LMFDB - Embedded newform 6084.2.a.y.1.2

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,-2,0,-6,0,0,0,5,0,0,0,0,0,-13,0,1,0,0,0,0,0,-9,0,0,0, 4,0,-2,0,0,0,4,0,2,0,0,0,11,0,-9] 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:	$0$
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 2028)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.24698$ 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)

$f(q)$	$=$	$q-0.554958 q^{5} +1.04892 q^{7} +2.91185 q^{11} -2.75302 q^{17} -4.63102 q^{19} +5.76271 q^{23} -4.69202 q^{25} +2.80194 q^{29} +4.18598 q^{31} -0.582105 q^{35} -0.466812 q^{37} -3.89977 q^{41} +9.19567 q^{43} +11.5211 q^{47} -5.89977 q^{49} -5.62565 q^{53} -1.61596 q^{55} +3.10992 q^{59} +10.9051 q^{61} -8.04892 q^{67} +13.6920 q^{71} -9.36658 q^{73} +3.05429 q^{77} -3.60925 q^{79} +1.65519 q^{83} +1.52781 q^{85} +17.9705 q^{89} +2.57002 q^{95} +1.31767 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{5} - 6 q^{7} + 5 q^{11} - 13 q^{17} + q^{19} - 9 q^{25} + 4 q^{29} - 2 q^{31} + 4 q^{35} + 2 q^{37} + 11 q^{41} - 9 q^{43} + 19 q^{47} + 5 q^{49} - 5 q^{53} - 15 q^{55} + 10 q^{59} + 7 q^{61}+ \cdots - 13 q^{97}+O(q^{100}) $ 3 * q - 2 * q^5 - 6 * q^7 + 5 * q^11 - 13 * q^17 + q^19 - 9 * q^25 + 4 * q^29 - 2 * q^31 + 4 * q^35 + 2 * q^37 + 11 * q^41 - 9 * q^43 + 19 * q^47 + 5 * q^49 - 5 * q^53 - 15 * q^55 + 10 * q^59 + 7 * q^61 - 15 * q^67 + 36 * q^71 - 2 * q^73 - 3 * q^77 + q^79 + 28 * q^83 + 11 * q^85 + 19 * q^89 - 17 * 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
$5$	−0.554958	−0.248185	−0.124092	−	0.992271i	$-0.539602\pi$
$5$	−0.554958	−0.248185	−0.124092	+	0.992271i	$0.539602\pi$
$6$	0	0
$7$	1.04892	0.396453	0.198227	−	0.980156i	$-0.436482\pi$
$7$	1.04892	0.396453	0.198227	+	0.980156i	$0.436482\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.91185	0.877957	0.438979	−	0.898498i	$-0.355340\pi$
$11$	2.91185	0.877957	0.438979	+	0.898498i	$0.355340\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.75302	−0.667706	−0.333853	−	0.942625i	$-0.608349\pi$
$17$	−2.75302	−0.667706	−0.333853	+	0.942625i	$0.608349\pi$
$18$	0	0
$19$	−4.63102	−1.06243	−0.531215	−	0.847237i	$-0.678264\pi$
$19$	−4.63102	−1.06243	−0.531215	+	0.847237i	$0.678264\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.76271	1.20161	0.600804	−	0.799396i	$-0.294847\pi$
$23$	5.76271	1.20161	0.600804	+	0.799396i	$0.294847\pi$
$24$	0	0
$25$	−4.69202	−0.938404
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.80194	0.520307	0.260153	−	0.965567i	$-0.416227\pi$
$29$	2.80194	0.520307	0.260153	+	0.965567i	$0.416227\pi$
$30$	0	0
$31$	4.18598	0.751824	0.375912	−	0.926655i	$-0.377329\pi$
$31$	4.18598	0.751824	0.375912	+	0.926655i	$0.377329\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.582105	−0.0983937
$36$	0	0
$37$	−0.466812	−0.0767434	−0.0383717	−	0.999264i	$-0.512217\pi$
$37$	−0.466812	−0.0767434	−0.0383717	+	0.999264i	$0.512217\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.89977	−0.609042	−0.304521	−	0.952506i	$-0.598496\pi$
$41$	−3.89977	−0.609042	−0.304521	+	0.952506i	$0.598496\pi$
$42$	0	0
$43$	9.19567	1.40233	0.701163	−	0.713001i	$-0.252664\pi$
$43$	9.19567	1.40233	0.701163	+	0.713001i	$0.252664\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.5211	1.68053	0.840263	−	0.542179i	$-0.182401\pi$
$47$	11.5211	1.68053	0.840263	+	0.542179i	$0.182401\pi$
$48$	0	0
$49$	−5.89977	−0.842825
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.62565	−0.772742	−0.386371	−	0.922343i	$-0.626272\pi$
$53$	−5.62565	−0.772742	−0.386371	+	0.922343i	$0.626272\pi$
$54$	0	0
$55$	−1.61596	−0.217896
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.10992	0.404877	0.202438	−	0.979295i	$-0.435113\pi$
$59$	3.10992	0.404877	0.202438	+	0.979295i	$0.435113\pi$
$60$	0	0
$61$	10.9051	1.39626	0.698131	−	0.715970i	$-0.254015\pi$
$61$	10.9051	1.39626	0.698131	+	0.715970i	$0.254015\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.04892	−0.983332	−0.491666	−	0.870784i	$-0.663612\pi$
$67$	−8.04892	−0.983332	−0.491666	+	0.870784i	$0.663612\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.6920	1.62494	0.812472	−	0.583000i	$-0.198121\pi$
$71$	13.6920	1.62494	0.812472	+	0.583000i	$0.198121\pi$
$72$	0	0
$73$	−9.36658	−1.09628	−0.548138	−	0.836388i	$-0.684663\pi$
$73$	−9.36658	−1.09628	−0.548138	+	0.836388i	$0.684663\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.05429	0.348069
$78$	0	0
$79$	−3.60925	−0.406073	−0.203036	−	0.979171i	$-0.565081\pi$
$79$	−3.60925	−0.406073	−0.203036	+	0.979171i	$0.565081\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.65519	0.181680	0.0908401	−	0.995865i	$-0.471045\pi$
$83$	1.65519	0.181680	0.0908401	+	0.995865i	$0.471045\pi$
$84$	0	0
$85$	1.52781	0.165714
$86$	0	0
$87$	0	0
$88$	0	0
$89$	17.9705	1.90486	0.952432	−	0.304750i	$-0.0985728\pi$
$89$	17.9705	1.90486	0.952432	+	0.304750i	$0.0985728\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.57002	0.263679
$96$	0	0
$97$	1.31767	0.133789	0.0668944	−	0.997760i	$-0.478691\pi$
$97$	1.31767	0.133789	0.0668944	+	0.997760i	$0.478691\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15.0248	1.49502	0.747509	−	0.664251i	$-0.231249\pi$
$101$	15.0248	1.49502	0.747509	+	0.664251i	$0.231249\pi$
$102$	0	0
$103$	−9.20775	−0.907267	−0.453633	−	0.891188i	$-0.649872\pi$
$103$	−9.20775	−0.907267	−0.453633	+	0.891188i	$0.649872\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.22952	−0.698904	−0.349452	−	0.936954i	$-0.613632\pi$
$107$	−7.22952	−0.698904	−0.349452	+	0.936954i	$0.613632\pi$
$108$	0	0
$109$	15.5036	1.48498	0.742490	−	0.669857i	$-0.233645\pi$
$109$	15.5036	1.48498	0.742490	+	0.669857i	$0.233645\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−13.8267	−1.30071	−0.650353	−	0.759632i	$-0.725379\pi$
$113$	−13.8267	−1.30071	−0.650353	+	0.759632i	$0.725379\pi$
$114$	0	0
$115$	−3.19806	−0.298221
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.88769	−0.264714
$120$	0	0
$121$	−2.52111	−0.229191
$122$	0	0
$123$	0	0
$124$	0	0
$125$	5.37867	0.481083
$126$	0	0
$127$	7.17629	0.636793	0.318396	−	0.947958i	$-0.396856\pi$
$127$	7.17629	0.636793	0.318396	+	0.947958i	$0.396856\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−13.1304	−1.14720	−0.573602	−	0.819134i	$-0.694455\pi$
$131$	−13.1304	−1.14720	−0.573602	+	0.819134i	$0.694455\pi$
$132$	0	0
$133$	−4.85756	−0.421204
$134$	0	0
$135$	0	0
$136$	0	0
$137$	21.8116	1.86349	0.931747	−	0.363109i	$-0.118285\pi$
$137$	21.8116	1.86349	0.931747	+	0.363109i	$0.118285\pi$
$138$	0	0
$139$	−2.96615	−0.251585	−0.125793	−	0.992057i	$-0.540147\pi$
$139$	−2.96615	−0.251585	−0.125793	+	0.992057i	$0.540147\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−1.55496	−0.129132
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−5.69633	−0.466662	−0.233331	−	0.972397i	$-0.574963\pi$
$149$	−5.69633	−0.466662	−0.233331	+	0.972397i	$0.574963\pi$
$150$	0	0
$151$	−19.1468	−1.55814	−0.779070	−	0.626937i	$-0.784309\pi$
$151$	−19.1468	−1.55814	−0.779070	+	0.626937i	$0.784309\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.32304	−0.186591
$156$	0	0
$157$	−8.38404	−0.669119	−0.334560	−	0.942375i	$-0.608588\pi$
$157$	−8.38404	−0.669119	−0.334560	+	0.942375i	$0.608588\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	6.04461	0.476382
$162$	0	0
$163$	1.97046	0.154338	0.0771692	−	0.997018i	$-0.475412\pi$
$163$	1.97046	0.154338	0.0771692	+	0.997018i	$0.475412\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	24.7995	1.91905	0.959523	−	0.281630i	$-0.0908749\pi$
$167$	24.7995	1.91905	0.959523	+	0.281630i	$0.0908749\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.57135	0.195496	0.0977481	−	0.995211i	$-0.468836\pi$
$173$	2.57135	0.195496	0.0977481	+	0.995211i	$0.468836\pi$
$174$	0	0
$175$	−4.92154	−0.372034
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3.00538	0.224632	0.112316	−	0.993673i	$-0.464173\pi$
$179$	3.00538	0.224632	0.112316	+	0.993673i	$0.464173\pi$
$180$	0	0
$181$	−14.6843	−1.09147	−0.545736	−	0.837957i	$-0.683750\pi$
$181$	−14.6843	−1.09147	−0.545736	+	0.837957i	$0.683750\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0.259061	0.0190466
$186$	0	0
$187$	−8.01639	−0.586217
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.0804	1.23589	0.617946	−	0.786220i	$-0.287965\pi$
$191$	17.0804	1.23589	0.617946	+	0.786220i	$0.287965\pi$
$192$	0	0
$193$	14.9758	1.07798	0.538992	−	0.842311i	$-0.318805\pi$
$193$	14.9758	1.07798	0.538992	+	0.842311i	$0.318805\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.192685	0.0137283	0.00686414	−	0.999976i	$-0.497815\pi$
$197$	0.192685	0.0137283	0.00686414	+	0.999976i	$0.497815\pi$
$198$	0	0
$199$	11.8726	0.841628	0.420814	−	0.907147i	$-0.361744\pi$
$199$	11.8726	0.841628	0.420814	+	0.907147i	$0.361744\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.93900	0.206277
$204$	0	0
$205$	2.16421	0.151155
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−13.4849	−0.932767
$210$	0	0
$211$	6.03385	0.415387	0.207694	−	0.978194i	$-0.433404\pi$
$211$	6.03385	0.415387	0.207694	+	0.978194i	$0.433404\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.10321	−0.348036
$216$	0	0
$217$	4.39075	0.298063
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−8.52648	−0.570976	−0.285488	−	0.958382i	$-0.592156\pi$
$223$	−8.52648	−0.570976	−0.285488	+	0.958382i	$0.592156\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.0000	0.730096	0.365048	−	0.930989i	$-0.381053\pi$
$227$	11.0000	0.730096	0.365048	+	0.930989i	$0.381053\pi$
$228$	0	0
$229$	17.9119	1.18365	0.591824	−	0.806067i	$-0.298408\pi$
$229$	17.9119	1.18365	0.591824	+	0.806067i	$0.298408\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−22.8159	−1.49472	−0.747361	−	0.664418i	$-0.768679\pi$
$233$	−22.8159	−1.49472	−0.747361	+	0.664418i	$0.768679\pi$
$234$	0	0
$235$	−6.39373	−0.417081
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8.34481	0.539781	0.269891	−	0.962891i	$-0.413012\pi$
$239$	8.34481	0.539781	0.269891	+	0.962891i	$0.413012\pi$
$240$	0	0
$241$	3.75541	0.241907	0.120954	−	0.992658i	$-0.461405\pi$
$241$	3.75541	0.241907	0.120954	+	0.992658i	$0.461405\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.27413	0.209176
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	30.3086	1.91306	0.956530	−	0.291634i	$-0.0941989\pi$
$251$	30.3086	1.91306	0.956530	+	0.291634i	$0.0941989\pi$
$252$	0	0
$253$	16.7802	1.05496
$254$	0	0
$255$	0	0
$256$	0	0
$257$	20.8412	1.30004	0.650018	−	0.759919i	$-0.274761\pi$
$257$	20.8412	1.30004	0.650018	+	0.759919i	$0.274761\pi$
$258$	0	0
$259$	−0.489647	−0.0304252
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−8.58642	−0.529461	−0.264731	−	0.964322i	$-0.585283\pi$
$263$	−8.58642	−0.529461	−0.264731	+	0.964322i	$0.585283\pi$
$264$	0	0
$265$	3.12200	0.191783
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−14.8267	−0.903999	−0.452000	−	0.892018i	$-0.649289\pi$
$269$	−14.8267	−0.903999	−0.452000	+	0.892018i	$0.649289\pi$
$270$	0	0
$271$	7.23490	0.439489	0.219744	−	0.975557i	$-0.429478\pi$
$271$	7.23490	0.439489	0.219744	+	0.975557i	$0.429478\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−13.6625	−0.823879
$276$	0	0
$277$	11.2054	0.673265	0.336632	−	0.941636i	$-0.390712\pi$
$277$	11.2054	0.673265	0.336632	+	0.941636i	$0.390712\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−12.3002	−0.733769	−0.366884	−	0.930267i	$-0.619576\pi$
$281$	−12.3002	−0.733769	−0.366884	+	0.930267i	$0.619576\pi$
$282$	0	0
$283$	24.3521	1.44758	0.723791	−	0.690019i	$-0.242398\pi$
$283$	24.3521	1.44758	0.723791	+	0.690019i	$0.242398\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.09054	−0.241457
$288$	0	0
$289$	−9.42088	−0.554169
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6.20237	0.362347	0.181173	−	0.983451i	$-0.442011\pi$
$293$	6.20237	0.362347	0.181173	+	0.983451i	$0.442011\pi$
$294$	0	0
$295$	−1.72587	−0.100484
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	9.64550	0.555957
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−6.05190	−0.346531
$306$	0	0
$307$	26.4795	1.51126	0.755632	−	0.654996i	$-0.227330\pi$
$307$	26.4795	1.51126	0.755632	+	0.654996i	$0.227330\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−15.5308	−0.880671	−0.440335	−	0.897833i	$-0.645140\pi$
$311$	−15.5308	−0.880671	−0.440335	+	0.897833i	$0.645140\pi$
$312$	0	0
$313$	16.9681	0.959092	0.479546	−	0.877517i	$-0.340801\pi$
$313$	16.9681	0.959092	0.479546	+	0.877517i	$0.340801\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	25.5066	1.43260	0.716298	−	0.697795i	$-0.245835\pi$
$317$	25.5066	1.43260	0.716298	+	0.697795i	$0.245835\pi$
$318$	0	0
$319$	8.15883	0.456807
$320$	0	0
$321$	0	0
$322$	0	0
$323$	12.7493	0.709390
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	12.0847	0.666250
$330$	0	0
$331$	−10.6136	−0.583374	−0.291687	−	0.956514i	$-0.594217\pi$
$331$	−10.6136	−0.583374	−0.291687	+	0.956514i	$0.594217\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4.46681	0.244048
$336$	0	0
$337$	29.2717	1.59453	0.797266	−	0.603628i	$-0.206279\pi$
$337$	29.2717	1.59453	0.797266	+	0.603628i	$0.206279\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	12.1890	0.660069
$342$	0	0
$343$	−13.5308	−0.730594
$344$	0	0
$345$	0	0
$346$	0	0
$347$	34.8146	1.86895	0.934473	−	0.356034i	$-0.115871\pi$
$347$	34.8146	1.86895	0.934473	+	0.356034i	$0.115871\pi$
$348$	0	0
$349$	30.1957	1.61634	0.808169	−	0.588951i	$-0.200459\pi$
$349$	30.1957	1.61634	0.808169	+	0.588951i	$0.200459\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−24.9691	−1.32897	−0.664486	−	0.747300i	$-0.731350\pi$
$353$	−24.9691	−1.32897	−0.664486	+	0.747300i	$0.731350\pi$
$354$	0	0
$355$	−7.59850	−0.403286
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−14.1661	−0.747660	−0.373830	−	0.927497i	$-0.621956\pi$
$359$	−14.1661	−0.747660	−0.373830	+	0.927497i	$0.621956\pi$
$360$	0	0
$361$	2.44637	0.128756
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5.19806	0.272079
$366$	0	0
$367$	8.78687	0.458671	0.229335	−	0.973347i	$-0.426345\pi$
$367$	8.78687	0.458671	0.229335	+	0.973347i	$0.426345\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−5.90084	−0.306356
$372$	0	0
$373$	−18.3260	−0.948886	−0.474443	−	0.880286i	$-0.657350\pi$
$373$	−18.3260	−0.948886	−0.474443	+	0.880286i	$0.657350\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	4.77586	0.245319	0.122660	−	0.992449i	$-0.460858\pi$
$379$	4.77586	0.245319	0.122660	+	0.992449i	$0.460858\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6.19029	0.316309	0.158155	−	0.987414i	$-0.449446\pi$
$383$	6.19029	0.316309	0.158155	+	0.987414i	$0.449446\pi$
$384$	0	0
$385$	−1.69501	−0.0863855
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−24.8780	−1.26136	−0.630682	−	0.776041i	$-0.717225\pi$
$389$	−24.8780	−1.26136	−0.630682	+	0.776041i	$0.717225\pi$
$390$	0	0
$391$	−15.8649	−0.802320
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.00298	0.100781
$396$	0	0
$397$	4.62027	0.231885	0.115942	−	0.993256i	$-0.463011\pi$
$397$	4.62027	0.231885	0.115942	+	0.993256i	$0.463011\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.64848	−0.0823212	−0.0411606	−	0.999153i	$-0.513106\pi$
$401$	−1.64848	−0.0823212	−0.0411606	+	0.999153i	$0.513106\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.35929	−0.0673774
$408$	0	0
$409$	34.8049	1.72099	0.860496	−	0.509457i	$-0.170154\pi$
$409$	34.8049	1.72099	0.860496	+	0.509457i	$0.170154\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3.26205	0.160515
$414$	0	0
$415$	−0.918559	−0.0450903
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−21.6353	−1.05696	−0.528478	−	0.848947i	$-0.677237\pi$
$419$	−21.6353	−1.05696	−0.528478	+	0.848947i	$0.677237\pi$
$420$	0	0
$421$	35.1008	1.71071	0.855355	−	0.518043i	$-0.173339\pi$
$421$	35.1008	1.71071	0.855355	+	0.518043i	$0.173339\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	12.9172	0.626578
$426$	0	0
$427$	11.4386	0.553553
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−20.0925	−0.967820	−0.483910	−	0.875118i	$-0.660784\pi$
$431$	−20.0925	−0.967820	−0.483910	+	0.875118i	$0.660784\pi$
$432$	0	0
$433$	−18.6939	−0.898373	−0.449187	−	0.893438i	$-0.648286\pi$
$433$	−18.6939	−0.898373	−0.449187	+	0.893438i	$0.648286\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−26.6872	−1.27662
$438$	0	0
$439$	1.35019	0.0644411	0.0322206	−	0.999481i	$-0.489742\pi$
$439$	1.35019	0.0644411	0.0322206	+	0.999481i	$0.489742\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0.400436	0.0190253	0.00951265	−	0.999955i	$-0.496972\pi$
$443$	0.400436	0.0190253	0.00951265	+	0.999955i	$0.496972\pi$
$444$	0	0
$445$	−9.97285	−0.472759
$446$	0	0
$447$	0	0
$448$	0	0
$449$	35.7904	1.68906	0.844528	−	0.535512i	$-0.179881\pi$
$449$	35.7904	1.68906	0.844528	+	0.535512i	$0.179881\pi$
$450$	0	0
$451$	−11.3556	−0.534713
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−5.17092	−0.241885	−0.120943	−	0.992660i	$-0.538592\pi$
$457$	−5.17092	−0.241885	−0.120943	+	0.992660i	$0.538592\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−20.8780	−0.972386	−0.486193	−	0.873852i	$-0.661615\pi$
$461$	−20.8780	−0.972386	−0.486193	+	0.873852i	$0.661615\pi$
$462$	0	0
$463$	30.7198	1.42767	0.713834	−	0.700315i	$-0.246957\pi$
$463$	30.7198	1.42767	0.713834	+	0.700315i	$0.246957\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−15.2741	−0.706802	−0.353401	−	0.935472i	$-0.614975\pi$
$467$	−15.2741	−0.706802	−0.353401	+	0.935472i	$0.614975\pi$
$468$	0	0
$469$	−8.44265	−0.389845
$470$	0	0
$471$	0	0
$472$	0	0
$473$	26.7764	1.23118
$474$	0	0
$475$	21.7289	0.996988
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3.05861	0.139751	0.0698756	−	0.997556i	$-0.477740\pi$
$479$	3.05861	0.139751	0.0698756	+	0.997556i	$0.477740\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−0.731250	−0.0332044
$486$	0	0
$487$	−15.1521	−0.686608	−0.343304	−	0.939224i	$-0.611546\pi$
$487$	−15.1521	−0.686608	−0.343304	+	0.939224i	$0.611546\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−23.4795	−1.05961	−0.529807	−	0.848118i	$-0.677736\pi$
$491$	−23.4795	−1.05961	−0.529807	+	0.848118i	$0.677736\pi$
$492$	0	0
$493$	−7.71379	−0.347412
$494$	0	0
$495$	0	0
$496$	0	0
$497$	14.3618	0.644215
$498$	0	0
$499$	−38.0689	−1.70420	−0.852099	−	0.523381i	$-0.824670\pi$
$499$	−38.0689	−1.70420	−0.852099	+	0.523381i	$0.824670\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	11.1274	0.496145	0.248073	−	0.968741i	$-0.420203\pi$
$503$	11.1274	0.496145	0.248073	+	0.968741i	$0.420203\pi$
$504$	0	0
$505$	−8.33811	−0.371041
$506$	0	0
$507$	0	0
$508$	0	0
$509$	26.7633	1.18626	0.593131	−	0.805106i	$-0.297892\pi$
$509$	26.7633	1.18626	0.593131	+	0.805106i	$0.297892\pi$
$510$	0	0
$511$	−9.82477	−0.434622
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.10992	0.225170
$516$	0	0
$517$	33.5478	1.47543
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−20.5797	−0.901614	−0.450807	−	0.892622i	$-0.648864\pi$
$521$	−20.5797	−0.901614	−0.450807	+	0.892622i	$0.648864\pi$
$522$	0	0
$523$	−6.77479	−0.296241	−0.148120	−	0.988969i	$-0.547322\pi$
$523$	−6.77479	−0.296241	−0.148120	+	0.988969i	$0.547322\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−11.5241	−0.501997
$528$	0	0
$529$	10.2088	0.443862
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	4.01208	0.173457
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−17.1793	−0.739964
$540$	0	0
$541$	−1.94331	−0.0835495	−0.0417748	−	0.999127i	$-0.513301\pi$
$541$	−1.94331	−0.0835495	−0.0417748	+	0.999127i	$0.513301\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−8.60388	−0.368550
$546$	0	0
$547$	−39.1323	−1.67318	−0.836588	−	0.547833i	$-0.815453\pi$
$547$	−39.1323	−1.67318	−0.836588	+	0.547833i	$0.815453\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−12.9758	−0.552789
$552$	0	0
$553$	−3.78581	−0.160989
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−24.4989	−1.03805	−0.519025	−	0.854759i	$-0.673705\pi$
$557$	−24.4989	−1.03805	−0.519025	+	0.854759i	$0.673705\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	21.8267	0.919885	0.459943	−	0.887949i	$-0.347870\pi$
$563$	21.8267	0.919885	0.459943	+	0.887949i	$0.347870\pi$
$564$	0	0
$565$	7.67324	0.322815
$566$	0	0
$567$	0	0
$568$	0	0
$569$	11.5627	0.484735	0.242367	−	0.970185i	$-0.422076\pi$
$569$	11.5627	0.484735	0.242367	+	0.970185i	$0.422076\pi$
$570$	0	0
$571$	24.7614	1.03623	0.518116	−	0.855310i	$-0.326634\pi$
$571$	24.7614	1.03623	0.518116	+	0.855310i	$0.326634\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−27.0388	−1.12759
$576$	0	0
$577$	−13.5090	−0.562388	−0.281194	−	0.959651i	$-0.590730\pi$
$577$	−13.5090	−0.562388	−0.281194	+	0.959651i	$0.590730\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.73615	0.0720278
$582$	0	0
$583$	−16.3811	−0.678434
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−6.70304	−0.276664	−0.138332	−	0.990386i	$-0.544174\pi$
$587$	−6.70304	−0.276664	−0.138332	+	0.990386i	$0.544174\pi$
$588$	0	0
$589$	−19.3854	−0.798760
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−16.0194	−0.657837	−0.328918	−	0.944358i	$-0.606684\pi$
$593$	−16.0194	−0.657837	−0.328918	+	0.944358i	$0.606684\pi$
$594$	0	0
$595$	1.60255	0.0656980
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−19.8243	−0.809999	−0.404999	−	0.914317i	$-0.632728\pi$
$599$	−19.8243	−0.809999	−0.404999	+	0.914317i	$0.632728\pi$
$600$	0	0
$601$	29.1909	1.19072	0.595360	−	0.803459i	$-0.297009\pi$
$601$	29.1909	1.19072	0.595360	+	0.803459i	$0.297009\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.39911	0.0568818
$606$	0	0
$607$	−6.36227	−0.258237	−0.129118	−	0.991629i	$-0.541215\pi$
$607$	−6.36227	−0.258237	−0.129118	+	0.991629i	$0.541215\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−30.7711	−1.24283	−0.621416	−	0.783481i	$-0.713442\pi$
$613$	−30.7711	−1.24283	−0.621416	+	0.783481i	$0.713442\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.0242	0.484075	0.242037	−	0.970267i	$-0.422184\pi$
$617$	12.0242	0.484075	0.242037	+	0.970267i	$0.422184\pi$
$618$	0	0
$619$	−9.02715	−0.362832	−0.181416	−	0.983406i	$-0.558068\pi$
$619$	−9.02715	−0.362832	−0.181416	+	0.983406i	$0.558068\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	18.8495	0.755190
$624$	0	0
$625$	20.4752	0.819007
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.28514	0.0512420
$630$	0	0
$631$	−15.5526	−0.619138	−0.309569	−	0.950877i	$-0.600185\pi$
$631$	−15.5526	−0.619138	−0.309569	+	0.950877i	$0.600185\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3.98254	−0.158042
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	19.4209	0.767079	0.383539	−	0.923525i	$-0.374705\pi$
$641$	19.4209	0.767079	0.383539	+	0.923525i	$0.374705\pi$
$642$	0	0
$643$	26.8267	1.05794	0.528971	−	0.848640i	$-0.322578\pi$
$643$	26.8267	1.05794	0.528971	+	0.848640i	$0.322578\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−5.08815	−0.200036	−0.100018	−	0.994986i	$-0.531890\pi$
$647$	−5.08815	−0.200036	−0.100018	+	0.994986i	$0.531890\pi$
$648$	0	0
$649$	9.05562	0.355464
$650$	0	0
$651$	0	0
$652$	0	0
$653$	19.7157	0.771535	0.385768	−	0.922596i	$-0.373937\pi$
$653$	19.7157	0.771535	0.385768	+	0.922596i	$0.373937\pi$
$654$	0	0
$655$	7.28680	0.284719
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−9.42519	−0.367153	−0.183577	−	0.983005i	$-0.558768\pi$
$659$	−9.42519	−0.367153	−0.183577	+	0.983005i	$0.558768\pi$
$660$	0	0
$661$	29.9946	1.16666	0.583328	−	0.812237i	$-0.301750\pi$
$661$	29.9946	1.16666	0.583328	+	0.812237i	$0.301750\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.69574	0.104536
$666$	0	0
$667$	16.1468	0.625205
$668$	0	0
$669$	0	0
$670$	0	0
$671$	31.7542	1.22586
$672$	0	0
$673$	−47.7294	−1.83984	−0.919918	−	0.392112i	$-0.871745\pi$
$673$	−47.7294	−1.83984	−0.919918	+	0.392112i	$0.871745\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	42.9124	1.64926	0.824630	−	0.565673i	$-0.191384\pi$
$677$	42.9124	1.64926	0.824630	+	0.565673i	$0.191384\pi$
$678$	0	0
$679$	1.38212	0.0530411
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0.970460	0.0371336	0.0185668	−	0.999828i	$-0.494090\pi$
$683$	0.970460	0.0371336	0.0185668	+	0.999828i	$0.494090\pi$
$684$	0	0
$685$	−12.1045	−0.462491
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−1.16660	−0.0443797	−0.0221898	−	0.999754i	$-0.507064\pi$
$691$	−1.16660	−0.0443797	−0.0221898	+	0.999754i	$0.507064\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.64609	0.0624397
$696$	0	0
$697$	10.7362	0.406661
$698$	0	0
$699$	0	0
$700$	0	0
$701$	17.3274	0.654445	0.327223	−	0.944947i	$-0.393887\pi$
$701$	17.3274	0.654445	0.327223	+	0.944947i	$0.393887\pi$
$702$	0	0
$703$	2.16182	0.0815345
$704$	0	0
$705$	0	0
$706$	0	0
$707$	15.7597	0.592705
$708$	0	0
$709$	−16.4601	−0.618172	−0.309086	−	0.951034i	$-0.600023\pi$
$709$	−16.4601	−0.618172	−0.309086	+	0.951034i	$0.600023\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	24.1226	0.903398
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−14.1715	−0.528508	−0.264254	−	0.964453i	$-0.585126\pi$
$719$	−14.1715	−0.528508	−0.264254	+	0.964453i	$0.585126\pi$
$720$	0	0
$721$	−9.65817	−0.359689
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−13.1468	−0.488258
$726$	0	0
$727$	−37.8810	−1.40493	−0.702464	−	0.711719i	$-0.747917\pi$
$727$	−37.8810	−1.40493	−0.702464	+	0.711719i	$0.747917\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−25.3159	−0.936341
$732$	0	0
$733$	−0.377338	−0.0139373	−0.00696865	−	0.999976i	$-0.502218\pi$
$733$	−0.377338	−0.0139373	−0.00696865	+	0.999976i	$0.502218\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−23.4373	−0.863323
$738$	0	0
$739$	−1.12631	−0.0414320	−0.0207160	−	0.999785i	$-0.506595\pi$
$739$	−1.12631	−0.0414320	−0.0207160	+	0.999785i	$0.506595\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−22.7047	−0.832954	−0.416477	−	0.909146i	$-0.636735\pi$
$743$	−22.7047	−0.832954	−0.416477	+	0.909146i	$0.636735\pi$
$744$	0	0
$745$	3.16123	0.115818
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−7.58317	−0.277083
$750$	0	0
$751$	13.9282	0.508249	0.254124	−	0.967172i	$-0.418213\pi$
$751$	13.9282	0.508249	0.254124	+	0.967172i	$0.418213\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	10.6256	0.386707
$756$	0	0
$757$	1.16660	0.0424009	0.0212005	−	0.999775i	$-0.493251\pi$
$757$	1.16660	0.0424009	0.0212005	+	0.999775i	$0.493251\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0.821789	0.0297898	0.0148949	−	0.999889i	$-0.495259\pi$
$761$	0.821789	0.0297898	0.0148949	+	0.999889i	$0.495259\pi$
$762$	0	0
$763$	16.2620	0.588726
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	33.7017	1.21531	0.607657	−	0.794199i	$-0.292109\pi$
$769$	33.7017	1.21531	0.607657	+	0.794199i	$0.292109\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−0.764037	−0.0274805	−0.0137403	−	0.999906i	$-0.504374\pi$
$773$	−0.764037	−0.0274805	−0.0137403	+	0.999906i	$0.504374\pi$
$774$	0	0
$775$	−19.6407	−0.705515
$776$	0	0
$777$	0	0
$778$	0	0
$779$	18.0599	0.647064
$780$	0	0
$781$	39.8692	1.42663
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.65279	0.166065
$786$	0	0
$787$	−12.1739	−0.433953	−0.216976	−	0.976177i	$-0.569619\pi$
$787$	−12.1739	−0.433953	−0.216976	+	0.976177i	$0.569619\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−14.5031	−0.515669
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−38.2887	−1.35626	−0.678128	−	0.734944i	$-0.737209\pi$
$797$	−38.2887	−1.35626	−0.678128	+	0.734944i	$0.737209\pi$
$798$	0	0
$799$	−31.7178	−1.12210
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−27.2741	−0.962483
$804$	0	0
$805$	−3.35450	−0.118231
$806$	0	0
$807$	0	0
$808$	0	0
$809$	29.9638	1.05347	0.526735	−	0.850030i	$-0.323416\pi$
$809$	29.9638	1.05347	0.526735	+	0.850030i	$0.323416\pi$
$810$	0	0
$811$	46.3521	1.62764	0.813821	−	0.581115i	$-0.197383\pi$
$811$	46.3521	1.62764	0.813821	+	0.581115i	$0.197383\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.09352	−0.0383044
$816$	0	0
$817$	−42.5854	−1.48987
$818$	0	0
$819$	0	0
$820$	0	0
$821$	48.3105	1.68605	0.843024	−	0.537876i	$-0.180773\pi$
$821$	48.3105	1.68605	0.843024	+	0.537876i	$0.180773\pi$
$822$	0	0
$823$	−5.31229	−0.185175	−0.0925874	−	0.995705i	$-0.529514\pi$
$823$	−5.31229	−0.185175	−0.0925874	+	0.995705i	$0.529514\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	22.0968	0.768380	0.384190	−	0.923254i	$-0.374481\pi$
$827$	22.0968	0.768380	0.384190	+	0.923254i	$0.374481\pi$
$828$	0	0
$829$	−1.61655	−0.0561450	−0.0280725	−	0.999606i	$-0.508937\pi$
$829$	−1.61655	−0.0561450	−0.0280725	+	0.999606i	$0.508937\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	16.2422	0.562759
$834$	0	0
$835$	−13.7627	−0.476278
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−41.0823	−1.41832	−0.709159	−	0.705048i	$-0.750925\pi$
$839$	−41.0823	−1.41832	−0.709159	+	0.705048i	$0.750925\pi$
$840$	0	0
$841$	−21.1491	−0.729281
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−2.64443	−0.0908638
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−2.69010	−0.0922155
$852$	0	0
$853$	−35.0441	−1.19989	−0.599944	−	0.800042i	$-0.704811\pi$
$853$	−35.0441	−1.19989	−0.599944	+	0.800042i	$0.704811\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.76079	0.128466	0.0642331	−	0.997935i	$-0.479540\pi$
$857$	3.76079	0.128466	0.0642331	+	0.997935i	$0.479540\pi$
$858$	0	0
$859$	7.11662	0.242816	0.121408	−	0.992603i	$-0.461259\pi$
$859$	7.11662	0.242816	0.121408	+	0.992603i	$0.461259\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−57.5900	−1.96039	−0.980193	−	0.198044i	$-0.936541\pi$
$863$	−57.5900	−1.96039	−0.980193	+	0.198044i	$0.936541\pi$
$864$	0	0
$865$	−1.42699	−0.0485192
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−10.5096	−0.356514
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5.64178	0.190727
$876$	0	0
$877$	−19.2150	−0.648846	−0.324423	−	0.945912i	$-0.605170\pi$
$877$	−19.2150	−0.648846	−0.324423	+	0.945912i	$0.605170\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	11.1444	0.375463	0.187731	−	0.982220i	$-0.439887\pi$
$881$	11.1444	0.375463	0.187731	+	0.982220i	$0.439887\pi$
$882$	0	0
$883$	−4.28919	−0.144343	−0.0721714	−	0.997392i	$-0.522993\pi$
$883$	−4.28919	−0.144343	−0.0721714	+	0.997392i	$0.522993\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	33.1739	1.11387	0.556935	−	0.830556i	$-0.311977\pi$
$887$	33.1739	1.11387	0.556935	+	0.830556i	$0.311977\pi$
$888$	0	0
$889$	7.52734	0.252459
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−53.3545	−1.78544
$894$	0	0
$895$	−1.66786	−0.0557504
$896$	0	0
$897$	0	0
$898$	0	0
$899$	11.7289	0.391179
$900$	0	0
$901$	15.4875	0.515964
$902$	0	0
$903$	0	0
$904$	0	0
$905$	8.14914	0.270887
$906$	0	0
$907$	−40.6282	−1.34904	−0.674518	−	0.738259i	$-0.735648\pi$
$907$	−40.6282	−1.34904	−0.674518	+	0.738259i	$0.735648\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−46.2731	−1.53309	−0.766547	−	0.642188i	$-0.778027\pi$
$911$	−46.2731	−1.53309	−0.766547	+	0.642188i	$0.778027\pi$
$912$	0	0
$913$	4.81966	0.159507
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−13.7727	−0.454813
$918$	0	0
$919$	50.4674	1.66477	0.832383	−	0.554201i	$-0.186976\pi$
$919$	50.4674	1.66477	0.832383	+	0.554201i	$0.186976\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	2.19029	0.0720164
$926$	0	0
$927$	0	0
$928$	0	0
$929$	17.9250	0.588100	0.294050	−	0.955790i	$-0.404997\pi$
$929$	17.9250	0.588100	0.294050	+	0.955790i	$0.404997\pi$
$930$	0	0
$931$	27.3220	0.895442
$932$	0	0
$933$	0	0
$934$	0	0
$935$	4.44876	0.145490
$936$	0	0
$937$	5.85325	0.191217	0.0956086	−	0.995419i	$-0.469520\pi$
$937$	5.85325	0.191217	0.0956086	+	0.995419i	$0.469520\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−56.7958	−1.85149	−0.925745	−	0.378147i	$-0.876561\pi$
$941$	−56.7958	−1.85149	−0.925745	+	0.378147i	$0.876561\pi$
$942$	0	0
$943$	−22.4733	−0.731830
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−6.14974	−0.199840	−0.0999198	−	0.994995i	$-0.531859\pi$
$947$	−6.14974	−0.199840	−0.0999198	+	0.994995i	$0.531859\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−48.9778	−1.58655	−0.793273	−	0.608867i	$-0.791624\pi$
$953$	−48.9778	−1.58655	−0.793273	+	0.608867i	$0.791624\pi$
$954$	0	0
$955$	−9.47889	−0.306730
$956$	0	0
$957$	0	0
$958$	0	0
$959$	22.8786	0.738788
$960$	0	0
$961$	−13.4776	−0.434760
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−8.31096	−0.267539
$966$	0	0
$967$	−54.2616	−1.74493	−0.872467	−	0.488673i	$-0.837481\pi$
$967$	−54.2616	−1.74493	−0.872467	+	0.488673i	$0.837481\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	36.3142	1.16538	0.582689	−	0.812695i	$-0.302000\pi$
$971$	36.3142	1.16538	0.582689	+	0.812695i	$0.302000\pi$
$972$	0	0
$973$	−3.11124	−0.0997419
$974$	0	0
$975$	0	0
$976$	0	0
$977$	49.7375	1.59124	0.795621	−	0.605794i	$-0.207144\pi$
$977$	49.7375	1.59124	0.795621	+	0.605794i	$0.207144\pi$
$978$	0	0
$979$	52.3274	1.67239
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−10.9981	−0.350784	−0.175392	−	0.984499i	$-0.556119\pi$
$983$	−10.9981	−0.350784	−0.175392	+	0.984499i	$0.556119\pi$
$984$	0	0
$985$	−0.106932	−0.00340715
$986$	0	0
$987$	0	0
$988$	0	0
$989$	52.9920	1.68505
$990$	0	0
$991$	6.73078	0.213810	0.106905	−	0.994269i	$-0.465906\pi$
$991$	6.73078	0.213810	0.106905	+	0.994269i	$0.465906\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−6.58881	−0.208879
$996$	0	0
$997$	43.3497	1.37290	0.686450	−	0.727177i	$-0.259168\pi$
$997$	43.3497	1.37290	0.686450	+	0.727177i	$0.259168\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6084.2.a.y.1.2		3
3.2	odd	2		2028.2.a.j.1.2	yes	3
12.11	even	2		8112.2.a.co.1.2		3
13.5	odd	4		6084.2.b.r.4393.4		6
13.8	odd	4		6084.2.b.r.4393.3		6
13.12	even	2		6084.2.a.bb.1.2		3
39.2	even	12		2028.2.q.j.1837.3		12
39.5	even	4		2028.2.b.f.337.3		6
39.8	even	4		2028.2.b.f.337.4		6
39.11	even	12		2028.2.q.j.1837.4		12
39.17	odd	6		2028.2.i.l.2005.2		6
39.20	even	12		2028.2.q.j.361.4		12
39.23	odd	6		2028.2.i.l.529.2		6
39.29	odd	6		2028.2.i.m.529.2		6
39.32	even	12		2028.2.q.j.361.3		12
39.35	odd	6		2028.2.i.m.2005.2		6
39.38	odd	2		2028.2.a.i.1.2	✓	3
156.155	even	2		8112.2.a.ch.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2028.2.a.i.1.2	✓	3	39.38	odd	2
2028.2.a.j.1.2	yes	3	3.2	odd	2
2028.2.b.f.337.3		6	39.5	even	4
2028.2.b.f.337.4		6	39.8	even	4
2028.2.i.l.529.2		6	39.23	odd	6
2028.2.i.l.2005.2		6	39.17	odd	6
2028.2.i.m.529.2		6	39.29	odd	6
2028.2.i.m.2005.2		6	39.35	odd	6
2028.2.q.j.361.3		12	39.32	even	12
2028.2.q.j.361.4		12	39.20	even	12
2028.2.q.j.1837.3		12	39.2	even	12
2028.2.q.j.1837.4		12	39.11	even	12
6084.2.a.y.1.2		3	1.1	even	1	trivial
6084.2.a.bb.1.2		3	13.12	even	2
6084.2.b.r.4393.3		6	13.8	odd	4
6084.2.b.r.4393.4		6	13.5	odd	4
8112.2.a.ch.1.2		3	156.155	even	2
8112.2.a.co.1.2		3	12.11	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:	\( N \)	\(=\)	\( 6084 = 2^{2} \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6084.a (trivial)

\(f(q)\)	\(=\)	\(q-0.554958 q^{5} +1.04892 q^{7} +2.91185 q^{11} -2.75302 q^{17} -4.63102 q^{19} +5.76271 q^{23} -4.69202 q^{25} +2.80194 q^{29} +4.18598 q^{31} -0.582105 q^{35} -0.466812 q^{37} -3.89977 q^{41} +9.19567 q^{43} +11.5211 q^{47} -5.89977 q^{49} -5.62565 q^{53} -1.61596 q^{55} +3.10992 q^{59} +10.9051 q^{61} -8.04892 q^{67} +13.6920 q^{71} -9.36658 q^{73} +3.05429 q^{77} -3.60925 q^{79} +1.65519 q^{83} +1.52781 q^{85} +17.9705 q^{89} +2.57002 q^{95} +1.31767 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{5} - 6 q^{7} + 5 q^{11} - 13 q^{17} + q^{19} - 9 q^{25} + 4 q^{29} - 2 q^{31} + 4 q^{35} + 2 q^{37} + 11 q^{41} - 9 q^{43} + 19 q^{47} + 5 q^{49} - 5 q^{53} - 15 q^{55} + 10 q^{59} + 7 q^{61}+ \cdots - 13 q^{97}+O(q^{100}) \) 3 * q - 2 * q^5 - 6 * q^7 + 5 * q^11 - 13 * q^17 + q^19 - 9 * q^25 + 4 * q^29 - 2 * q^31 + 4 * q^35 + 2 * q^37 + 11 * q^41 - 9 * q^43 + 19 * q^47 + 5 * q^49 - 5 * q^53 - 15 * q^55 + 10 * q^59 + 7 * q^61 - 15 * q^67 + 36 * q^71 - 2 * q^73 - 3 * q^77 + q^79 + 28 * q^83 + 11 * q^85 + 19 * q^89 - 17 * q^95 - 13 * q^97

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