LMFDB - Embedded newform 6084.2.a.x.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6084,2,Mod(1,6084)] 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("6084.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6084, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6084 = 2^{2} \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6084.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,-8,0,-1,0,0,0,-6,0,0,0,0,0,10,0,4,0,0,0,-3,0,11,0,0, 0,1,0,5,0,0,0,12,0,4,0,0,0,-25,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(43)] == traces)

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

Self dual:	yes
Analytic conductor:	$48.5809845897$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 676)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	6084.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.19806 q^{5} +4.29590 q^{7} -3.69202 q^{11} +7.85086 q^{17} -4.65279 q^{19} -3.71379 q^{23} -3.56465 q^{25} -4.29590 q^{29} +2.91185 q^{31} -5.14675 q^{35} -8.03684 q^{37} -8.10992 q^{41} +0.884707 q^{43} +4.58211 q^{47} +11.4547 q^{49} -5.43296 q^{53} +4.42327 q^{55} -2.32975 q^{59} -6.25667 q^{61} -3.33513 q^{67} +4.35690 q^{71} +3.82908 q^{73} -15.8605 q^{77} -10.5526 q^{79} -5.24698 q^{83} -9.40581 q^{85} -9.46681 q^{89} +5.57434 q^{95} +3.67994 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 8 q^{5} - q^{7} - 6 q^{11} + 10 q^{17} + 4 q^{19} - 3 q^{23} + 11 q^{25} + q^{29} + 5 q^{31} + 12 q^{35} + 4 q^{37} - 25 q^{41} + 5 q^{43} + 8 q^{47} + 12 q^{49} + 3 q^{53} + 16 q^{55} - 9 q^{59}+ \cdots - 13 q^{97}+O(q^{100}) $ 3 * q - 8 * q^5 - q^7 - 6 * q^11 + 10 * q^17 + 4 * q^19 - 3 * q^23 + 11 * q^25 + q^29 + 5 * q^31 + 12 * q^35 + 4 * q^37 - 25 * q^41 + 5 * q^43 + 8 * q^47 + 12 * q^49 + 3 * q^53 + 16 * q^55 - 9 * q^59 + 8 * q^61 - 9 * q^67 + 9 * q^71 + q^73 - 12 * q^77 + 9 * q^79 - 11 * q^83 - 15 * q^85 - 25 * q^89 - 27 * q^95 - 13 * q^97

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.19806	−0.535790	−0.267895	−	0.963448i	$-0.586328\pi$
$5$	−1.19806	−0.535790	−0.267895	+	0.963448i	$0.586328\pi$
$6$	0	0
$7$	4.29590	1.62370	0.811848	−	0.583869i	$-0.198462\pi$
$7$	4.29590	1.62370	0.811848	+	0.583869i	$0.198462\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.69202	−1.11319	−0.556593	−	0.830785i	$-0.687892\pi$
$11$	−3.69202	−1.11319	−0.556593	+	0.830785i	$0.687892\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.85086	1.90411	0.952056	−	0.305924i	$-0.0989653\pi$
$17$	7.85086	1.90411	0.952056	+	0.305924i	$0.0989653\pi$
$18$	0	0
$19$	−4.65279	−1.06742	−0.533712	−	0.845666i	$-0.679203\pi$
$19$	−4.65279	−1.06742	−0.533712	+	0.845666i	$0.679203\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.71379	−0.774379	−0.387190	−	0.922000i	$-0.626554\pi$
$23$	−3.71379	−0.774379	−0.387190	+	0.922000i	$0.626554\pi$
$24$	0	0
$25$	−3.56465	−0.712929
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.29590	−0.797728	−0.398864	−	0.917010i	$-0.630595\pi$
$29$	−4.29590	−0.797728	−0.398864	+	0.917010i	$0.630595\pi$
$30$	0	0
$31$	2.91185	0.522984	0.261492	−	0.965206i	$-0.415785\pi$
$31$	2.91185	0.522984	0.261492	+	0.965206i	$0.415785\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−5.14675	−0.869960
$36$	0	0
$37$	−8.03684	−1.32125	−0.660624	−	0.750717i	$-0.729708\pi$
$37$	−8.03684	−1.32125	−0.660624	+	0.750717i	$0.729708\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.10992	−1.26656	−0.633278	−	0.773924i	$-0.718291\pi$
$41$	−8.10992	−1.26656	−0.633278	+	0.773924i	$0.718291\pi$
$42$	0	0
$43$	0.884707	0.134917	0.0674583	−	0.997722i	$-0.478511\pi$
$43$	0.884707	0.134917	0.0674583	+	0.997722i	$0.478511\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.58211	0.668369	0.334184	−	0.942508i	$-0.391539\pi$
$47$	4.58211	0.668369	0.334184	+	0.942508i	$0.391539\pi$
$48$	0	0
$49$	11.4547	1.63639
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.43296	−0.746274	−0.373137	−	0.927776i	$-0.621718\pi$
$53$	−5.43296	−0.746274	−0.373137	+	0.927776i	$0.621718\pi$
$54$	0	0
$55$	4.42327	0.596434
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.32975	−0.303307	−0.151654	−	0.988434i	$-0.548460\pi$
$59$	−2.32975	−0.303307	−0.151654	+	0.988434i	$0.548460\pi$
$60$	0	0
$61$	−6.25667	−0.801084	−0.400542	−	0.916278i	$-0.631178\pi$
$61$	−6.25667	−0.801084	−0.400542	+	0.916278i	$0.631178\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−3.33513	−0.407450	−0.203725	−	0.979028i	$-0.565305\pi$
$67$	−3.33513	−0.407450	−0.203725	+	0.979028i	$0.565305\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.35690	0.517068	0.258534	−	0.966002i	$-0.416761\pi$
$71$	4.35690	0.517068	0.258534	+	0.966002i	$0.416761\pi$
$72$	0	0
$73$	3.82908	0.448160	0.224080	−	0.974571i	$-0.428062\pi$
$73$	3.82908	0.448160	0.224080	+	0.974571i	$0.428062\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−15.8605	−1.80748
$78$	0	0
$79$	−10.5526	−1.18726	−0.593628	−	0.804739i	$-0.702305\pi$
$79$	−10.5526	−1.18726	−0.593628	+	0.804739i	$0.702305\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−5.24698	−0.575931	−0.287965	−	0.957641i	$-0.592979\pi$
$83$	−5.24698	−0.575931	−0.287965	+	0.957641i	$0.592979\pi$
$84$	0	0
$85$	−9.40581	−1.02020
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−9.46681	−1.00348	−0.501740	−	0.865018i	$-0.667307\pi$
$89$	−9.46681	−1.00348	−0.501740	+	0.865018i	$0.667307\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.57434	0.571915
$96$	0	0
$97$	3.67994	0.373641	0.186821	−	0.982394i	$-0.440182\pi$
$97$	3.67994	0.373641	0.186821	+	0.982394i	$0.440182\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	17.1250	1.70400	0.852000	−	0.523542i	$-0.175390\pi$
$101$	17.1250	1.70400	0.852000	+	0.523542i	$0.175390\pi$
$102$	0	0
$103$	8.14675	0.802723	0.401362	−	0.915920i	$-0.368537\pi$
$103$	8.14675	0.802723	0.401362	+	0.915920i	$0.368537\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.564647	−0.0545865	−0.0272932	−	0.999627i	$-0.508689\pi$
$107$	−0.564647	−0.0545865	−0.0272932	+	0.999627i	$0.508689\pi$
$108$	0	0
$109$	2.04354	0.195736	0.0978678	−	0.995199i	$-0.468798\pi$
$109$	2.04354	0.195736	0.0978678	+	0.995199i	$0.468798\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	13.7845	1.29673	0.648367	−	0.761328i	$-0.275452\pi$
$113$	13.7845	1.29673	0.648367	+	0.761328i	$0.275452\pi$
$114$	0	0
$115$	4.44935	0.414904
$116$	0	0
$117$	0	0
$118$	0	0
$119$	33.7265	3.09170
$120$	0	0
$121$	2.63102	0.239184
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10.2610	0.917770
$126$	0	0
$127$	−6.30798	−0.559743	−0.279871	−	0.960038i	$-0.590292\pi$
$127$	−6.30798	−0.559743	−0.279871	+	0.960038i	$0.590292\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−14.2959	−1.24904	−0.624519	−	0.781009i	$-0.714705\pi$
$131$	−14.2959	−1.24904	−0.624519	+	0.781009i	$0.714705\pi$
$132$	0	0
$133$	−19.9879	−1.73317
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−11.2078	−0.957543	−0.478771	−	0.877940i	$-0.658918\pi$
$137$	−11.2078	−0.957543	−0.478771	+	0.877940i	$0.658918\pi$
$138$	0	0
$139$	4.56465	0.387168	0.193584	−	0.981084i	$-0.437989\pi$
$139$	4.56465	0.387168	0.193584	+	0.981084i	$0.437989\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	5.14675	0.427414
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−21.6746	−1.77565	−0.887825	−	0.460182i	$-0.847784\pi$
$149$	−21.6746	−1.77565	−0.887825	+	0.460182i	$0.847784\pi$
$150$	0	0
$151$	−20.9758	−1.70699	−0.853495	−	0.521102i	$-0.825521\pi$
$151$	−20.9758	−1.70699	−0.853495	+	0.521102i	$0.825521\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.48858	−0.280210
$156$	0	0
$157$	3.62565	0.289358	0.144679	−	0.989479i	$-0.453785\pi$
$157$	3.62565	0.289358	0.144679	+	0.989479i	$0.453785\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−15.9541	−1.25736
$162$	0	0
$163$	−21.7724	−1.70535	−0.852673	−	0.522445i	$-0.825020\pi$
$163$	−21.7724	−1.70535	−0.852673	+	0.522445i	$0.825020\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.81163	−0.217570	−0.108785	−	0.994065i	$-0.534696\pi$
$167$	−2.81163	−0.217570	−0.108785	+	0.994065i	$0.534696\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−7.01208	−0.533119	−0.266559	−	0.963819i	$-0.585887\pi$
$173$	−7.01208	−0.533119	−0.266559	+	0.963819i	$0.585887\pi$
$174$	0	0
$175$	−15.3134	−1.15758
$176$	0	0
$177$	0	0
$178$	0	0
$179$	15.4862	1.15749	0.578746	−	0.815508i	$-0.303542\pi$
$179$	15.4862	1.15749	0.578746	+	0.815508i	$0.303542\pi$
$180$	0	0
$181$	5.67456	0.421787	0.210893	−	0.977509i	$-0.432363\pi$
$181$	5.67456	0.421787	0.210893	+	0.977509i	$0.432363\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	9.62863	0.707911
$186$	0	0
$187$	−28.9855	−2.11963
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.5864	1.48958	0.744790	−	0.667298i	$-0.232549\pi$
$191$	20.5864	1.48958	0.744790	+	0.667298i	$0.232549\pi$
$192$	0	0
$193$	−18.9530	−1.36427	−0.682133	−	0.731228i	$-0.738948\pi$
$193$	−18.9530	−1.36427	−0.682133	+	0.731228i	$0.738948\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.34050	0.166754	0.0833769	−	0.996518i	$-0.473429\pi$
$197$	2.34050	0.166754	0.0833769	+	0.996518i	$0.473429\pi$
$198$	0	0
$199$	5.64071	0.399859	0.199930	−	0.979810i	$-0.435929\pi$
$199$	5.64071	0.399859	0.199930	+	0.979810i	$0.435929\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−18.4547	−1.29527
$204$	0	0
$205$	9.71618	0.678608
$206$	0	0
$207$	0	0
$208$	0	0
$209$	17.1782	1.18824
$210$	0	0
$211$	0.582105	0.0400738	0.0200369	−	0.999799i	$-0.493622\pi$
$211$	0.582105	0.0400738	0.0200369	+	0.999799i	$0.493622\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.05993	−0.0722869
$216$	0	0
$217$	12.5090	0.849168
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	15.2295	1.01984	0.509922	−	0.860221i	$-0.329674\pi$
$223$	15.2295	1.01984	0.509922	+	0.860221i	$0.329674\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.4523	0.892863	0.446432	−	0.894818i	$-0.352695\pi$
$227$	13.4523	0.892863	0.446432	+	0.894818i	$0.352695\pi$
$228$	0	0
$229$	−7.11529	−0.470192	−0.235096	−	0.971972i	$-0.575540\pi$
$229$	−7.11529	−0.470192	−0.235096	+	0.971972i	$0.575540\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−7.36898	−0.482758	−0.241379	−	0.970431i	$-0.577600\pi$
$233$	−7.36898	−0.482758	−0.241379	+	0.970431i	$0.577600\pi$
$234$	0	0
$235$	−5.48965	−0.358105
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−8.34481	−0.539781	−0.269891	−	0.962891i	$-0.586988\pi$
$239$	−8.34481	−0.539781	−0.269891	+	0.962891i	$0.586988\pi$
$240$	0	0
$241$	6.48427	0.417689	0.208844	−	0.977949i	$-0.433030\pi$
$241$	6.48427	0.417689	0.208844	+	0.977949i	$0.433030\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−13.7235	−0.876761
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	7.26205	0.458376	0.229188	−	0.973382i	$-0.426393\pi$
$251$	7.26205	0.458376	0.229188	+	0.973382i	$0.426393\pi$
$252$	0	0
$253$	13.7114	0.862028
$254$	0	0
$255$	0	0
$256$	0	0
$257$	13.6364	0.850615	0.425308	−	0.905049i	$-0.360166\pi$
$257$	13.6364	0.850615	0.425308	+	0.905049i	$0.360166\pi$
$258$	0	0
$259$	−34.5254	−2.14531
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−11.3123	−0.697546	−0.348773	−	0.937207i	$-0.613402\pi$
$263$	−11.3123	−0.697546	−0.348773	+	0.937207i	$0.613402\pi$
$264$	0	0
$265$	6.50902	0.399846
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−0.669186	−0.0408010	−0.0204005	−	0.999792i	$-0.506494\pi$
$269$	−0.669186	−0.0408010	−0.0204005	+	0.999792i	$0.506494\pi$
$270$	0	0
$271$	−20.4101	−1.23983	−0.619913	−	0.784670i	$-0.712832\pi$
$271$	−20.4101	−1.23983	−0.619913	+	0.784670i	$0.712832\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	13.1608	0.793623
$276$	0	0
$277$	−2.48858	−0.149524	−0.0747622	−	0.997201i	$-0.523820\pi$
$277$	−2.48858	−0.149524	−0.0747622	+	0.997201i	$0.523820\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−16.6310	−0.992124	−0.496062	−	0.868287i	$-0.665221\pi$
$281$	−16.6310	−0.992124	−0.496062	+	0.868287i	$0.665221\pi$
$282$	0	0
$283$	2.86592	0.170361	0.0851806	−	0.996366i	$-0.472853\pi$
$283$	2.86592	0.170361	0.0851806	+	0.996366i	$0.472853\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−34.8394	−2.05650
$288$	0	0
$289$	44.6359	2.62564
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−30.7711	−1.79767	−0.898833	−	0.438292i	$-0.855584\pi$
$293$	−30.7711	−1.79767	−0.898833	+	0.438292i	$0.855584\pi$
$294$	0	0
$295$	2.79118	0.162509
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	3.80061	0.219064
$302$	0	0
$303$	0	0
$304$	0	0
$305$	7.49588	0.429213
$306$	0	0
$307$	20.2392	1.15511	0.577556	−	0.816351i	$-0.304006\pi$
$307$	20.2392	1.15511	0.577556	+	0.816351i	$0.304006\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−18.8442	−1.06855	−0.534277	−	0.845310i	$-0.679416\pi$
$311$	−18.8442	−1.06855	−0.534277	+	0.845310i	$0.679416\pi$
$312$	0	0
$313$	4.87263	0.275417	0.137709	−	0.990473i	$-0.456026\pi$
$313$	4.87263	0.275417	0.137709	+	0.990473i	$0.456026\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−13.2185	−0.742425	−0.371213	−	0.928548i	$-0.621058\pi$
$317$	−13.2185	−0.742425	−0.371213	+	0.928548i	$0.621058\pi$
$318$	0	0
$319$	15.8605	0.888020
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−36.5284	−2.03249
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	19.6843	1.08523
$330$	0	0
$331$	21.8049	1.19851	0.599253	−	0.800559i	$-0.295464\pi$
$331$	21.8049	1.19851	0.599253	+	0.800559i	$0.295464\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3.99569	0.218308
$336$	0	0
$337$	−10.1564	−0.553257	−0.276628	−	0.960977i	$-0.589217\pi$
$337$	−10.1564	−0.553257	−0.276628	+	0.960977i	$0.589217\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−10.7506	−0.582179
$342$	0	0
$343$	19.1371	1.03330
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−27.8834	−1.49686	−0.748429	−	0.663215i	$-0.769192\pi$
$347$	−27.8834	−1.49686	−0.748429	+	0.663215i	$0.769192\pi$
$348$	0	0
$349$	35.4209	1.89604	0.948018	−	0.318216i	$-0.103084\pi$
$349$	35.4209	1.89604	0.948018	+	0.318216i	$0.103084\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−8.99031	−0.478506	−0.239253	−	0.970957i	$-0.576903\pi$
$353$	−8.99031	−0.478506	−0.239253	+	0.970957i	$0.576903\pi$
$354$	0	0
$355$	−5.21983	−0.277040
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4.83446	−0.255153	−0.127577	−	0.991829i	$-0.540720\pi$
$359$	−4.83446	−0.255153	−0.127577	+	0.991829i	$0.540720\pi$
$360$	0	0
$361$	2.64848	0.139394
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4.58748	−0.240120
$366$	0	0
$367$	−28.4077	−1.48287	−0.741436	−	0.671024i	$-0.765855\pi$
$367$	−28.4077	−1.48287	−0.741436	+	0.671024i	$0.765855\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−23.3394	−1.21172
$372$	0	0
$373$	−26.3967	−1.36677	−0.683385	−	0.730058i	$-0.739493\pi$
$373$	−26.3967	−1.36677	−0.683385	+	0.730058i	$0.739493\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	19.8616	1.02022	0.510111	−	0.860108i	$-0.329604\pi$
$379$	19.8616	1.02022	0.510111	+	0.860108i	$0.329604\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−0.219833	−0.0112329	−0.00561646	−	0.999984i	$-0.501788\pi$
$383$	−0.219833	−0.0112329	−0.00561646	+	0.999984i	$0.501788\pi$
$384$	0	0
$385$	19.0019	0.968427
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10.0543	0.509773	0.254886	−	0.966971i	$-0.417962\pi$
$389$	10.0543	0.509773	0.254886	+	0.966971i	$0.417962\pi$
$390$	0	0
$391$	−29.1564	−1.47450
$392$	0	0
$393$	0	0
$394$	0	0
$395$	12.6426	0.636120
$396$	0	0
$397$	5.76032	0.289102	0.144551	−	0.989497i	$-0.453826\pi$
$397$	5.76032	0.289102	0.144551	+	0.989497i	$0.453826\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−34.9299	−1.74432	−0.872158	−	0.489224i	$-0.837280\pi$
$401$	−34.9299	−1.74432	−0.872158	+	0.489224i	$0.837280\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	29.6722	1.47079
$408$	0	0
$409$	−11.8442	−0.585656	−0.292828	−	0.956165i	$-0.594596\pi$
$409$	−11.8442	−0.585656	−0.292828	+	0.956165i	$0.594596\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−10.0084	−0.492479
$414$	0	0
$415$	6.28621	0.308578
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−10.7530	−0.525319	−0.262660	−	0.964889i	$-0.584600\pi$
$419$	−10.7530	−0.525319	−0.262660	+	0.964889i	$0.584600\pi$
$420$	0	0
$421$	−37.6722	−1.83603	−0.918015	−	0.396547i	$-0.870209\pi$
$421$	−37.6722	−1.83603	−0.918015	+	0.396547i	$0.870209\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−27.9855	−1.35750
$426$	0	0
$427$	−26.8780	−1.30072
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−23.3129	−1.12294	−0.561471	−	0.827496i	$-0.689764\pi$
$431$	−23.3129	−1.12294	−0.561471	+	0.827496i	$0.689764\pi$
$432$	0	0
$433$	23.2368	1.11669	0.558345	−	0.829609i	$-0.311437\pi$
$433$	23.2368	1.11669	0.558345	+	0.829609i	$0.311437\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	17.2795	0.826591
$438$	0	0
$439$	18.6189	0.888634	0.444317	−	0.895870i	$-0.353446\pi$
$439$	18.6189	0.888634	0.444317	+	0.895870i	$0.353446\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−27.0180	−1.28367	−0.641833	−	0.766844i	$-0.721826\pi$
$443$	−27.0180	−1.28367	−0.641833	+	0.766844i	$0.721826\pi$
$444$	0	0
$445$	11.3418	0.537654
$446$	0	0
$447$	0	0
$448$	0	0
$449$	30.6708	1.44745	0.723723	−	0.690090i	$-0.242429\pi$
$449$	30.6708	1.44745	0.723723	+	0.690090i	$0.242429\pi$
$450$	0	0
$451$	29.9420	1.40991
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−17.4426	−0.815933	−0.407966	−	0.912997i	$-0.633762\pi$
$457$	−17.4426	−0.815933	−0.407966	+	0.912997i	$0.633762\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−27.0368	−1.25923	−0.629615	−	0.776907i	$-0.716788\pi$
$461$	−27.0368	−1.25923	−0.629615	+	0.776907i	$0.716788\pi$
$462$	0	0
$463$	−7.90408	−0.367334	−0.183667	−	0.982989i	$-0.558797\pi$
$463$	−7.90408	−0.367334	−0.183667	+	0.982989i	$0.558797\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	14.7845	0.684144	0.342072	−	0.939674i	$-0.388871\pi$
$467$	14.7845	0.684144	0.342072	+	0.939674i	$0.388871\pi$
$468$	0	0
$469$	−14.3274	−0.661576
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3.26636	−0.150187
$474$	0	0
$475$	16.5856	0.760998
$476$	0	0
$477$	0	0
$478$	0	0
$479$	11.5224	0.526473	0.263237	−	0.964731i	$-0.415210\pi$
$479$	11.5224	0.526473	0.263237	+	0.964731i	$0.415210\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.40880	−0.200193
$486$	0	0
$487$	−32.7415	−1.48366	−0.741830	−	0.670588i	$-0.766042\pi$
$487$	−32.7415	−1.48366	−0.741830	+	0.670588i	$0.766042\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−0.106932	−0.00482579	−0.00241289	−	0.999997i	$-0.500768\pi$
$491$	−0.106932	−0.00482579	−0.00241289	+	0.999997i	$0.500768\pi$
$492$	0	0
$493$	−33.7265	−1.51896
$494$	0	0
$495$	0	0
$496$	0	0
$497$	18.7168	0.839562
$498$	0	0
$499$	5.58402	0.249975	0.124988	−	0.992158i	$-0.460111\pi$
$499$	5.58402	0.249975	0.124988	+	0.992158i	$0.460111\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	13.3134	0.593613	0.296807	−	0.954938i	$-0.404078\pi$
$503$	13.3134	0.593613	0.296807	+	0.954938i	$0.404078\pi$
$504$	0	0
$505$	−20.5168	−0.912985
$506$	0	0
$507$	0	0
$508$	0	0
$509$	23.5260	1.04277	0.521386	−	0.853321i	$-0.325415\pi$
$509$	23.5260	1.04277	0.521386	+	0.853321i	$0.325415\pi$
$510$	0	0
$511$	16.4494	0.727677
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−9.76032	−0.430091
$516$	0	0
$517$	−16.9172	−0.744019
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9.20046	0.403079	0.201540	−	0.979480i	$-0.435406\pi$
$521$	9.20046	0.403079	0.201540	+	0.979480i	$0.435406\pi$
$522$	0	0
$523$	−35.5609	−1.55497	−0.777485	−	0.628901i	$-0.783505\pi$
$523$	−35.5609	−1.55497	−0.777485	+	0.628901i	$0.783505\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	22.8605	0.995821
$528$	0	0
$529$	−9.20775	−0.400337
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0.676482	0.0292469
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−42.2911	−1.82161
$540$	0	0
$541$	20.1943	0.868223	0.434111	−	0.900859i	$-0.357062\pi$
$541$	20.1943	0.868223	0.434111	+	0.900859i	$0.357062\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2.44829	−0.104873
$546$	0	0
$547$	20.2218	0.864620	0.432310	−	0.901725i	$-0.357699\pi$
$547$	20.2218	0.864620	0.432310	+	0.901725i	$0.357699\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	19.9879	0.851514
$552$	0	0
$553$	−45.3327	−1.92774
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−5.57673	−0.236294	−0.118147	−	0.992996i	$-0.537695\pi$
$557$	−5.57673	−0.236294	−0.118147	+	0.992996i	$0.537695\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−14.7041	−0.619704	−0.309852	−	0.950785i	$-0.600280\pi$
$563$	−14.7041	−0.619704	−0.309852	+	0.950785i	$0.600280\pi$
$564$	0	0
$565$	−16.5147	−0.694777
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.79417	0.0752154	0.0376077	−	0.999293i	$-0.488026\pi$
$569$	1.79417	0.0752154	0.0376077	+	0.999293i	$0.488026\pi$
$570$	0	0
$571$	38.8025	1.62384	0.811918	−	0.583772i	$-0.198424\pi$
$571$	38.8025	1.62384	0.811918	+	0.583772i	$0.198424\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	13.2384	0.552078
$576$	0	0
$577$	−19.3230	−0.804429	−0.402214	−	0.915545i	$-0.631759\pi$
$577$	−19.3230	−0.804429	−0.402214	+	0.915545i	$0.631759\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−22.5405	−0.935137
$582$	0	0
$583$	20.0586	0.830743
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−26.2054	−1.08161	−0.540805	−	0.841148i	$-0.681880\pi$
$587$	−26.2054	−1.08161	−0.540805	+	0.841148i	$0.681880\pi$
$588$	0	0
$589$	−13.5483	−0.558246
$590$	0	0
$591$	0	0
$592$	0	0
$593$	10.5676	0.433961	0.216980	−	0.976176i	$-0.430379\pi$
$593$	10.5676	0.433961	0.216980	+	0.976176i	$0.430379\pi$
$594$	0	0
$595$	−40.4064	−1.65650
$596$	0	0
$597$	0	0
$598$	0	0
$599$	2.50663	0.102418	0.0512091	−	0.998688i	$-0.483693\pi$
$599$	2.50663	0.102418	0.0512091	+	0.998688i	$0.483693\pi$
$600$	0	0
$601$	−25.3370	−1.03352	−0.516760	−	0.856130i	$-0.672862\pi$
$601$	−25.3370	−1.03352	−0.516760	+	0.856130i	$0.672862\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.15213	−0.128152
$606$	0	0
$607$	−4.62624	−0.187773	−0.0938866	−	0.995583i	$-0.529929\pi$
$607$	−4.62624	−0.187773	−0.0938866	+	0.995583i	$0.529929\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−20.5429	−0.829719	−0.414859	−	0.909885i	$-0.636169\pi$
$613$	−20.5429	−0.829719	−0.414859	+	0.909885i	$0.636169\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	23.9879	0.965717	0.482859	−	0.875698i	$-0.339599\pi$
$617$	23.9879	0.965717	0.482859	+	0.875698i	$0.339599\pi$
$618$	0	0
$619$	33.1153	1.33102	0.665508	−	0.746391i	$-0.268215\pi$
$619$	33.1153	1.33102	0.665508	+	0.746391i	$0.268215\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−40.6684	−1.62935
$624$	0	0
$625$	5.52994	0.221198
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−63.0960	−2.51580
$630$	0	0
$631$	−14.3134	−0.569806	−0.284903	−	0.958556i	$-0.591961\pi$
$631$	−14.3134	−0.569806	−0.284903	+	0.958556i	$0.591961\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	7.55735	0.299904
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−42.7372	−1.68802	−0.844009	−	0.536329i	$-0.819811\pi$
$641$	−42.7372	−1.68802	−0.844009	+	0.536329i	$0.819811\pi$
$642$	0	0
$643$	28.9661	1.14231	0.571157	−	0.820841i	$-0.306495\pi$
$643$	28.9661	1.14231	0.571157	+	0.820841i	$0.306495\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	46.6112	1.83247	0.916237	−	0.400636i	$-0.131211\pi$
$647$	46.6112	1.83247	0.916237	+	0.400636i	$0.131211\pi$
$648$	0	0
$649$	8.60148	0.337638
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−34.1008	−1.33447	−0.667234	−	0.744848i	$-0.732522\pi$
$653$	−34.1008	−1.33447	−0.667234	+	0.744848i	$0.732522\pi$
$654$	0	0
$655$	17.1274	0.669222
$656$	0	0
$657$	0	0
$658$	0	0
$659$	41.5967	1.62038	0.810189	−	0.586169i	$-0.199365\pi$
$659$	41.5967	1.62038	0.810189	+	0.586169i	$0.199365\pi$
$660$	0	0
$661$	10.7536	0.418267	0.209134	−	0.977887i	$-0.432936\pi$
$661$	10.7536	0.418267	0.209134	+	0.977887i	$0.432936\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	23.9468	0.928616
$666$	0	0
$667$	15.9541	0.617744
$668$	0	0
$669$	0	0
$670$	0	0
$671$	23.0998	0.891756
$672$	0	0
$673$	−3.16613	−0.122045	−0.0610227	−	0.998136i	$-0.519436\pi$
$673$	−3.16613	−0.122045	−0.0610227	+	0.998136i	$0.519436\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−25.8323	−0.992817	−0.496409	−	0.868089i	$-0.665348\pi$
$677$	−25.8323	−0.992817	−0.496409	+	0.868089i	$0.665348\pi$
$678$	0	0
$679$	15.8086	0.606680
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.66487	−0.0637046	−0.0318523	−	0.999493i	$-0.510141\pi$
$683$	−1.66487	−0.0637046	−0.0318523	+	0.999493i	$0.510141\pi$
$684$	0	0
$685$	13.4276	0.513042
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−25.0006	−0.951067	−0.475534	−	0.879698i	$-0.657745\pi$
$691$	−25.0006	−0.951067	−0.475534	+	0.879698i	$0.657745\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5.46873	−0.207441
$696$	0	0
$697$	−63.6698	−2.41166
$698$	0	0
$699$	0	0
$700$	0	0
$701$	6.01208	0.227073	0.113537	−	0.993534i	$-0.463782\pi$
$701$	6.01208	0.227073	0.113537	+	0.993534i	$0.463782\pi$
$702$	0	0
$703$	37.3937	1.41033
$704$	0	0
$705$	0	0
$706$	0	0
$707$	73.5672	2.76678
$708$	0	0
$709$	−16.1317	−0.605838	−0.302919	−	0.953016i	$-0.597961\pi$
$709$	−16.1317	−0.605838	−0.302919	+	0.953016i	$0.597961\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−10.8140	−0.404988
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	17.5808	0.655652	0.327826	−	0.944738i	$-0.393684\pi$
$719$	17.5808	0.655652	0.327826	+	0.944738i	$0.393684\pi$
$720$	0	0
$721$	34.9976	1.30338
$722$	0	0
$723$	0	0
$724$	0	0
$725$	15.3134	0.568724
$726$	0	0
$727$	4.18731	0.155299	0.0776493	−	0.996981i	$-0.475259\pi$
$727$	4.18731	0.155299	0.0776493	+	0.996981i	$0.475259\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	6.94571	0.256896
$732$	0	0
$733$	24.7821	0.915347	0.457674	−	0.889120i	$-0.348683\pi$
$733$	24.7821	0.915347	0.457674	+	0.889120i	$0.348683\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12.3134	0.453568
$738$	0	0
$739$	22.8358	0.840028	0.420014	−	0.907518i	$-0.362025\pi$
$739$	22.8358	0.840028	0.420014	+	0.907518i	$0.362025\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	35.2924	1.29475	0.647377	−	0.762170i	$-0.275866\pi$
$743$	35.2924	1.29475	0.647377	+	0.762170i	$0.275866\pi$
$744$	0	0
$745$	25.9675	0.951375
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.42566	−0.0886319
$750$	0	0
$751$	51.4965	1.87913	0.939566	−	0.342367i	$-0.111228\pi$
$751$	51.4965	1.87913	0.939566	+	0.342367i	$0.111228\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	25.1304	0.914587
$756$	0	0
$757$	16.8538	0.612563	0.306282	−	0.951941i	$-0.400915\pi$
$757$	16.8538	0.612563	0.306282	+	0.951941i	$0.400915\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−7.91962	−0.287086	−0.143543	−	0.989644i	$-0.545850\pi$
$761$	−7.91962	−0.287086	−0.143543	+	0.989644i	$0.545850\pi$
$762$	0	0
$763$	8.77884	0.317815
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	16.0043	0.577130	0.288565	−	0.957460i	$-0.406822\pi$
$769$	16.0043	0.577130	0.288565	+	0.957460i	$0.406822\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	38.5080	1.38504	0.692518	−	0.721401i	$-0.256502\pi$
$773$	38.5080	1.38504	0.692518	+	0.721401i	$0.256502\pi$
$774$	0	0
$775$	−10.3797	−0.372851
$776$	0	0
$777$	0	0
$778$	0	0
$779$	37.7338	1.35195
$780$	0	0
$781$	−16.0858	−0.575594
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−4.34375	−0.155035
$786$	0	0
$787$	16.5840	0.591157	0.295578	−	0.955319i	$-0.404488\pi$
$787$	16.5840	0.591157	0.295578	+	0.955319i	$0.404488\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	59.2167	2.10550
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−37.8864	−1.34200	−0.671002	−	0.741456i	$-0.734136\pi$
$797$	−37.8864	−1.34200	−0.671002	+	0.741456i	$0.734136\pi$
$798$	0	0
$799$	35.9734	1.27265
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−14.1371	−0.498886
$804$	0	0
$805$	19.1140	0.673679
$806$	0	0
$807$	0	0
$808$	0	0
$809$	48.1366	1.69239	0.846196	−	0.532871i	$-0.178887\pi$
$809$	48.1366	1.69239	0.846196	+	0.532871i	$0.178887\pi$
$810$	0	0
$811$	38.7743	1.36155	0.680775	−	0.732492i	$-0.261643\pi$
$811$	38.7743	1.36155	0.680775	+	0.732492i	$0.261643\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	26.0847	0.913707
$816$	0	0
$817$	−4.11636	−0.144013
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−25.5730	−0.892504	−0.446252	−	0.894907i	$-0.647242\pi$
$821$	−25.5730	−0.892504	−0.446252	+	0.894907i	$0.647242\pi$
$822$	0	0
$823$	−35.9965	−1.25476	−0.627380	−	0.778713i	$-0.715873\pi$
$823$	−35.9965	−1.25476	−0.627380	+	0.778713i	$0.715873\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−46.2495	−1.60825	−0.804126	−	0.594459i	$-0.797366\pi$
$827$	−46.2495	−1.60825	−0.804126	+	0.594459i	$0.797366\pi$
$828$	0	0
$829$	39.1963	1.36134	0.680671	−	0.732589i	$-0.261688\pi$
$829$	39.1963	1.36134	0.680671	+	0.732589i	$0.261688\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	89.9294	3.11587
$834$	0	0
$835$	3.36850	0.116572
$836$	0	0
$837$	0	0
$838$	0	0
$839$	29.2271	1.00903	0.504516	−	0.863402i	$-0.331671\pi$
$839$	29.2271	1.00903	0.504516	+	0.863402i	$0.331671\pi$
$840$	0	0
$841$	−10.5453	−0.363630
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	11.3026	0.388362
$848$	0	0
$849$	0	0
$850$	0	0
$851$	29.8471	1.02315
$852$	0	0
$853$	14.9638	0.512349	0.256175	−	0.966631i	$-0.417538\pi$
$853$	14.9638	0.512349	0.256175	+	0.966631i	$0.417538\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−10.0537	−0.343428	−0.171714	−	0.985147i	$-0.554930\pi$
$857$	−10.0537	−0.343428	−0.171714	+	0.985147i	$0.554930\pi$
$858$	0	0
$859$	4.52409	0.154360	0.0771800	−	0.997017i	$-0.475408\pi$
$859$	4.52409	0.154360	0.0771800	+	0.997017i	$0.475408\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	38.6582	1.31594	0.657970	−	0.753044i	$-0.271415\pi$
$863$	38.6582	1.31594	0.657970	+	0.753044i	$0.271415\pi$
$864$	0	0
$865$	8.40091	0.285640
$866$	0	0
$867$	0	0
$868$	0	0
$869$	38.9603	1.32164
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	44.0801	1.49018
$876$	0	0
$877$	−44.8200	−1.51346	−0.756732	−	0.653726i	$-0.773205\pi$
$877$	−44.8200	−1.51346	−0.756732	+	0.653726i	$0.773205\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−35.1890	−1.18555	−0.592773	−	0.805369i	$-0.701967\pi$
$881$	−35.1890	−1.18555	−0.592773	+	0.805369i	$0.701967\pi$
$882$	0	0
$883$	−34.6262	−1.16527	−0.582633	−	0.812736i	$-0.697977\pi$
$883$	−34.6262	−1.16527	−0.582633	+	0.812736i	$0.697977\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.10560	−0.0706993	−0.0353496	−	0.999375i	$-0.511254\pi$
$887$	−2.10560	−0.0706993	−0.0353496	+	0.999375i	$0.511254\pi$
$888$	0	0
$889$	−27.0984	−0.908852
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−21.3196	−0.713433
$894$	0	0
$895$	−18.5534	−0.620172
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−12.5090	−0.417199
$900$	0	0
$901$	−42.6534	−1.42099
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6.79848	−0.225989
$906$	0	0
$907$	−16.4373	−0.545791	−0.272895	−	0.962044i	$-0.587981\pi$
$907$	−16.4373	−0.545791	−0.272895	+	0.962044i	$0.587981\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	12.5623	0.416206	0.208103	−	0.978107i	$-0.433271\pi$
$911$	12.5623	0.416206	0.208103	+	0.978107i	$0.433271\pi$
$912$	0	0
$913$	19.3720	0.641118
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−61.4137	−2.02806
$918$	0	0
$919$	−36.2881	−1.19703	−0.598517	−	0.801110i	$-0.704243\pi$
$919$	−36.2881	−1.19703	−0.598517	+	0.801110i	$0.704243\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	28.6485	0.941956
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−21.5719	−0.707752	−0.353876	−	0.935292i	$-0.615137\pi$
$929$	−21.5719	−0.707752	−0.353876	+	0.935292i	$0.615137\pi$
$930$	0	0
$931$	−53.2965	−1.74672
$932$	0	0
$933$	0	0
$934$	0	0
$935$	34.7265	1.13568
$936$	0	0
$937$	5.95599	0.194574	0.0972868	−	0.995256i	$-0.468984\pi$
$937$	5.95599	0.194574	0.0972868	+	0.995256i	$0.468984\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−46.8431	−1.52704	−0.763520	−	0.645784i	$-0.776531\pi$
$941$	−46.8431	−1.52704	−0.763520	+	0.645784i	$0.776531\pi$
$942$	0	0
$943$	30.1185	0.980795
$944$	0	0
$945$	0	0
$946$	0	0
$947$	43.1823	1.40323	0.701617	−	0.712554i	$-0.252462\pi$
$947$	43.1823	1.40323	0.701617	+	0.712554i	$0.252462\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−35.8407	−1.16099	−0.580497	−	0.814262i	$-0.697142\pi$
$953$	−35.8407	−1.16099	−0.580497	+	0.814262i	$0.697142\pi$
$954$	0	0
$955$	−24.6638	−0.798102
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−48.1473	−1.55476
$960$	0	0
$961$	−22.5211	−0.726487
$962$	0	0
$963$	0	0
$964$	0	0
$965$	22.7069	0.730960
$966$	0	0
$967$	−13.9815	−0.449614	−0.224807	−	0.974403i	$-0.572175\pi$
$967$	−13.9815	−0.449614	−0.224807	+	0.974403i	$0.572175\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	40.6098	1.30323	0.651616	−	0.758549i	$-0.274091\pi$
$971$	40.6098	1.30323	0.651616	+	0.758549i	$0.274091\pi$
$972$	0	0
$973$	19.6093	0.628644
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−14.9597	−0.478603	−0.239302	−	0.970945i	$-0.576919\pi$
$977$	−14.9597	−0.478603	−0.239302	+	0.970945i	$0.576919\pi$
$978$	0	0
$979$	34.9517	1.11706
$980$	0	0
$981$	0	0
$982$	0	0
$983$	25.4722	0.812437	0.406218	−	0.913776i	$-0.366847\pi$
$983$	25.4722	0.812437	0.406218	+	0.913776i	$0.366847\pi$
$984$	0	0
$985$	−2.80407	−0.0893450
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.28562	−0.104477
$990$	0	0
$991$	21.1803	0.672816	0.336408	−	0.941716i	$-0.390788\pi$
$991$	21.1803	0.672816	0.336408	+	0.941716i	$0.390788\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−6.75792	−0.214241
$996$	0	0
$997$	52.5954	1.66571	0.832856	−	0.553490i	$-0.186704\pi$
$997$	52.5954	1.66571	0.832856	+	0.553490i	$0.186704\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6084.2.a.x.1.3		3
3.2	odd	2		676.2.a.h.1.3	yes	3
12.11	even	2		2704.2.a.y.1.1		3
13.5	odd	4		6084.2.b.p.4393.4		6
13.8	odd	4		6084.2.b.p.4393.3		6
13.12	even	2		6084.2.a.bc.1.1		3
39.2	even	12		676.2.h.e.485.1		12
39.5	even	4		676.2.d.e.337.5		6
39.8	even	4		676.2.d.e.337.6		6
39.11	even	12		676.2.h.e.485.2		12
39.17	odd	6		676.2.e.f.653.1		6
39.20	even	12		676.2.h.e.361.2		12
39.23	odd	6		676.2.e.f.529.1		6
39.29	odd	6		676.2.e.g.529.1		6
39.32	even	12		676.2.h.e.361.1		12
39.35	odd	6		676.2.e.g.653.1		6
39.38	odd	2		676.2.a.g.1.3	✓	3
156.47	odd	4		2704.2.f.n.337.2		6
156.83	odd	4		2704.2.f.n.337.1		6
156.155	even	2		2704.2.a.x.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
676.2.a.g.1.3	✓	3	39.38	odd	2
676.2.a.h.1.3	yes	3	3.2	odd	2
676.2.d.e.337.5		6	39.5	even	4
676.2.d.e.337.6		6	39.8	even	4
676.2.e.f.529.1		6	39.23	odd	6
676.2.e.f.653.1		6	39.17	odd	6
676.2.e.g.529.1		6	39.29	odd	6
676.2.e.g.653.1		6	39.35	odd	6
676.2.h.e.361.1		12	39.32	even	12
676.2.h.e.361.2		12	39.20	even	12
676.2.h.e.485.1		12	39.2	even	12
676.2.h.e.485.2		12	39.11	even	12
2704.2.a.x.1.1		3	156.155	even	2
2704.2.a.y.1.1		3	12.11	even	2
2704.2.f.n.337.1		6	156.83	odd	4
2704.2.f.n.337.2		6	156.47	odd	4
6084.2.a.x.1.3		3	1.1	even	1	trivial
6084.2.a.bc.1.1		3	13.12	even	2
6084.2.b.p.4393.3		6	13.8	odd	4
6084.2.b.p.4393.4		6	13.5	odd	4

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:	\( N \)	\(=\)	\( 6084 = 2^{2} \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6084.a (trivial)

\(f(q)\)	\(=\)	\(q-1.19806 q^{5} +4.29590 q^{7} -3.69202 q^{11} +7.85086 q^{17} -4.65279 q^{19} -3.71379 q^{23} -3.56465 q^{25} -4.29590 q^{29} +2.91185 q^{31} -5.14675 q^{35} -8.03684 q^{37} -8.10992 q^{41} +0.884707 q^{43} +4.58211 q^{47} +11.4547 q^{49} -5.43296 q^{53} +4.42327 q^{55} -2.32975 q^{59} -6.25667 q^{61} -3.33513 q^{67} +4.35690 q^{71} +3.82908 q^{73} -15.8605 q^{77} -10.5526 q^{79} -5.24698 q^{83} -9.40581 q^{85} -9.46681 q^{89} +5.57434 q^{95} +3.67994 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 8 q^{5} - q^{7} - 6 q^{11} + 10 q^{17} + 4 q^{19} - 3 q^{23} + 11 q^{25} + q^{29} + 5 q^{31} + 12 q^{35} + 4 q^{37} - 25 q^{41} + 5 q^{43} + 8 q^{47} + 12 q^{49} + 3 q^{53} + 16 q^{55} - 9 q^{59}+ \cdots - 13 q^{97}+O(q^{100}) \) 3 * q - 8 * q^5 - q^7 - 6 * q^11 + 10 * q^17 + 4 * q^19 - 3 * q^23 + 11 * q^25 + q^29 + 5 * q^31 + 12 * q^35 + 4 * q^37 - 25 * q^41 + 5 * q^43 + 8 * q^47 + 12 * q^49 + 3 * q^53 + 16 * q^55 - 9 * q^59 + 8 * q^61 - 9 * q^67 + 9 * q^71 + q^73 - 12 * q^77 + 9 * q^79 - 11 * q^83 - 15 * q^85 - 25 * q^89 - 27 * q^95 - 13 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more