$838$	0	0
$839$	−17.6966	−0.610954	−0.305477	−	0.952199i	$-0.598816\pi$
$839$	−17.6966	−0.610954	−0.305477	+	0.952199i	$0.598816\pi$
$840$	0	0
$841$	−14.4480	−0.498207
$842$	0	0
$843$	−38.2459	−1.31726
$844$	0	0
$845$	−6.38040	−0.219492
$846$	0	0
$847$	22.1737	0.761899
$848$	0	0
$849$	−12.3145	−0.422634
$850$	0	0
$851$	−29.9296	−1.02597
$852$	0	0
$853$	17.2272	0.589849	0.294924	−	0.955521i	$-0.404706\pi$
$853$	17.2272	0.589849	0.294924	+	0.955521i	$0.404706\pi$
$854$	0	0
$855$	−2.61050	−0.0892772
$856$	0	0
$857$	−36.3377	−1.24127	−0.620637	−	0.784098i	$-0.713126\pi$
$857$	−36.3377	−1.24127	−0.620637	+	0.784098i	$0.713126\pi$
$858$	0	0
$859$	−35.7915	−1.22119	−0.610595	−	0.791943i	$-0.709070\pi$
$859$	−35.7915	−1.22119	−0.610595	+	0.791943i	$0.709070\pi$
$860$	0	0
$861$	−40.5134	−1.38069
$862$	0	0
$863$	−11.9672	−0.407368	−0.203684	−	0.979037i	$-0.565292\pi$
$863$	−11.9672	−0.407368	−0.203684	+	0.979037i	$0.565292\pi$
$864$	0	0
$865$	−18.7879	−0.638807
$866$	0	0
$867$	38.2509	1.29907
$868$	0	0
$869$	−10.6913	−0.362679
$870$	0	0
$871$	35.2553	1.19458
$872$	0	0
$873$	7.20015	0.243688
$874$	0	0
$875$	−3.61050	−0.122057
$876$	0	0
$877$	−53.5704	−1.80894	−0.904471	−	0.426534i	$-0.859734\pi$
$877$	−53.5704	−1.80894	−0.904471	+	0.426534i	$0.859734\pi$
$878$	0	0
$879$	18.4609	0.622671
$880$	0	0
$881$	−52.7852	−1.77838	−0.889189	−	0.457540i	$-0.848731\pi$
$881$	−52.7852	−1.77838	−0.889189	+	0.457540i	$0.848731\pi$
$882$	0	0
$883$	47.2665	1.59065	0.795323	−	0.606186i	$-0.207301\pi$
$883$	47.2665	1.59065	0.795323	+	0.606186i	$0.207301\pi$
$884$	0	0
$885$	−3.15269	−0.105977
$886$	0	0
$887$	30.3904	1.02041	0.510204	−	0.860053i	$-0.329570\pi$
$887$	30.3904	1.02041	0.510204	+	0.860053i	$0.329570\pi$
$888$	0	0
$889$	−6.96740	−0.233679
$890$	0	0
$891$	22.0791	0.739678
$892$	0	0
$893$	−0.279491	−0.00935282
$894$	0	0
$895$	8.91136	0.297874
$896$	0	0
$897$	−44.1215	−1.47317
$898$	0	0
$899$	−17.1040	−0.570451
$900$	0	0
$901$	9.10516	0.303337
$902$	0	0
$903$	46.3964	1.54398
$904$	0	0
$905$	6.96740	0.231604
$906$	0	0
$907$	22.1389	0.735111	0.367556	−	0.930001i	$-0.380195\pi$
$907$	22.1389	0.735111	0.367556	+	0.930001i	$0.380195\pi$
$908$	0	0
$909$	48.3705	1.60435
$910$	0	0
$911$	−43.3159	−1.43512	−0.717560	−	0.696497i	$-0.754741\pi$
$911$	−43.3159	−1.43512	−0.717560	+	0.696497i	$0.754741\pi$
$912$	0	0
$913$	20.7751	0.687557
$914$	0	0
$915$	26.3629	0.871529
$916$	0	0
$917$	−53.4669	−1.76563
$918$	0	0
$919$	−1.68725	−0.0556571	−0.0278286	−	0.999613i	$-0.508859\pi$
$919$	−1.68725	−0.0556571	−0.0278286	+	0.999613i	$0.508859\pi$
$920$	0	0
$921$	14.8585	0.489604
$922$	0	0
$923$	−28.6764	−0.943894
$924$	0	0
$925$	−4.13397	−0.135924
$926$	0	0
$927$	−14.2169	−0.466946
$928$	0	0
$929$	−29.4360	−0.965764	−0.482882	−	0.875685i	$-0.660410\pi$
$929$	−29.4360	−0.965764	−0.482882	+	0.875685i	$0.660410\pi$
$930$	0	0
$931$	−6.03571	−0.197812
$932$	0	0
$933$	69.3148	2.26926
$934$	0	0
$935$	−2.03358	−0.0665052
$936$	0	0
$937$	−49.5540	−1.61886	−0.809430	−	0.587217i	$-0.800224\pi$
$937$	−49.5540	−1.61886	−0.809430	+	0.587217i	$0.800224\pi$
$938$	0	0
$939$	48.4921	1.58248
$940$	0	0
$941$	−57.9201	−1.88814	−0.944071	−	0.329743i	$-0.893038\pi$
$941$	−57.9201	−1.88814	−0.944071	+	0.329743i	$0.893038\pi$
$942$	0	0
$943$	−34.2977	−1.11689
$944$	0	0
$945$	−3.33101	−0.108358
$946$	0	0
$947$	26.6304	0.865370	0.432685	−	0.901545i	$-0.357566\pi$
$947$	26.6304	0.865370	0.432685	+	0.901545i	$0.357566\pi$
$948$	0	0
$949$	−23.9001	−0.775829
$950$	0	0
$951$	11.0098	0.357018
$952$	0	0
$953$	−0.0289722	−0.000938500 0	−0.000469250	−	1.00000i	$-0.500149\pi$
$953$	−0.0289722	−0.000938500 0	−0.000469250		1.00000i	$0.500149\pi$
$954$	0	0
$955$	−23.9278	−0.774286
$956$	0	0
$957$	19.9166	0.643812
$958$	0	0
$959$	−63.3904	−2.04698
$960$	0	0
$961$	−10.8964	−0.351498
$962$	0	0
$963$	−8.51224	−0.274303
$964$	0	0
$965$	−9.89961	−0.318680
$966$	0	0
$967$	23.9367	0.769751	0.384876	−	0.922968i	$-0.374244\pi$
$967$	23.9367	0.769751	0.384876	+	0.922968i	$0.374244\pi$
$968$	0	0
$969$	2.18529	0.0702016
$970$	0	0
$971$	−18.5867	−0.596477	−0.298238	−	0.954491i	$-0.596399\pi$
$971$	−18.5867	−0.596477	−0.298238	+	0.954491i	$0.596399\pi$
$972$	0	0
$973$	15.3005	0.490513
$974$	0	0
$975$	−6.09420	−0.195171
$976$	0	0
$977$	−23.6982	−0.758172	−0.379086	−	0.925361i	$-0.623762\pi$
$977$	−23.6982	−0.758172	−0.379086	+	0.925361i	$0.623762\pi$
$978$	0	0
$979$	22.8504	0.730303
$980$	0	0
$981$	−40.7713	−1.30173
$982$	0	0
$983$	−31.0827	−0.991385	−0.495693	−	0.868498i	$-0.665086\pi$
$983$	−31.0827	−0.991385	−0.495693	+	0.868498i	$0.665086\pi$
$984$	0	0
$985$	−12.3015	−0.391959
$986$	0	0
$987$	2.39021	0.0760813
$988$	0	0
$989$	39.2781	1.24897
$990$	0	0
$991$	−3.58772	−0.113968	−0.0569838	−	0.998375i	$-0.518148\pi$
$991$	−3.58772	−0.113968	−0.0569838	+	0.998375i	$0.518148\pi$
$992$	0	0
$993$	−62.5064	−1.98358
$994$	0	0
$995$	−9.07741	−0.287773
$996$	0	0
$997$	−13.8409	−0.438346	−0.219173	−	0.975686i	$-0.570336\pi$
$997$	−13.8409	−0.438346	−0.219173	+	0.975686i	$0.570336\pi$
$998$	0	0
$999$	−3.81396	−0.120668

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3040.2.a.r.1.1	✓	4
4.3	odd	2		3040.2.a.t.1.4	yes	4
8.3	odd	2		6080.2.a.cd.1.1		4
8.5	even	2		6080.2.a.cf.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3040.2.a.r.1.1	✓	4	1.1	even	1	trivial
3040.2.a.t.1.4	yes	4	4.3	odd	2
6080.2.a.cd.1.1		4	8.3	odd	2
6080.2.a.cf.1.4		4	8.5	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3040.a (trivial)

\(f(q)\)	\(=\)	\(q-2.36865 q^{3} +1.00000 q^{5} -3.61050 q^{7} +2.61050 q^{9} -2.20421 q^{11} +2.57286 q^{13} -2.36865 q^{15} +0.922589 q^{17} -1.00000 q^{19} +8.55201 q^{21} +7.23992 q^{23} +1.00000 q^{25} +0.922589 q^{27} +3.81471 q^{29} -4.48370 q^{31} +5.22100 q^{33} -3.61050 q^{35} -4.13397 q^{37} -6.09420 q^{39} -4.73730 q^{41} +5.42521 q^{43} +2.61050 q^{45} +0.279491 q^{47} +6.03571 q^{49} -2.18529 q^{51} +9.86914 q^{53} -2.20421 q^{55} +2.36865 q^{57} +1.33101 q^{59} -11.1299 q^{61} -9.42521 q^{63} +2.57286 q^{65} +13.7028 q^{67} -17.1488 q^{69} -11.1457 q^{71} -9.28931 q^{73} -2.36865 q^{75} +7.95830 q^{77} +4.85042 q^{79} -10.0168 q^{81} -9.42521 q^{83} +0.922589 q^{85} -9.03571 q^{87} -10.3667 q^{89} -9.28931 q^{91} +10.6203 q^{93} -1.00000 q^{95} +2.75815 q^{97} -5.75409 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} + 4 q^{5} - 5 q^{7} + q^{9} - 6 q^{11} - q^{13} - q^{15} - q^{17} - 4 q^{19} + 5 q^{21} - 5 q^{23} + 4 q^{25} - q^{27} + 3 q^{29} - 16 q^{31} + 2 q^{33} - 5 q^{35} - 8 q^{37} - 13 q^{39}+ \cdots + 10 q^{99}+O(q^{100}) \) 4 * q - q^3 + 4 * q^5 - 5 * q^7 + q^9 - 6 * q^11 - q^13 - q^15 - q^17 - 4 * q^19 + 5 * q^21 - 5 * q^23 + 4 * q^25 - q^27 + 3 * q^29 - 16 * q^31 + 2 * q^33 - 5 * q^35 - 8 * q^37 - 13 * q^39 - 2 * q^41 + q^45 + 2 * q^47 - 7 * q^49 - 21 * q^51 + 13 * q^53 - 6 * q^55 + q^57 - 5 * q^59 - 2 * q^61 - 16 * q^63 - q^65 + q^67 - 11 * q^69 - 22 * q^71 + 9 * q^73 - q^75 - 4 * q^77 - 24 * q^79 - 24 * q^81 - 16 * q^83 - q^85 - 5 * q^87 + 9 * q^91 - 14 * q^93 - 4 * q^95 + 12 * q^97 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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