LMFDB - Embedded newform 4840.2.a.bc.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4840,2,Mod(1,4840)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4840.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4840 = 2^{3} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4840.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-2,0,6,0,-4,0,4,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$38.6475945783$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.22733568.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 8x^{4} - 2x^{3} + 16x^{2} + 8x - 2 $ x^6 - 8x^4 - 2x^3 + 16x^2 + 8x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.27997$ of defining polynomial
Character	$\chi$	$=$	4840.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.19828 q^{3} +1.00000 q^{5} -2.58485 q^{7} +1.83244 q^{9} -1.66906 q^{13} +2.19828 q^{15} -6.39656 q^{17} -0.330943 q^{19} -5.68224 q^{21} +1.33727 q^{23} +1.00000 q^{25} -2.56662 q^{27} -0.941551 q^{29} +5.78263 q^{31} -2.58485 q^{35} -7.41429 q^{37} -3.66906 q^{39} -5.49063 q^{41} +4.54413 q^{43} +1.83244 q^{45} -7.21601 q^{47} -0.318527 q^{49} -14.0614 q^{51} +6.55577 q^{53} -0.727506 q^{57} -7.74191 q^{59} -3.88821 q^{61} -4.73660 q^{63} -1.66906 q^{65} +9.68736 q^{67} +2.93969 q^{69} -11.4882 q^{71} +12.1229 q^{73} +2.19828 q^{75} -7.61507 q^{79} -11.1395 q^{81} -9.29886 q^{83} -6.39656 q^{85} -2.06979 q^{87} -17.7651 q^{89} +4.31427 q^{91} +12.7118 q^{93} -0.330943 q^{95} +4.03731 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} + 6 q^{5} - 4 q^{7} + 4 q^{9} - 2 q^{15} - 8 q^{17} - 12 q^{19} + 8 q^{21} - 8 q^{23} + 6 q^{25} - 14 q^{27} - 16 q^{29} - 4 q^{31} - 4 q^{35} + 8 q^{37} - 12 q^{39} - 32 q^{41} + 4 q^{43}+ \cdots - 12 q^{95}+O(q^{100}) $ 6 * q - 2 * q^3 + 6 * q^5 - 4 * q^7 + 4 * q^9 - 2 * q^15 - 8 * q^17 - 12 * q^19 + 8 * q^21 - 8 * q^23 + 6 * q^25 - 14 * q^27 - 16 * q^29 - 4 * q^31 - 4 * q^35 + 8 * q^37 - 12 * q^39 - 32 * q^41 + 4 * q^43 + 4 * q^45 - 6 * q^47 + 16 * q^49 - 40 * q^51 + 8 * q^53 + 16 * q^57 + 4 * q^59 - 16 * q^61 - 28 * q^63 - 2 * q^67 + 8 * q^69 - 28 * q^71 - 16 * q^73 - 2 * q^75 - 10 * q^81 - 12 * q^83 - 8 * q^85 + 24 * q^87 + 18 * q^89 - 24 * q^91 + 20 * q^93 - 12 * q^95

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
$3$	2.19828	1.26918	0.634589	−	0.772850i	$-0.281169\pi$
$3$	2.19828	1.26918	0.634589	+	0.772850i	$0.281169\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.58485	−0.976983	−0.488492	−	0.872569i	$-0.662453\pi$
$7$	−2.58485	−0.976983	−0.488492	+	0.872569i	$0.662453\pi$
$8$	0	0
$9$	1.83244	0.610814
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.66906	−0.462913	−0.231457	−	0.972845i	$-0.574349\pi$
$13$	−1.66906	−0.462913	−0.231457	+	0.972845i	$0.574349\pi$
$14$	0	0
$15$	2.19828	0.567594
$16$	0	0
$17$	−6.39656	−1.55139	−0.775697	−	0.631105i	$-0.782602\pi$
$17$	−6.39656	−1.55139	−0.775697	+	0.631105i	$0.782602\pi$
$18$	0	0
$19$	−0.330943	−0.0759236	−0.0379618	−	0.999279i	$-0.512087\pi$
$19$	−0.330943	−0.0759236	−0.0379618	+	0.999279i	$0.512087\pi$
$20$	0	0
$21$	−5.68224	−1.23997
$22$	0	0
$23$	1.33727	0.278839	0.139420	−	0.990233i	$-0.455476\pi$
$23$	1.33727	0.278839	0.139420	+	0.990233i	$0.455476\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.56662	−0.493947
$28$	0	0
$29$	−0.941551	−0.174842	−0.0874208	−	0.996171i	$-0.527862\pi$
$29$	−0.941551	−0.174842	−0.0874208	+	0.996171i	$0.527862\pi$
$30$	0	0
$31$	5.78263	1.03859	0.519295	−	0.854595i	$-0.326194\pi$
$31$	5.78263	1.03859	0.519295	+	0.854595i	$0.326194\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.58485	−0.436920
$36$	0	0
$37$	−7.41429	−1.21890	−0.609451	−	0.792824i	$-0.708610\pi$
$37$	−7.41429	−1.21890	−0.609451	+	0.792824i	$0.708610\pi$
$38$	0	0
$39$	−3.66906	−0.587519
$40$	0	0
$41$	−5.49063	−0.857492	−0.428746	−	0.903425i	$-0.641045\pi$
$41$	−5.49063	−0.857492	−0.428746	+	0.903425i	$0.641045\pi$
$42$	0	0
$43$	4.54413	0.692973	0.346487	−	0.938055i	$-0.387375\pi$
$43$	4.54413	0.692973	0.346487	+	0.938055i	$0.387375\pi$
$44$	0	0
$45$	1.83244	0.273164
$46$	0	0
$47$	−7.21601	−1.05256	−0.526281	−	0.850310i	$-0.676414\pi$
$47$	−7.21601	−1.05256	−0.526281	+	0.850310i	$0.676414\pi$
$48$	0	0
$49$	−0.318527	−0.0455039
$50$	0	0
$51$	−14.0614	−1.96900
$52$	0	0
$53$	6.55577	0.900505	0.450252	−	0.892901i	$-0.351334\pi$
$53$	6.55577	0.900505	0.450252	+	0.892901i	$0.351334\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.727506	−0.0963606
$58$	0	0
$59$	−7.74191	−1.00791	−0.503955	−	0.863730i	$-0.668122\pi$
$59$	−7.74191	−1.00791	−0.503955	+	0.863730i	$0.668122\pi$
$60$	0	0
$61$	−3.88821	−0.497834	−0.248917	−	0.968525i	$-0.580075\pi$
$61$	−3.88821	−0.497834	−0.248917	+	0.968525i	$0.580075\pi$
$62$	0	0
$63$	−4.73660	−0.596755
$64$	0	0
$65$	−1.66906	−0.207021
$66$	0	0
$67$	9.68736	1.18350	0.591750	−	0.806122i	$-0.298437\pi$
$67$	9.68736	1.18350	0.591750	+	0.806122i	$0.298437\pi$
$68$	0	0
$69$	2.93969	0.353897
$70$	0	0
$71$	−11.4882	−1.36340	−0.681701	−	0.731631i	$-0.738760\pi$
$71$	−11.4882	−1.36340	−0.681701	+	0.731631i	$0.738760\pi$
$72$	0	0
$73$	12.1229	1.41888	0.709439	−	0.704767i	$-0.248949\pi$
$73$	12.1229	1.41888	0.709439	+	0.704767i	$0.248949\pi$
$74$	0	0
$75$	2.19828	0.253836
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−7.61507	−0.856762	−0.428381	−	0.903598i	$-0.640916\pi$
$79$	−7.61507	−0.856762	−0.428381	+	0.903598i	$0.640916\pi$
$80$	0	0
$81$	−11.1395	−1.23772
$82$	0	0
$83$	−9.29886	−1.02068	−0.510341	−	0.859972i	$-0.670481\pi$
$83$	−9.29886	−1.02068	−0.510341	+	0.859972i	$0.670481\pi$
$84$	0	0
$85$	−6.39656	−0.693805
$86$	0	0
$87$	−2.06979	−0.221905
$88$	0	0
$89$	−17.7651	−1.88309	−0.941546	−	0.336885i	$-0.890627\pi$
$89$	−17.7651	−1.88309	−0.941546	+	0.336885i	$0.890627\pi$
$90$	0	0
$91$	4.31427	0.452258
$92$	0	0
$93$	12.7118	1.31816
$94$	0	0
$95$	−0.330943	−0.0339541
$96$	0	0
$97$	4.03731	0.409927	0.204963	−	0.978770i	$-0.434292\pi$
$97$	4.03731	0.409927	0.204963	+	0.978770i	$0.434292\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	9.90476	0.985560	0.492780	−	0.870154i	$-0.335981\pi$
$101$	9.90476	0.985560	0.492780	+	0.870154i	$0.335981\pi$
$102$	0	0
$103$	−11.9304	−1.17553	−0.587766	−	0.809031i	$-0.699993\pi$
$103$	−11.9304	−1.17553	−0.587766	+	0.809031i	$0.699993\pi$
$104$	0	0
$105$	−5.68224	−0.554530
$106$	0	0
$107$	−11.4320	−1.10517	−0.552585	−	0.833457i	$-0.686359\pi$
$107$	−11.4320	−1.10517	−0.552585	+	0.833457i	$0.686359\pi$
$108$	0	0
$109$	−13.2534	−1.26945	−0.634724	−	0.772739i	$-0.718886\pi$
$109$	−13.2534	−1.26945	−0.634724	+	0.772739i	$0.718886\pi$
$110$	0	0
$111$	−16.2987	−1.54700
$112$	0	0
$113$	14.4580	1.36010	0.680048	−	0.733168i	$-0.261959\pi$
$113$	14.4580	1.36010	0.680048	+	0.733168i	$0.261959\pi$
$114$	0	0
$115$	1.33727	0.124701
$116$	0	0
$117$	−3.05845	−0.282754
$118$	0	0
$119$	16.5342	1.51569
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−12.0700	−1.08831
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−17.9795	−1.59542	−0.797711	−	0.603040i	$-0.793956\pi$
$127$	−17.9795	−1.59542	−0.797711	+	0.603040i	$0.793956\pi$
$128$	0	0
$129$	9.98928	0.879507
$130$	0	0
$131$	9.70451	0.847887	0.423943	−	0.905689i	$-0.360646\pi$
$131$	9.70451	0.847887	0.423943	+	0.905689i	$0.360646\pi$
$132$	0	0
$133$	0.855440	0.0741761
$134$	0	0
$135$	−2.56662	−0.220900
$136$	0	0
$137$	19.4516	1.66186	0.830931	−	0.556375i	$-0.187808\pi$
$137$	19.4516	1.66186	0.830931	+	0.556375i	$0.187808\pi$
$138$	0	0
$139$	−1.88457	−0.159847	−0.0799235	−	0.996801i	$-0.525468\pi$
$139$	−1.88457	−0.159847	−0.0799235	+	0.996801i	$0.525468\pi$
$140$	0	0
$141$	−15.8628	−1.33589
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−0.941551	−0.0781915
$146$	0	0
$147$	−0.700213	−0.0577526
$148$	0	0
$149$	8.19423	0.671298	0.335649	−	0.941987i	$-0.391044\pi$
$149$	8.19423	0.671298	0.335649	+	0.941987i	$0.391044\pi$
$150$	0	0
$151$	−3.64408	−0.296551	−0.148275	−	0.988946i	$-0.547372\pi$
$151$	−3.64408	−0.296551	−0.148275	+	0.988946i	$0.547372\pi$
$152$	0	0
$153$	−11.7213	−0.947613
$154$	0	0
$155$	5.78263	0.464472
$156$	0	0
$157$	−8.12289	−0.648277	−0.324139	−	0.946010i	$-0.605074\pi$
$157$	−8.12289	−0.648277	−0.324139	+	0.946010i	$0.605074\pi$
$158$	0	0
$159$	14.4114	1.14290
$160$	0	0
$161$	−3.45664	−0.272421
$162$	0	0
$163$	12.1858	0.954464	0.477232	−	0.878777i	$-0.341640\pi$
$163$	12.1858	0.954464	0.477232	+	0.878777i	$0.341640\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	22.9941	1.77933	0.889667	−	0.456610i	$-0.150936\pi$
$167$	22.9941	1.77933	0.889667	+	0.456610i	$0.150936\pi$
$168$	0	0
$169$	−10.2142	−0.785711
$170$	0	0
$171$	−0.606434	−0.0463752
$172$	0	0
$173$	−13.0803	−0.994475	−0.497237	−	0.867615i	$-0.665652\pi$
$173$	−13.0803	−0.994475	−0.497237	+	0.867615i	$0.665652\pi$
$174$	0	0
$175$	−2.58485	−0.195397
$176$	0	0
$177$	−17.0189	−1.27922
$178$	0	0
$179$	2.75740	0.206098	0.103049	−	0.994676i	$-0.467140\pi$
$179$	2.75740	0.206098	0.103049	+	0.994676i	$0.467140\pi$
$180$	0	0
$181$	3.38392	0.251524	0.125762	−	0.992060i	$-0.459862\pi$
$181$	3.38392	0.251524	0.125762	+	0.992060i	$0.459862\pi$
$182$	0	0
$183$	−8.54737	−0.631840
$184$	0	0
$185$	−7.41429	−0.545109
$186$	0	0
$187$	0	0
$188$	0	0
$189$	6.63434	0.482577
$190$	0	0
$191$	−0.0917500	−0.00663880	−0.00331940	−	0.999994i	$-0.501057\pi$
$191$	−0.0917500	−0.00663880	−0.00331940	+	0.999994i	$0.501057\pi$
$192$	0	0
$193$	24.9932	1.79905	0.899525	−	0.436869i	$-0.143913\pi$
$193$	24.9932	1.79905	0.899525	+	0.436869i	$0.143913\pi$
$194$	0	0
$195$	−3.66906	−0.262747
$196$	0	0
$197$	−18.6454	−1.32843	−0.664216	−	0.747540i	$-0.731235\pi$
$197$	−18.6454	−1.32843	−0.664216	+	0.747540i	$0.731235\pi$
$198$	0	0
$199$	−21.9376	−1.55511	−0.777557	−	0.628812i	$-0.783541\pi$
$199$	−21.9376	−1.55511	−0.777557	+	0.628812i	$0.783541\pi$
$200$	0	0
$201$	21.2956	1.50207
$202$	0	0
$203$	2.43377	0.170817
$204$	0	0
$205$	−5.49063	−0.383482
$206$	0	0
$207$	2.45046	0.170319
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−16.8128	−1.15744	−0.578719	−	0.815527i	$-0.696447\pi$
$211$	−16.8128	−1.15744	−0.578719	+	0.815527i	$0.696447\pi$
$212$	0	0
$213$	−25.2544	−1.73040
$214$	0	0
$215$	4.54413	0.309907
$216$	0	0
$217$	−14.9473	−1.01469
$218$	0	0
$219$	26.6495	1.80081
$220$	0	0
$221$	10.6762	0.718161
$222$	0	0
$223$	0.156711	0.0104942	0.00524708	−	0.999986i	$-0.498330\pi$
$223$	0.156711	0.0104942	0.00524708	+	0.999986i	$0.498330\pi$
$224$	0	0
$225$	1.83244	0.122163
$226$	0	0
$227$	−4.26616	−0.283155	−0.141577	−	0.989927i	$-0.545217\pi$
$227$	−4.26616	−0.283155	−0.141577	+	0.989927i	$0.545217\pi$
$228$	0	0
$229$	−3.55280	−0.234776	−0.117388	−	0.993086i	$-0.537452\pi$
$229$	−3.55280	−0.234776	−0.117388	+	0.993086i	$0.537452\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.20904	0.144719	0.0723596	−	0.997379i	$-0.476947\pi$
$233$	2.20904	0.144719	0.0723596	+	0.997379i	$0.476947\pi$
$234$	0	0
$235$	−7.21601	−0.470720
$236$	0	0
$237$	−16.7401	−1.08738
$238$	0	0
$239$	−12.7323	−0.823586	−0.411793	−	0.911277i	$-0.635097\pi$
$239$	−12.7323	−0.823586	−0.411793	+	0.911277i	$0.635097\pi$
$240$	0	0
$241$	13.6950	0.882171	0.441085	−	0.897465i	$-0.354594\pi$
$241$	13.6950	0.882171	0.441085	+	0.897465i	$0.354594\pi$
$242$	0	0
$243$	−16.7879	−1.07694
$244$	0	0
$245$	−0.318527	−0.0203500
$246$	0	0
$247$	0.552363	0.0351460
$248$	0	0
$249$	−20.4415	−1.29543
$250$	0	0
$251$	−9.12212	−0.575783	−0.287892	−	0.957663i	$-0.592954\pi$
$251$	−9.12212	−0.575783	−0.287892	+	0.957663i	$0.592954\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−14.0614	−0.880562
$256$	0	0
$257$	23.2138	1.44804	0.724019	−	0.689780i	$-0.242293\pi$
$257$	23.2138	1.44804	0.724019	+	0.689780i	$0.242293\pi$
$258$	0	0
$259$	19.1649	1.19085
$260$	0	0
$261$	−1.72534	−0.106796
$262$	0	0
$263$	30.5012	1.88078	0.940392	−	0.340092i	$-0.110458\pi$
$263$	30.5012	1.88078	0.940392	+	0.340092i	$0.110458\pi$
$264$	0	0
$265$	6.55577	0.402718
$266$	0	0
$267$	−39.0526	−2.38998
$268$	0	0
$269$	−5.13465	−0.313065	−0.156533	−	0.987673i	$-0.550032\pi$
$269$	−5.13465	−0.313065	−0.156533	+	0.987673i	$0.550032\pi$
$270$	0	0
$271$	−12.6879	−0.770734	−0.385367	−	0.922763i	$-0.625925\pi$
$271$	−12.6879	−0.770734	−0.385367	+	0.922763i	$0.625925\pi$
$272$	0	0
$273$	9.48398	0.573996
$274$	0	0
$275$	0	0
$276$	0	0
$277$	12.3144	0.739899	0.369949	−	0.929052i	$-0.379375\pi$
$277$	12.3144	0.739899	0.369949	+	0.929052i	$0.379375\pi$
$278$	0	0
$279$	10.5963	0.634386
$280$	0	0
$281$	−13.3736	−0.797800	−0.398900	−	0.916994i	$-0.630608\pi$
$281$	−13.3736	−0.797800	−0.398900	+	0.916994i	$0.630608\pi$
$282$	0	0
$283$	11.0470	0.656678	0.328339	−	0.944560i	$-0.393511\pi$
$283$	11.0470	0.656678	0.328339	+	0.944560i	$0.393511\pi$
$284$	0	0
$285$	−0.727506	−0.0430938
$286$	0	0
$287$	14.1925	0.837756
$288$	0	0
$289$	23.9160	1.40682
$290$	0	0
$291$	8.87515	0.520270
$292$	0	0
$293$	−25.4592	−1.48734	−0.743672	−	0.668545i	$-0.766918\pi$
$293$	−25.4592	−1.48734	−0.743672	+	0.668545i	$0.766918\pi$
$294$	0	0
$295$	−7.74191	−0.450751
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.23197	−0.129078
$300$	0	0
$301$	−11.7459	−0.677023
$302$	0	0
$303$	21.7734	1.25085
$304$	0	0
$305$	−3.88821	−0.222638
$306$	0	0
$307$	30.1355	1.71992	0.859961	−	0.510360i	$-0.170488\pi$
$307$	30.1355	1.71992	0.859961	+	0.510360i	$0.170488\pi$
$308$	0	0
$309$	−26.2263	−1.49196
$310$	0	0
$311$	12.3904	0.702598	0.351299	−	0.936263i	$-0.385740\pi$
$311$	12.3904	0.702598	0.351299	+	0.936263i	$0.385740\pi$
$312$	0	0
$313$	26.2056	1.48123	0.740613	−	0.671932i	$-0.234535\pi$
$313$	26.2056	1.48123	0.740613	+	0.671932i	$0.234535\pi$
$314$	0	0
$315$	−4.73660	−0.266877
$316$	0	0
$317$	−16.9782	−0.953589	−0.476794	−	0.879015i	$-0.658201\pi$
$317$	−16.9782	−0.953589	−0.476794	+	0.879015i	$0.658201\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−25.1307	−1.40266
$322$	0	0
$323$	2.11690	0.117787
$324$	0	0
$325$	−1.66906	−0.0925826
$326$	0	0
$327$	−29.1348	−1.61116
$328$	0	0
$329$	18.6523	1.02834
$330$	0	0
$331$	−13.7482	−0.755668	−0.377834	−	0.925873i	$-0.623331\pi$
$331$	−13.7482	−0.755668	−0.377834	+	0.925873i	$0.623331\pi$
$332$	0	0
$333$	−13.5863	−0.744522
$334$	0	0
$335$	9.68736	0.529277
$336$	0	0
$337$	16.9287	0.922163	0.461081	−	0.887358i	$-0.347462\pi$
$337$	16.9287	0.922163	0.461081	+	0.887358i	$0.347462\pi$
$338$	0	0
$339$	31.7828	1.72620
$340$	0	0
$341$	0	0
$342$	0	0
$343$	18.9173	1.02144
$344$	0	0
$345$	2.93969	0.158268
$346$	0	0
$347$	−9.03447	−0.484996	−0.242498	−	0.970152i	$-0.577967\pi$
$347$	−9.03447	−0.484996	−0.242498	+	0.970152i	$0.577967\pi$
$348$	0	0
$349$	27.5646	1.47550	0.737748	−	0.675076i	$-0.235889\pi$
$349$	27.5646	1.47550	0.737748	+	0.675076i	$0.235889\pi$
$350$	0	0
$351$	4.28384	0.228654
$352$	0	0
$353$	9.53305	0.507393	0.253696	−	0.967284i	$-0.418354\pi$
$353$	9.53305	0.507393	0.253696	+	0.967284i	$0.418354\pi$
$354$	0	0
$355$	−11.4882	−0.609732
$356$	0	0
$357$	36.3468	1.92368
$358$	0	0
$359$	26.2842	1.38723	0.693615	−	0.720346i	$-0.256017\pi$
$359$	26.2842	1.38723	0.693615	+	0.720346i	$0.256017\pi$
$360$	0	0
$361$	−18.8905	−0.994236
$362$	0	0
$363$	0	0
$364$	0	0
$365$	12.1229	0.634541
$366$	0	0
$367$	−21.0794	−1.10033	−0.550167	−	0.835054i	$-0.685436\pi$
$367$	−21.0794	−1.10033	−0.550167	+	0.835054i	$0.685436\pi$
$368$	0	0
$369$	−10.0613	−0.523768
$370$	0	0
$371$	−16.9457	−0.879778
$372$	0	0
$373$	−4.86333	−0.251813	−0.125907	−	0.992042i	$-0.540184\pi$
$373$	−4.86333	−0.251813	−0.125907	+	0.992042i	$0.540184\pi$
$374$	0	0
$375$	2.19828	0.113519
$376$	0	0
$377$	1.57150	0.0809364
$378$	0	0
$379$	17.6659	0.907435	0.453718	−	0.891146i	$-0.350097\pi$
$379$	17.6659	0.907435	0.453718	+	0.891146i	$0.350097\pi$
$380$	0	0
$381$	−39.5240	−2.02488
$382$	0	0
$383$	−20.9804	−1.07205	−0.536024	−	0.844203i	$-0.680074\pi$
$383$	−20.9804	−1.07205	−0.536024	+	0.844203i	$0.680074\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.32685	0.423278
$388$	0	0
$389$	−9.45185	−0.479228	−0.239614	−	0.970868i	$-0.577021\pi$
$389$	−9.45185	−0.479228	−0.239614	+	0.970868i	$0.577021\pi$
$390$	0	0
$391$	−8.55391	−0.432590
$392$	0	0
$393$	21.3332	1.07612
$394$	0	0
$395$	−7.61507	−0.383156
$396$	0	0
$397$	26.5417	1.33209	0.666046	−	0.745911i	$-0.267986\pi$
$397$	26.5417	1.33209	0.666046	+	0.745911i	$0.267986\pi$
$398$	0	0
$399$	1.88050	0.0941426
$400$	0	0
$401$	9.91003	0.494883	0.247442	−	0.968903i	$-0.420410\pi$
$401$	9.91003	0.494883	0.247442	+	0.968903i	$0.420410\pi$
$402$	0	0
$403$	−9.65154	−0.480777
$404$	0	0
$405$	−11.1395	−0.553525
$406$	0	0
$407$	0	0
$408$	0	0
$409$	24.9962	1.23598	0.617991	−	0.786185i	$-0.287947\pi$
$409$	24.9962	1.23598	0.617991	+	0.786185i	$0.287947\pi$
$410$	0	0
$411$	42.7601	2.10920
$412$	0	0
$413$	20.0117	0.984711
$414$	0	0
$415$	−9.29886	−0.456463
$416$	0	0
$417$	−4.14281	−0.202874
$418$	0	0
$419$	31.6860	1.54796	0.773982	−	0.633208i	$-0.218262\pi$
$419$	31.6860	1.54796	0.773982	+	0.633208i	$0.218262\pi$
$420$	0	0
$421$	27.6659	1.34835	0.674176	−	0.738571i	$-0.264499\pi$
$421$	27.6659	1.34835	0.674176	+	0.738571i	$0.264499\pi$
$422$	0	0
$423$	−13.2229	−0.642920
$424$	0	0
$425$	−6.39656	−0.310279
$426$	0	0
$427$	10.0505	0.486375
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−8.37658	−0.403486	−0.201743	−	0.979439i	$-0.564661\pi$
$431$	−8.37658	−0.403486	−0.201743	+	0.979439i	$0.564661\pi$
$432$	0	0
$433$	16.9179	0.813021	0.406511	−	0.913646i	$-0.366745\pi$
$433$	16.9179	0.813021	0.406511	+	0.913646i	$0.366745\pi$
$434$	0	0
$435$	−2.06979	−0.0992390
$436$	0	0
$437$	−0.442559	−0.0211705
$438$	0	0
$439$	15.9993	0.763605	0.381802	−	0.924244i	$-0.375303\pi$
$439$	15.9993	0.763605	0.381802	+	0.924244i	$0.375303\pi$
$440$	0	0
$441$	−0.583683	−0.0277944
$442$	0	0
$443$	−21.6509	−1.02867	−0.514333	−	0.857591i	$-0.671960\pi$
$443$	−21.6509	−1.02867	−0.514333	+	0.857591i	$0.671960\pi$
$444$	0	0
$445$	−17.7651	−0.842144
$446$	0	0
$447$	18.0132	0.851997
$448$	0	0
$449$	10.9199	0.515341	0.257671	−	0.966233i	$-0.417045\pi$
$449$	10.9199	0.515341	0.257671	+	0.966233i	$0.417045\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−8.01070	−0.376376
$454$	0	0
$455$	4.31427	0.202256
$456$	0	0
$457$	−32.1926	−1.50591	−0.752953	−	0.658074i	$-0.771371\pi$
$457$	−32.1926	−1.50591	−0.752953	+	0.658074i	$0.771371\pi$
$458$	0	0
$459$	16.4176	0.766306
$460$	0	0
$461$	−1.21757	−0.0567081	−0.0283540	−	0.999598i	$-0.509027\pi$
$461$	−1.21757	−0.0567081	−0.0283540	+	0.999598i	$0.509027\pi$
$462$	0	0
$463$	−25.9855	−1.20765	−0.603824	−	0.797118i	$-0.706357\pi$
$463$	−25.9855	−1.20765	−0.603824	+	0.797118i	$0.706357\pi$
$464$	0	0
$465$	12.7118	0.589498
$466$	0	0
$467$	20.3664	0.942442	0.471221	−	0.882015i	$-0.343813\pi$
$467$	20.3664	0.942442	0.471221	+	0.882015i	$0.343813\pi$
$468$	0	0
$469$	−25.0404	−1.15626
$470$	0	0
$471$	−17.8564	−0.822780
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−0.330943	−0.0151847
$476$	0	0
$477$	12.0131	0.550041
$478$	0	0
$479$	29.9861	1.37010	0.685050	−	0.728496i	$-0.259780\pi$
$479$	29.9861	1.37010	0.685050	+	0.728496i	$0.259780\pi$
$480$	0	0
$481$	12.3749	0.564245
$482$	0	0
$483$	−7.59867	−0.345751
$484$	0	0
$485$	4.03731	0.183325
$486$	0	0
$487$	−28.7529	−1.30292	−0.651460	−	0.758683i	$-0.725843\pi$
$487$	−28.7529	−1.30292	−0.651460	+	0.758683i	$0.725843\pi$
$488$	0	0
$489$	26.7878	1.21139
$490$	0	0
$491$	−16.9472	−0.764816	−0.382408	−	0.923994i	$-0.624905\pi$
$491$	−16.9472	−0.764816	−0.382408	+	0.923994i	$0.624905\pi$
$492$	0	0
$493$	6.02269	0.271248
$494$	0	0
$495$	0	0
$496$	0	0
$497$	29.6954	1.33202
$498$	0	0
$499$	−35.5491	−1.59140	−0.795699	−	0.605692i	$-0.792896\pi$
$499$	−35.5491	−1.59140	−0.795699	+	0.605692i	$0.792896\pi$
$500$	0	0
$501$	50.5474	2.25829
$502$	0	0
$503$	−15.3213	−0.683142	−0.341571	−	0.939856i	$-0.610959\pi$
$503$	−15.3213	−0.683142	−0.341571	+	0.939856i	$0.610959\pi$
$504$	0	0
$505$	9.90476	0.440756
$506$	0	0
$507$	−22.4538	−0.997208
$508$	0	0
$509$	−34.5306	−1.53054	−0.765272	−	0.643707i	$-0.777395\pi$
$509$	−34.5306	−1.53054	−0.765272	+	0.643707i	$0.777395\pi$
$510$	0	0
$511$	−31.3359	−1.38622
$512$	0	0
$513$	0.849406	0.0375022
$514$	0	0
$515$	−11.9304	−0.525714
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−28.7541	−1.26217
$520$	0	0
$521$	7.89583	0.345922	0.172961	−	0.984929i	$-0.444666\pi$
$521$	7.89583	0.345922	0.172961	+	0.984929i	$0.444666\pi$
$522$	0	0
$523$	−39.2227	−1.71509	−0.857545	−	0.514410i	$-0.828011\pi$
$523$	−39.2227	−1.71509	−0.857545	+	0.514410i	$0.828011\pi$
$524$	0	0
$525$	−5.68224	−0.247993
$526$	0	0
$527$	−36.9890	−1.61126
$528$	0	0
$529$	−21.2117	−0.922249
$530$	0	0
$531$	−14.1866	−0.615646
$532$	0	0
$533$	9.16417	0.396944
$534$	0	0
$535$	−11.4320	−0.494247
$536$	0	0
$537$	6.06155	0.261575
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−4.28640	−0.184287	−0.0921433	−	0.995746i	$-0.529372\pi$
$541$	−4.28640	−0.184287	−0.0921433	+	0.995746i	$0.529372\pi$
$542$	0	0
$543$	7.43880	0.319229
$544$	0	0
$545$	−13.2534	−0.567714
$546$	0	0
$547$	0.942165	0.0402841	0.0201420	−	0.999797i	$-0.493588\pi$
$547$	0.942165	0.0402841	0.0201420	+	0.999797i	$0.493588\pi$
$548$	0	0
$549$	−7.12491	−0.304084
$550$	0	0
$551$	0.311600	0.0132746
$552$	0	0
$553$	19.6838	0.837042
$554$	0	0
$555$	−16.2987	−0.691841
$556$	0	0
$557$	20.7709	0.880089	0.440045	−	0.897976i	$-0.354963\pi$
$557$	20.7709	0.880089	0.440045	+	0.897976i	$0.354963\pi$
$558$	0	0
$559$	−7.58441	−0.320786
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.09795	0.130563	0.0652815	−	0.997867i	$-0.479205\pi$
$563$	3.09795	0.130563	0.0652815	+	0.997867i	$0.479205\pi$
$564$	0	0
$565$	14.4580	0.608253
$566$	0	0
$567$	28.7939	1.20923
$568$	0	0
$569$	4.83283	0.202603	0.101301	−	0.994856i	$-0.467699\pi$
$569$	4.83283	0.202603	0.101301	+	0.994856i	$0.467699\pi$
$570$	0	0
$571$	−9.40074	−0.393409	−0.196704	−	0.980463i	$-0.563024\pi$
$571$	−9.40074	−0.393409	−0.196704	+	0.980463i	$0.563024\pi$
$572$	0	0
$573$	−0.201692	−0.00842582
$574$	0	0
$575$	1.33727	0.0557679
$576$	0	0
$577$	6.62302	0.275720	0.137860	−	0.990452i	$-0.455978\pi$
$577$	6.62302	0.275720	0.137860	+	0.990452i	$0.455978\pi$
$578$	0	0
$579$	54.9421	2.28332
$580$	0	0
$581$	24.0362	0.997189
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−3.05845	−0.126451
$586$	0	0
$587$	−28.2488	−1.16595	−0.582976	−	0.812489i	$-0.698112\pi$
$587$	−28.2488	−1.16595	−0.582976	+	0.812489i	$0.698112\pi$
$588$	0	0
$589$	−1.91372	−0.0788535
$590$	0	0
$591$	−40.9879	−1.68602
$592$	0	0
$593$	1.97934	0.0812818	0.0406409	−	0.999174i	$-0.487060\pi$
$593$	1.97934	0.0812818	0.0406409	+	0.999174i	$0.487060\pi$
$594$	0	0
$595$	16.5342	0.677835
$596$	0	0
$597$	−48.2250	−1.97372
$598$	0	0
$599$	−20.3610	−0.831927	−0.415964	−	0.909381i	$-0.636556\pi$
$599$	−20.3610	−0.831927	−0.415964	+	0.909381i	$0.636556\pi$
$600$	0	0
$601$	25.0897	1.02343	0.511716	−	0.859155i	$-0.329010\pi$
$601$	25.0897	1.02343	0.511716	+	0.859155i	$0.329010\pi$
$602$	0	0
$603$	17.7515	0.722898
$604$	0	0
$605$	0	0
$606$	0	0
$607$	9.61170	0.390127	0.195063	−	0.980791i	$-0.437509\pi$
$607$	9.61170	0.390127	0.195063	+	0.980791i	$0.437509\pi$
$608$	0	0
$609$	5.35011	0.216798
$610$	0	0
$611$	12.0439	0.487245
$612$	0	0
$613$	16.4057	0.662618	0.331309	−	0.943522i	$-0.392510\pi$
$613$	16.4057	0.662618	0.331309	+	0.943522i	$0.392510\pi$
$614$	0	0
$615$	−12.0700	−0.486707
$616$	0	0
$617$	36.5821	1.47274	0.736370	−	0.676579i	$-0.236538\pi$
$617$	36.5821	1.47274	0.736370	+	0.676579i	$0.236538\pi$
$618$	0	0
$619$	31.6865	1.27359	0.636793	−	0.771035i	$-0.280261\pi$
$619$	31.6865	1.27359	0.636793	+	0.771035i	$0.280261\pi$
$620$	0	0
$621$	−3.43226	−0.137732
$622$	0	0
$623$	45.9201	1.83975
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	47.4260	1.89100
$630$	0	0
$631$	−22.3894	−0.891307	−0.445654	−	0.895206i	$-0.647029\pi$
$631$	−22.3894	−0.891307	−0.445654	+	0.895206i	$0.647029\pi$
$632$	0	0
$633$	−36.9592	−1.46900
$634$	0	0
$635$	−17.9795	−0.713495
$636$	0	0
$637$	0.531640	0.0210643
$638$	0	0
$639$	−21.0515	−0.832785
$640$	0	0
$641$	25.8587	1.02136	0.510678	−	0.859772i	$-0.329394\pi$
$641$	25.8587	1.02136	0.510678	+	0.859772i	$0.329394\pi$
$642$	0	0
$643$	−46.6628	−1.84020	−0.920101	−	0.391681i	$-0.871894\pi$
$643$	−46.6628	−1.84020	−0.920101	+	0.391681i	$0.871894\pi$
$644$	0	0
$645$	9.98928	0.393327
$646$	0	0
$647$	−25.9855	−1.02159	−0.510797	−	0.859701i	$-0.670650\pi$
$647$	−25.9855	−1.02159	−0.510797	+	0.859701i	$0.670650\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−32.8583	−1.28782
$652$	0	0
$653$	−3.44698	−0.134891	−0.0674453	−	0.997723i	$-0.521485\pi$
$653$	−3.44698	−0.134891	−0.0674453	+	0.997723i	$0.521485\pi$
$654$	0	0
$655$	9.70451	0.379187
$656$	0	0
$657$	22.2145	0.866670
$658$	0	0
$659$	15.6066	0.607945	0.303973	−	0.952681i	$-0.401687\pi$
$659$	15.6066	0.607945	0.303973	+	0.952681i	$0.401687\pi$
$660$	0	0
$661$	39.5126	1.53686	0.768431	−	0.639933i	$-0.221038\pi$
$661$	39.5126	1.53686	0.768431	+	0.639933i	$0.221038\pi$
$662$	0	0
$663$	23.4694	0.911474
$664$	0	0
$665$	0.855440	0.0331725
$666$	0	0
$667$	−1.25910	−0.0487527
$668$	0	0
$669$	0.344496	0.0133190
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−4.33800	−0.167218	−0.0836089	−	0.996499i	$-0.526645\pi$
$673$	−4.33800	−0.167218	−0.0836089	+	0.996499i	$0.526645\pi$
$674$	0	0
$675$	−2.56662	−0.0987893
$676$	0	0
$677$	−41.6655	−1.60134	−0.800668	−	0.599109i	$-0.795522\pi$
$677$	−41.6655	−1.60134	−0.800668	+	0.599109i	$0.795522\pi$
$678$	0	0
$679$	−10.4359	−0.400492
$680$	0	0
$681$	−9.37822	−0.359374
$682$	0	0
$683$	−43.4283	−1.66174	−0.830869	−	0.556468i	$-0.812156\pi$
$683$	−43.4283	−1.66174	−0.830869	+	0.556468i	$0.812156\pi$
$684$	0	0
$685$	19.4516	0.743207
$686$	0	0
$687$	−7.81007	−0.297973
$688$	0	0
$689$	−10.9420	−0.416856
$690$	0	0
$691$	−19.6046	−0.745793	−0.372897	−	0.927873i	$-0.621635\pi$
$691$	−19.6046	−0.745793	−0.372897	+	0.927873i	$0.621635\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.88457	−0.0714857
$696$	0	0
$697$	35.1212	1.33031
$698$	0	0
$699$	4.85610	0.183675
$700$	0	0
$701$	−36.9648	−1.39614	−0.698071	−	0.716029i	$-0.745958\pi$
$701$	−36.9648	−1.39614	−0.698071	+	0.716029i	$0.745958\pi$
$702$	0	0
$703$	2.45371	0.0925434
$704$	0	0
$705$	−15.8628	−0.597428
$706$	0	0
$707$	−25.6024	−0.962876
$708$	0	0
$709$	−41.3706	−1.55371	−0.776854	−	0.629681i	$-0.783185\pi$
$709$	−41.3706	−1.55371	−0.776854	+	0.629681i	$0.783185\pi$
$710$	0	0
$711$	−13.9542	−0.523322
$712$	0	0
$713$	7.73292	0.289600
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−27.9893	−1.04528
$718$	0	0
$719$	2.88409	0.107558	0.0537792	−	0.998553i	$-0.482873\pi$
$719$	2.88409	0.107558	0.0537792	+	0.998553i	$0.482873\pi$
$720$	0	0
$721$	30.8382	1.14848
$722$	0	0
$723$	30.1054	1.11963
$724$	0	0
$725$	−0.941551	−0.0349683
$726$	0	0
$727$	−17.9716	−0.666530	−0.333265	−	0.942833i	$-0.608150\pi$
$727$	−17.9716	−0.666530	−0.333265	+	0.942833i	$0.608150\pi$
$728$	0	0
$729$	−3.48598	−0.129110
$730$	0	0
$731$	−29.0668	−1.07508
$732$	0	0
$733$	15.3188	0.565812	0.282906	−	0.959148i	$-0.408702\pi$
$733$	15.3188	0.565812	0.282906	+	0.959148i	$0.408702\pi$
$734$	0	0
$735$	−0.700213	−0.0258277
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−6.38055	−0.234712	−0.117356	−	0.993090i	$-0.537442\pi$
$739$	−6.38055	−0.234712	−0.117356	+	0.993090i	$0.537442\pi$
$740$	0	0
$741$	1.21425	0.0446066
$742$	0	0
$743$	−32.3860	−1.18813	−0.594064	−	0.804418i	$-0.702478\pi$
$743$	−32.3860	−1.18813	−0.594064	+	0.804418i	$0.702478\pi$
$744$	0	0
$745$	8.19423	0.300213
$746$	0	0
$747$	−17.0396	−0.623447
$748$	0	0
$749$	29.5500	1.07973
$750$	0	0
$751$	32.5871	1.18912	0.594560	−	0.804051i	$-0.297326\pi$
$751$	32.5871	1.18912	0.594560	+	0.804051i	$0.297326\pi$
$752$	0	0
$753$	−20.0530	−0.730772
$754$	0	0
$755$	−3.64408	−0.132621
$756$	0	0
$757$	32.7857	1.19162	0.595808	−	0.803127i	$-0.296832\pi$
$757$	32.7857	1.19162	0.595808	+	0.803127i	$0.296832\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−41.3176	−1.49776	−0.748881	−	0.662705i	$-0.769408\pi$
$761$	−41.3176	−1.49776	−0.748881	+	0.662705i	$0.769408\pi$
$762$	0	0
$763$	34.2582	1.24023
$764$	0	0
$765$	−11.7213	−0.423786
$766$	0	0
$767$	12.9217	0.466575
$768$	0	0
$769$	−20.4922	−0.738967	−0.369483	−	0.929237i	$-0.620465\pi$
$769$	−20.4922	−0.738967	−0.369483	+	0.929237i	$0.620465\pi$
$770$	0	0
$771$	51.0305	1.83782
$772$	0	0
$773$	−1.88634	−0.0678470	−0.0339235	−	0.999424i	$-0.510800\pi$
$773$	−1.88634	−0.0678470	−0.0339235	+	0.999424i	$0.510800\pi$
$774$	0	0
$775$	5.78263	0.207718
$776$	0	0
$777$	42.1298	1.51140
$778$	0	0
$779$	1.81709	0.0651039
$780$	0	0
$781$	0	0
$782$	0	0
$783$	2.41660	0.0863624
$784$	0	0
$785$	−8.12289	−0.289918
$786$	0	0
$787$	20.6178	0.734946	0.367473	−	0.930034i	$-0.380223\pi$
$787$	20.6178	0.734946	0.367473	+	0.930034i	$0.380223\pi$
$788$	0	0
$789$	67.0502	2.38705
$790$	0	0
$791$	−37.3719	−1.32879
$792$	0	0
$793$	6.48964	0.230454
$794$	0	0
$795$	14.4114	0.511121
$796$	0	0
$797$	5.96779	0.211390	0.105695	−	0.994399i	$-0.466293\pi$
$797$	5.96779	0.211390	0.105695	+	0.994399i	$0.466293\pi$
$798$	0	0
$799$	46.1576	1.63294
$800$	0	0
$801$	−32.5534	−1.15022
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−3.45664	−0.121831
$806$	0	0
$807$	−11.2874	−0.397336
$808$	0	0
$809$	−27.3857	−0.962829	−0.481415	−	0.876493i	$-0.659877\pi$
$809$	−27.3857	−0.962829	−0.481415	+	0.876493i	$0.659877\pi$
$810$	0	0
$811$	10.5465	0.370336	0.185168	−	0.982707i	$-0.440717\pi$
$811$	10.5465	0.370336	0.185168	+	0.982707i	$0.440717\pi$
$812$	0	0
$813$	−27.8915	−0.978199
$814$	0	0
$815$	12.1858	0.426849
$816$	0	0
$817$	−1.50385	−0.0526130
$818$	0	0
$819$	7.90565	0.276246
$820$	0	0
$821$	−41.0152	−1.43144	−0.715720	−	0.698387i	$-0.753901\pi$
$821$	−41.0152	−1.43144	−0.715720	+	0.698387i	$0.753901\pi$
$822$	0	0
$823$	−26.8483	−0.935871	−0.467936	−	0.883763i	$-0.655002\pi$
$823$	−26.8483	−0.935871	−0.467936	+	0.883763i	$0.655002\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.1904	0.423903	0.211951	−	0.977280i	$-0.432018\pi$
$827$	12.1904	0.423903	0.211951	+	0.977280i	$0.432018\pi$
$828$	0	0
$829$	−34.5013	−1.19828	−0.599139	−	0.800645i	$-0.704490\pi$
$829$	−34.5013	−1.19828	−0.599139	+	0.800645i	$0.704490\pi$
$830$	0	0
$831$	27.0705	0.939064
$832$	0	0
$833$	2.03748	0.0705945
$834$	0	0
$835$	22.9941	0.795742
$836$	0	0
$837$	−14.8418	−0.513008
$838$	0	0
$839$	−13.3549	−0.461064	−0.230532	−	0.973065i	$-0.574047\pi$
$839$	−13.3549	−0.461064	−0.230532	+	0.973065i	$0.574047\pi$
$840$	0	0
$841$	−28.1135	−0.969430
$842$	0	0
$843$	−29.3989	−1.01255
$844$	0	0
$845$	−10.2142	−0.351381
$846$	0	0
$847$	0	0
$848$	0	0
$849$	24.2845	0.833442
$850$	0	0
$851$	−9.91489	−0.339878
$852$	0	0
$853$	39.5614	1.35456	0.677278	−	0.735727i	$-0.263160\pi$
$853$	39.5614	1.35456	0.677278	+	0.735727i	$0.263160\pi$
$854$	0	0
$855$	−0.606434	−0.0207396
$856$	0	0
$857$	−10.6805	−0.364838	−0.182419	−	0.983221i	$-0.558393\pi$
$857$	−10.6805	−0.364838	−0.182419	+	0.983221i	$0.558393\pi$
$858$	0	0
$859$	2.20217	0.0751369	0.0375685	−	0.999294i	$-0.488039\pi$
$859$	2.20217	0.0751369	0.0375685	+	0.999294i	$0.488039\pi$
$860$	0	0
$861$	31.1991	1.06326
$862$	0	0
$863$	4.30423	0.146518	0.0732588	−	0.997313i	$-0.476660\pi$
$863$	4.30423	0.146518	0.0732588	+	0.997313i	$0.476660\pi$
$864$	0	0
$865$	−13.0803	−0.444743
$866$	0	0
$867$	52.5741	1.78551
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−16.1688	−0.547858
$872$	0	0
$873$	7.39814	0.250389
$874$	0	0
$875$	−2.58485	−0.0873840
$876$	0	0
$877$	41.8693	1.41383	0.706913	−	0.707300i	$-0.250087\pi$
$877$	41.8693	1.41383	0.706913	+	0.707300i	$0.250087\pi$
$878$	0	0
$879$	−55.9666	−1.88770
$880$	0	0
$881$	55.2076	1.85999	0.929996	−	0.367570i	$-0.119810\pi$
$881$	55.2076	1.85999	0.929996	+	0.367570i	$0.119810\pi$
$882$	0	0
$883$	27.4993	0.925426	0.462713	−	0.886508i	$-0.346876\pi$
$883$	27.4993	0.925426	0.462713	+	0.886508i	$0.346876\pi$
$884$	0	0
$885$	−17.0189	−0.572084
$886$	0	0
$887$	28.9410	0.971743	0.485871	−	0.874030i	$-0.338502\pi$
$887$	28.9410	0.971743	0.485871	+	0.874030i	$0.338502\pi$
$888$	0	0
$889$	46.4744	1.55870
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2.38809	0.0799143
$894$	0	0
$895$	2.75740	0.0921698
$896$	0	0
$897$	−4.90651	−0.163824
$898$	0	0
$899$	−5.44464	−0.181589
$900$	0	0
$901$	−41.9344	−1.39704
$902$	0	0
$903$	−25.8208	−0.859263
$904$	0	0
$905$	3.38392	0.112485
$906$	0	0
$907$	−55.6297	−1.84715	−0.923577	−	0.383412i	$-0.874749\pi$
$907$	−55.6297	−1.84715	−0.923577	+	0.383412i	$0.874749\pi$
$908$	0	0
$909$	18.1499	0.601994
$910$	0	0
$911$	−49.2558	−1.63192	−0.815959	−	0.578110i	$-0.803790\pi$
$911$	−49.2558	−1.63192	−0.815959	+	0.578110i	$0.803790\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−8.54737	−0.282567
$916$	0	0
$917$	−25.0847	−0.828371
$918$	0	0
$919$	−2.39222	−0.0789121	−0.0394560	−	0.999221i	$-0.512563\pi$
$919$	−2.39222	−0.0789121	−0.0394560	+	0.999221i	$0.512563\pi$
$920$	0	0
$921$	66.2462	2.18289
$922$	0	0
$923$	19.1745	0.631137
$924$	0	0
$925$	−7.41429	−0.243780
$926$	0	0
$927$	−21.8617	−0.718032
$928$	0	0
$929$	−36.1713	−1.18674	−0.593371	−	0.804929i	$-0.702203\pi$
$929$	−36.1713	−1.18674	−0.593371	+	0.804929i	$0.702203\pi$
$930$	0	0
$931$	0.105414	0.00345482
$932$	0	0
$933$	27.2377	0.891722
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−4.30586	−0.140666	−0.0703331	−	0.997524i	$-0.522406\pi$
$937$	−4.30586	−0.140666	−0.0703331	+	0.997524i	$0.522406\pi$
$938$	0	0
$939$	57.6072	1.87994
$940$	0	0
$941$	−0.649571	−0.0211754	−0.0105877	−	0.999944i	$-0.503370\pi$
$941$	−0.649571	−0.0211754	−0.0105877	+	0.999944i	$0.503370\pi$
$942$	0	0
$943$	−7.34244	−0.239103
$944$	0	0
$945$	6.63434	0.215815
$946$	0	0
$947$	39.0346	1.26845	0.634226	−	0.773147i	$-0.281319\pi$
$947$	39.0346	1.26845	0.634226	+	0.773147i	$0.281319\pi$
$948$	0	0
$949$	−20.2338	−0.656817
$950$	0	0
$951$	−37.3228	−1.21027
$952$	0	0
$953$	11.1295	0.360521	0.180260	−	0.983619i	$-0.442306\pi$
$953$	11.1295	0.360521	0.180260	+	0.983619i	$0.442306\pi$
$954$	0	0
$955$	−0.0917500	−0.00296896
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−50.2796	−1.62361
$960$	0	0
$961$	2.43880	0.0786709
$962$	0	0
$963$	−20.9484	−0.675053
$964$	0	0
$965$	24.9932	0.804560
$966$	0	0
$967$	−5.30703	−0.170663	−0.0853313	−	0.996353i	$-0.527195\pi$
$967$	−5.30703	−0.170663	−0.0853313	+	0.996353i	$0.527195\pi$
$968$	0	0
$969$	4.65354	0.149493
$970$	0	0
$971$	−4.61394	−0.148068	−0.0740342	−	0.997256i	$-0.523587\pi$
$971$	−4.61394	−0.148068	−0.0740342	+	0.997256i	$0.523587\pi$
$972$	0	0
$973$	4.87133	0.156168
$974$	0	0
$975$	−3.66906	−0.117504
$976$	0	0
$977$	55.8973	1.78831	0.894156	−	0.447756i	$-0.147777\pi$
$977$	55.8973	1.78831	0.894156	+	0.447756i	$0.147777\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−24.2861	−0.775396
$982$	0	0
$983$	7.55085	0.240835	0.120417	−	0.992723i	$-0.461577\pi$
$983$	7.55085	0.240835	0.120417	+	0.992723i	$0.461577\pi$
$984$	0	0
$985$	−18.6454	−0.594093
$986$	0	0
$987$	41.0031	1.30514
$988$	0	0
$989$	6.07672	0.193228
$990$	0	0
$991$	−23.2874	−0.739749	−0.369874	−	0.929082i	$-0.620599\pi$
$991$	−23.2874	−0.739749	−0.369874	+	0.929082i	$0.620599\pi$
$992$	0	0
$993$	−30.2224	−0.959077
$994$	0	0
$995$	−21.9376	−0.695468
$996$	0	0
$997$	21.1584	0.670092	0.335046	−	0.942202i	$-0.391248\pi$
$997$	21.1584	0.670092	0.335046	+	0.942202i	$0.391248\pi$
$998$	0	0
$999$	19.0297	0.602072

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4840.2.a.bc.1.6	✓	6
4.3	odd	2		9680.2.a.db.1.1		6
11.10	odd	2		4840.2.a.bd.1.6	yes	6
44.43	even	2		9680.2.a.da.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4840.2.a.bc.1.6	✓	6	1.1	even	1	trivial
4840.2.a.bd.1.6	yes	6	11.10	odd	2
9680.2.a.da.1.1		6	44.43	even	2
9680.2.a.db.1.1		6	4.3	odd	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:	\( N \)	\(=\)	\( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4840.a (trivial)

\(f(q)\)	\(=\)	\(q+2.19828 q^{3} +1.00000 q^{5} -2.58485 q^{7} +1.83244 q^{9} -1.66906 q^{13} +2.19828 q^{15} -6.39656 q^{17} -0.330943 q^{19} -5.68224 q^{21} +1.33727 q^{23} +1.00000 q^{25} -2.56662 q^{27} -0.941551 q^{29} +5.78263 q^{31} -2.58485 q^{35} -7.41429 q^{37} -3.66906 q^{39} -5.49063 q^{41} +4.54413 q^{43} +1.83244 q^{45} -7.21601 q^{47} -0.318527 q^{49} -14.0614 q^{51} +6.55577 q^{53} -0.727506 q^{57} -7.74191 q^{59} -3.88821 q^{61} -4.73660 q^{63} -1.66906 q^{65} +9.68736 q^{67} +2.93969 q^{69} -11.4882 q^{71} +12.1229 q^{73} +2.19828 q^{75} -7.61507 q^{79} -11.1395 q^{81} -9.29886 q^{83} -6.39656 q^{85} -2.06979 q^{87} -17.7651 q^{89} +4.31427 q^{91} +12.7118 q^{93} -0.330943 q^{95} +4.03731 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} + 6 q^{5} - 4 q^{7} + 4 q^{9} - 2 q^{15} - 8 q^{17} - 12 q^{19} + 8 q^{21} - 8 q^{23} + 6 q^{25} - 14 q^{27} - 16 q^{29} - 4 q^{31} - 4 q^{35} + 8 q^{37} - 12 q^{39} - 32 q^{41} + 4 q^{43}+ \cdots - 12 q^{95}+O(q^{100}) \) 6 * q - 2 * q^3 + 6 * q^5 - 4 * q^7 + 4 * q^9 - 2 * q^15 - 8 * q^17 - 12 * q^19 + 8 * q^21 - 8 * q^23 + 6 * q^25 - 14 * q^27 - 16 * q^29 - 4 * q^31 - 4 * q^35 + 8 * q^37 - 12 * q^39 - 32 * q^41 + 4 * q^43 + 4 * q^45 - 6 * q^47 + 16 * q^49 - 40 * q^51 + 8 * q^53 + 16 * q^57 + 4 * q^59 - 16 * q^61 - 28 * q^63 - 2 * q^67 + 8 * q^69 - 28 * q^71 - 16 * q^73 - 2 * q^75 - 10 * q^81 - 12 * q^83 - 8 * q^85 + 24 * q^87 + 18 * q^89 - 24 * q^91 + 20 * q^93 - 12 * q^95

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