LMFDB - Embedded newform 1584.4.a.bc.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1584,4,Mod(1,1584)] 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("1584.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1584, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1584 = 2^{4} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1584.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-2,0,-20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$93.4590254491$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 11)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	1584.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+12.8564 q^{5} -16.9282 q^{7} -11.0000 q^{11} +74.6410 q^{13} +82.7846 q^{17} +67.9230 q^{19} +13.3538 q^{23} +40.2872 q^{25} -168.995 q^{29} +65.4974 q^{31} -217.636 q^{35} +40.8564 q^{37} -274.928 q^{41} +2.28719 q^{43} +71.8461 q^{47} -56.4359 q^{49} +149.005 q^{53} -141.420 q^{55} +545.631 q^{59} +101.303 q^{61} +959.615 q^{65} -411.641 q^{67} -470.636 q^{71} +610.600 q^{73} +186.210 q^{77} +978.225 q^{79} +26.1539 q^{83} +1064.31 q^{85} +352.887 q^{89} -1263.54 q^{91} +873.246 q^{95} +847.585 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 20 q^{7} - 22 q^{11} + 80 q^{13} + 124 q^{17} - 72 q^{19} - 98 q^{23} + 136 q^{25} - 144 q^{29} + 34 q^{31} - 172 q^{35} + 54 q^{37} - 536 q^{41} + 60 q^{43} - 272 q^{47} - 390 q^{49} + 492 q^{53}+ \cdots + 2194 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 20 * q^7 - 22 * q^11 + 80 * q^13 + 124 * q^17 - 72 * q^19 - 98 * q^23 + 136 * q^25 - 144 * q^29 + 34 * q^31 - 172 * q^35 + 54 * q^37 - 536 * q^41 + 60 * q^43 - 272 * q^47 - 390 * q^49 + 492 * q^53 + 22 * q^55 + 634 * q^59 + 840 * q^61 + 880 * q^65 - 754 * q^67 - 678 * q^71 - 400 * q^73 + 220 * q^77 - 316 * q^79 + 468 * q^83 + 452 * q^85 + 1842 * q^89 - 1280 * q^91 + 2952 * q^95 + 2194 * 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
$5$	12.8564	1.14991	0.574956	−	0.818184i	$-0.305019\pi$
$5$	12.8564	1.14991	0.574956	+	0.818184i	$0.305019\pi$
$6$	0	0
$7$	−16.9282	−0.914037	−0.457019	−	0.889457i	$-0.651083\pi$
$7$	−16.9282	−0.914037	−0.457019	+	0.889457i	$0.651083\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	74.6410	1.59244	0.796219	−	0.605009i	$-0.206830\pi$
$13$	74.6410	1.59244	0.796219	+	0.605009i	$0.206830\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	82.7846	1.18107	0.590536	−	0.807011i	$-0.298916\pi$
$17$	82.7846	1.18107	0.590536	+	0.807011i	$0.298916\pi$
$18$	0	0
$19$	67.9230	0.820138	0.410069	−	0.912055i	$-0.365505\pi$
$19$	67.9230	0.820138	0.410069	+	0.912055i	$0.365505\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	13.3538	0.121064	0.0605319	−	0.998166i	$-0.480720\pi$
$23$	13.3538	0.121064	0.0605319	+	0.998166i	$0.480720\pi$
$24$	0	0
$25$	40.2872	0.322297
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−168.995	−1.08212	−0.541061	−	0.840983i	$-0.681977\pi$
$29$	−168.995	−1.08212	−0.541061	+	0.840983i	$0.681977\pi$
$30$	0	0
$31$	65.4974	0.379474	0.189737	−	0.981835i	$-0.439237\pi$
$31$	65.4974	0.379474	0.189737	+	0.981835i	$0.439237\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−217.636	−1.05106
$36$	0	0
$37$	40.8564	0.181534	0.0907669	−	0.995872i	$-0.471068\pi$
$37$	40.8564	0.181534	0.0907669	+	0.995872i	$0.471068\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−274.928	−1.04723	−0.523617	−	0.851954i	$-0.675418\pi$
$41$	−274.928	−1.04723	−0.523617	+	0.851954i	$0.675418\pi$
$42$	0	0
$43$	2.28719	0.00811146	0.00405573	−	0.999992i	$-0.498709\pi$
$43$	2.28719	0.00811146	0.00405573	+	0.999992i	$0.498709\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	71.8461	0.222975	0.111488	−	0.993766i	$-0.464438\pi$
$47$	71.8461	0.222975	0.111488	+	0.993766i	$0.464438\pi$
$48$	0	0
$49$	−56.4359	−0.164536
$50$	0	0
$51$	0	0
$52$	0	0
$53$	149.005	0.386178	0.193089	−	0.981181i	$-0.438149\pi$
$53$	149.005	0.386178	0.193089	+	0.981181i	$0.438149\pi$
$54$	0	0
$55$	−141.420	−0.346711
$56$	0	0
$57$	0	0
$58$	0	0
$59$	545.631	1.20398	0.601992	−	0.798502i	$-0.294374\pi$
$59$	545.631	1.20398	0.601992	+	0.798502i	$0.294374\pi$
$60$	0	0
$61$	101.303	0.212631	0.106315	−	0.994332i	$-0.466095\pi$
$61$	101.303	0.212631	0.106315	+	0.994332i	$0.466095\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	959.615	1.83116
$66$	0	0
$67$	−411.641	−0.750596	−0.375298	−	0.926904i	$-0.622460\pi$
$67$	−411.641	−0.750596	−0.375298	+	0.926904i	$0.622460\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−470.636	−0.786679	−0.393339	−	0.919393i	$-0.628680\pi$
$71$	−470.636	−0.786679	−0.393339	+	0.919393i	$0.628680\pi$
$72$	0	0
$73$	610.600	0.978977	0.489488	−	0.872010i	$-0.337184\pi$
$73$	610.600	0.978977	0.489488	+	0.872010i	$0.337184\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	186.210	0.275593
$78$	0	0
$79$	978.225	1.39315	0.696576	−	0.717483i	$-0.254706\pi$
$79$	978.225	1.39315	0.696576	+	0.717483i	$0.254706\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	26.1539	0.0345875	0.0172938	−	0.999850i	$-0.494495\pi$
$83$	26.1539	0.0345875	0.0172938	+	0.999850i	$0.494495\pi$
$84$	0	0
$85$	1064.31	1.35813
$86$	0	0
$87$	0	0
$88$	0	0
$89$	352.887	0.420292	0.210146	−	0.977670i	$-0.432606\pi$
$89$	352.887	0.420292	0.210146	+	0.977670i	$0.432606\pi$
$90$	0	0
$91$	−1263.54	−1.45555
$92$	0	0
$93$	0	0
$94$	0	0
$95$	873.246	0.943086
$96$	0	0
$97$	847.585	0.887208	0.443604	−	0.896223i	$-0.353700\pi$
$97$	847.585	0.887208	0.443604	+	0.896223i	$0.353700\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1293.46	−1.27430	−0.637150	−	0.770740i	$-0.719887\pi$
$101$	−1293.46	−1.27430	−0.637150	+	0.770740i	$0.719887\pi$
$102$	0	0
$103$	1725.24	1.65042	0.825209	−	0.564828i	$-0.191057\pi$
$103$	1725.24	1.65042	0.825209	+	0.564828i	$0.191057\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−484.179	−0.437452	−0.218726	−	0.975786i	$-0.570190\pi$
$107$	−484.179	−0.437452	−0.218726	+	0.975786i	$0.570190\pi$
$108$	0	0
$109$	−64.2563	−0.0564645	−0.0282323	−	0.999601i	$-0.508988\pi$
$109$	−64.2563	−0.0564645	−0.0282323	+	0.999601i	$0.508988\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2005.08	1.66922	0.834612	−	0.550839i	$-0.185692\pi$
$113$	2005.08	1.66922	0.834612	+	0.550839i	$0.185692\pi$
$114$	0	0
$115$	171.682	0.139213
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1401.39	−1.07954
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1089.10	−0.779298
$126$	0	0
$127$	−109.605	−0.0765816	−0.0382908	−	0.999267i	$-0.512191\pi$
$127$	−109.605	−0.0765816	−0.0382908	+	0.999267i	$0.512191\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1156.71	0.771469	0.385734	−	0.922610i	$-0.373948\pi$
$131$	1156.71	0.771469	0.385734	+	0.922610i	$0.373948\pi$
$132$	0	0
$133$	−1149.82	−0.749636
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−198.323	−0.123678	−0.0618391	−	0.998086i	$-0.519697\pi$
$137$	−198.323	−0.123678	−0.0618391	+	0.998086i	$0.519697\pi$
$138$	0	0
$139$	2900.14	1.76969	0.884844	−	0.465888i	$-0.154265\pi$
$139$	2900.14	1.76969	0.884844	+	0.465888i	$0.154265\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−821.051	−0.480138
$144$	0	0
$145$	−2172.67	−1.24435
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3488.34	−1.91796	−0.958980	−	0.283472i	$-0.908514\pi$
$149$	−3488.34	−1.91796	−0.958980	+	0.283472i	$0.908514\pi$
$150$	0	0
$151$	1163.32	0.626953	0.313477	−	0.949596i	$-0.398506\pi$
$151$	1163.32	0.626953	0.313477	+	0.949596i	$0.398506\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	842.061	0.436361
$156$	0	0
$157$	342.057	0.173880	0.0869398	−	0.996214i	$-0.472291\pi$
$157$	342.057	0.173880	0.0869398	+	0.996214i	$0.472291\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−226.056	−0.110657
$162$	0	0
$163$	1394.89	0.670285	0.335142	−	0.942167i	$-0.391216\pi$
$163$	1394.89	0.670285	0.335142	+	0.942167i	$0.391216\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	478.703	0.221815	0.110908	−	0.993831i	$-0.464624\pi$
$167$	478.703	0.221815	0.110908	+	0.993831i	$0.464624\pi$
$168$	0	0
$169$	3374.28	1.53586
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1808.58	−0.794822	−0.397411	−	0.917641i	$-0.630091\pi$
$173$	−1808.58	−0.794822	−0.397411	+	0.917641i	$0.630091\pi$
$174$	0	0
$175$	−681.990	−0.294592
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4429.85	−1.84973	−0.924867	−	0.380292i	$-0.875824\pi$
$179$	−4429.85	−1.84973	−0.924867	+	0.380292i	$0.875824\pi$
$180$	0	0
$181$	3409.17	1.40001	0.700005	−	0.714138i	$-0.253181\pi$
$181$	3409.17	1.40001	0.700005	+	0.714138i	$0.253181\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	525.267	0.208748
$186$	0	0
$187$	−910.631	−0.356106
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2923.75	1.10762	0.553810	−	0.832643i	$-0.313173\pi$
$191$	2923.75	1.10762	0.553810	+	0.832643i	$0.313173\pi$
$192$	0	0
$193$	−2484.18	−0.926505	−0.463253	−	0.886226i	$-0.653318\pi$
$193$	−2484.18	−0.926505	−0.463253	+	0.886226i	$0.653318\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5125.67	1.85375	0.926876	−	0.375369i	$-0.122484\pi$
$197$	5125.67	1.85375	0.926876	+	0.375369i	$0.122484\pi$
$198$	0	0
$199$	7.69219	0.00274013	0.00137006	−	0.999999i	$-0.499564\pi$
$199$	7.69219	0.00274013	0.00137006	+	0.999999i	$0.499564\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2860.78	0.989100
$204$	0	0
$205$	−3534.59	−1.20423
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−747.154	−0.247281
$210$	0	0
$211$	−3107.34	−1.01383	−0.506915	−	0.861996i	$-0.669214\pi$
$211$	−3107.34	−1.01383	−0.506915	+	0.861996i	$0.669214\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	29.4050	0.00932746
$216$	0	0
$217$	−1108.75	−0.346853
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6179.13	1.88078
$222$	0	0
$223$	12.3185	0.00369913	0.00184957	−	0.999998i	$-0.499411\pi$
$223$	12.3185	0.00369913	0.00184957	+	0.999998i	$0.499411\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4615.90	1.34964	0.674820	−	0.737983i	$-0.264221\pi$
$227$	4615.90	1.34964	0.674820	+	0.737983i	$0.264221\pi$
$228$	0	0
$229$	5074.63	1.46437	0.732186	−	0.681105i	$-0.238500\pi$
$229$	5074.63	1.46437	0.732186	+	0.681105i	$0.238500\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−211.683	−0.0595184	−0.0297592	−	0.999557i	$-0.509474\pi$
$233$	−211.683	−0.0595184	−0.0297592	+	0.999557i	$0.509474\pi$
$234$	0	0
$235$	923.683	0.256402
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4312.49	1.16716	0.583581	−	0.812055i	$-0.301651\pi$
$239$	4312.49	1.16716	0.583581	+	0.812055i	$0.301651\pi$
$240$	0	0
$241$	−996.584	−0.266372	−0.133186	−	0.991091i	$-0.542521\pi$
$241$	−996.584	−0.266372	−0.133186	+	0.991091i	$0.542521\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−725.563	−0.189202
$246$	0	0
$247$	5069.85	1.30602
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−276.892	−0.0696306	−0.0348153	−	0.999394i	$-0.511084\pi$
$251$	−276.892	−0.0696306	−0.0348153	+	0.999394i	$0.511084\pi$
$252$	0	0
$253$	−146.892	−0.0365021
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3235.18	0.785233	0.392617	−	0.919702i	$-0.371570\pi$
$257$	3235.18	0.785233	0.392617	+	0.919702i	$0.371570\pi$
$258$	0	0
$259$	−691.626	−0.165929
$260$	0	0
$261$	0	0
$262$	0	0
$263$	207.944	0.0487544	0.0243772	−	0.999703i	$-0.492240\pi$
$263$	207.944	0.0487544	0.0243772	+	0.999703i	$0.492240\pi$
$264$	0	0
$265$	1915.67	0.444071
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5033.04	−1.14078	−0.570390	−	0.821374i	$-0.693208\pi$
$269$	−5033.04	−1.14078	−0.570390	+	0.821374i	$0.693208\pi$
$270$	0	0
$271$	−1487.01	−0.333319	−0.166660	−	0.986015i	$-0.553298\pi$
$271$	−1487.01	−0.333319	−0.166660	+	0.986015i	$0.553298\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−443.159	−0.0971764
$276$	0	0
$277$	−235.836	−0.0511552	−0.0255776	−	0.999673i	$-0.508142\pi$
$277$	−235.836	−0.0511552	−0.0255776	+	0.999673i	$0.508142\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4915.01	1.04343	0.521717	−	0.853118i	$-0.325292\pi$
$281$	4915.01	1.04343	0.521717	+	0.853118i	$0.325292\pi$
$282$	0	0
$283$	5199.56	1.09216	0.546081	−	0.837733i	$-0.316119\pi$
$283$	5199.56	1.09216	0.546081	+	0.837733i	$0.316119\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4654.04	0.957210
$288$	0	0
$289$	1940.29	0.394930
$290$	0	0
$291$	0	0
$292$	0	0
$293$	8880.92	1.77075	0.885373	−	0.464881i	$-0.153903\pi$
$293$	8880.92	1.77075	0.885373	+	0.464881i	$0.153903\pi$
$294$	0	0
$295$	7014.85	1.38448
$296$	0	0
$297$	0	0
$298$	0	0
$299$	996.743	0.192786
$300$	0	0
$301$	−38.7180	−0.00741417
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1302.39	0.244507
$306$	0	0
$307$	1497.93	0.278474	0.139237	−	0.990259i	$-0.455535\pi$
$307$	1497.93	0.278474	0.139237	+	0.990259i	$0.455535\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−7484.71	−1.36469	−0.682345	−	0.731030i	$-0.739040\pi$
$311$	−7484.71	−1.36469	−0.682345	+	0.731030i	$0.739040\pi$
$312$	0	0
$313$	−658.363	−0.118891	−0.0594455	−	0.998232i	$-0.518933\pi$
$313$	−658.363	−0.118891	−0.0594455	+	0.998232i	$0.518933\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−233.708	−0.0414080	−0.0207040	−	0.999786i	$-0.506591\pi$
$317$	−233.708	−0.0414080	−0.0207040	+	0.999786i	$0.506591\pi$
$318$	0	0
$319$	1858.94	0.326272
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5622.98	0.968641
$324$	0	0
$325$	3007.08	0.513239
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1216.23	−0.203808
$330$	0	0
$331$	−8532.95	−1.41696	−0.708480	−	0.705731i	$-0.750619\pi$
$331$	−8532.95	−1.41696	−0.708480	+	0.705731i	$0.750619\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5292.22	−0.863120
$336$	0	0
$337$	11691.2	1.88979	0.944895	−	0.327373i	$-0.106163\pi$
$337$	11691.2	1.88979	0.944895	+	0.327373i	$0.106163\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−720.472	−0.114416
$342$	0	0
$343$	6761.73	1.06443
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4598.79	0.711459	0.355729	−	0.934589i	$-0.384232\pi$
$347$	4598.79	0.711459	0.355729	+	0.934589i	$0.384232\pi$
$348$	0	0
$349$	6720.27	1.03074	0.515369	−	0.856968i	$-0.327655\pi$
$349$	6720.27	1.03074	0.515369	+	0.856968i	$0.327655\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−5738.70	−0.865270	−0.432635	−	0.901569i	$-0.642416\pi$
$353$	−5738.70	−0.865270	−0.432635	+	0.901569i	$0.642416\pi$
$354$	0	0
$355$	−6050.69	−0.904611
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4115.27	−0.605001	−0.302501	−	0.953149i	$-0.597821\pi$
$359$	−4115.27	−0.605001	−0.302501	+	0.953149i	$0.597821\pi$
$360$	0	0
$361$	−2245.46	−0.327374
$362$	0	0
$363$	0	0
$364$	0	0
$365$	7850.12	1.12574
$366$	0	0
$367$	−9662.99	−1.37440	−0.687199	−	0.726469i	$-0.741160\pi$
$367$	−9662.99	−1.37440	−0.687199	+	0.726469i	$0.741160\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2522.39	−0.352981
$372$	0	0
$373$	−141.780	−0.0196812	−0.00984062	−	0.999952i	$-0.503132\pi$
$373$	−141.780	−0.0196812	−0.00984062	+	0.999952i	$0.503132\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−12613.9	−1.72321
$378$	0	0
$379$	2819.73	0.382163	0.191082	−	0.981574i	$-0.438800\pi$
$379$	2819.73	0.382163	0.191082	+	0.981574i	$0.438800\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−6337.84	−0.845557	−0.422778	−	0.906233i	$-0.638945\pi$
$383$	−6337.84	−0.845557	−0.422778	+	0.906233i	$0.638945\pi$
$384$	0	0
$385$	2393.99	0.316907
$386$	0	0
$387$	0	0
$388$	0	0
$389$	8805.25	1.14767	0.573836	−	0.818970i	$-0.305455\pi$
$389$	8805.25	1.14767	0.573836	+	0.818970i	$0.305455\pi$
$390$	0	0
$391$	1105.49	0.142985
$392$	0	0
$393$	0	0
$394$	0	0
$395$	12576.5	1.60200
$396$	0	0
$397$	4315.26	0.545534	0.272767	−	0.962080i	$-0.412061\pi$
$397$	4315.26	0.545534	0.272767	+	0.962080i	$0.412061\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−361.681	−0.0450411	−0.0225206	−	0.999746i	$-0.507169\pi$
$401$	−361.681	−0.0450411	−0.0225206	+	0.999746i	$0.507169\pi$
$402$	0	0
$403$	4888.79	0.604288
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−449.420	−0.0547345
$408$	0	0
$409$	9220.50	1.11473	0.557365	−	0.830268i	$-0.311812\pi$
$409$	9220.50	1.11473	0.557365	+	0.830268i	$0.311812\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−9236.55	−1.10049
$414$	0	0
$415$	336.245	0.0397726
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−14912.9	−1.73876	−0.869380	−	0.494144i	$-0.835481\pi$
$419$	−14912.9	−1.73876	−0.869380	+	0.494144i	$0.835481\pi$
$420$	0	0
$421$	−13486.0	−1.56121	−0.780603	−	0.625027i	$-0.785088\pi$
$421$	−13486.0	−1.56121	−0.780603	+	0.625027i	$0.785088\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3335.16	0.380656
$426$	0	0
$427$	−1714.87	−0.194352
$428$	0	0
$429$	0	0
$430$	0	0
$431$	406.334	0.0454116	0.0227058	−	0.999742i	$-0.492772\pi$
$431$	406.334	0.0454116	0.0227058	+	0.999742i	$0.492772\pi$
$432$	0	0
$433$	−1766.69	−0.196078	−0.0980391	−	0.995183i	$-0.531257\pi$
$433$	−1766.69	−0.196078	−0.0980391	+	0.995183i	$0.531257\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	907.033	0.0992889
$438$	0	0
$439$	−7824.19	−0.850634	−0.425317	−	0.905044i	$-0.639837\pi$
$439$	−7824.19	−0.850634	−0.425317	+	0.905044i	$0.639837\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	11667.9	1.25137	0.625686	−	0.780075i	$-0.284819\pi$
$443$	11667.9	1.25137	0.625686	+	0.780075i	$0.284819\pi$
$444$	0	0
$445$	4536.86	0.483299
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−16975.3	−1.78421	−0.892107	−	0.451825i	$-0.850773\pi$
$449$	−16975.3	−1.78421	−0.892107	+	0.451825i	$0.850773\pi$
$450$	0	0
$451$	3024.21	0.315753
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−16244.6	−1.67375
$456$	0	0
$457$	−16192.9	−1.65748	−0.828741	−	0.559632i	$-0.810943\pi$
$457$	−16192.9	−1.65748	−0.828741	+	0.559632i	$0.810943\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−8586.04	−0.867444	−0.433722	−	0.901047i	$-0.642800\pi$
$461$	−8586.04	−0.867444	−0.433722	+	0.901047i	$0.642800\pi$
$462$	0	0
$463$	7917.20	0.794694	0.397347	−	0.917668i	$-0.369931\pi$
$463$	7917.20	0.794694	0.397347	+	0.917668i	$0.369931\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−15155.0	−1.50169	−0.750844	−	0.660480i	$-0.770353\pi$
$467$	−15155.0	−1.50169	−0.750844	+	0.660480i	$0.770353\pi$
$468$	0	0
$469$	6968.34	0.686073
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−25.1591	−0.00244570
$474$	0	0
$475$	2736.43	0.264328
$476$	0	0
$477$	0	0
$478$	0	0
$479$	10001.1	0.953993	0.476996	−	0.878905i	$-0.341725\pi$
$479$	10001.1	0.953993	0.476996	+	0.878905i	$0.341725\pi$
$480$	0	0
$481$	3049.56	0.289081
$482$	0	0
$483$	0	0
$484$	0	0
$485$	10896.9	1.02021
$486$	0	0
$487$	−7044.54	−0.655480	−0.327740	−	0.944768i	$-0.606287\pi$
$487$	−7044.54	−0.655480	−0.327740	+	0.944768i	$0.606287\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−13326.4	−1.22487	−0.612437	−	0.790520i	$-0.709811\pi$
$491$	−13326.4	−1.22487	−0.612437	+	0.790520i	$0.709811\pi$
$492$	0	0
$493$	−13990.2	−1.27806
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7967.02	0.719054
$498$	0	0
$499$	20069.1	1.80044	0.900218	−	0.435440i	$-0.143407\pi$
$499$	20069.1	1.80044	0.900218	+	0.435440i	$0.143407\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	7782.35	0.689856	0.344928	−	0.938629i	$-0.387903\pi$
$503$	7782.35	0.689856	0.344928	+	0.938629i	$0.387903\pi$
$504$	0	0
$505$	−16629.3	−1.46533
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1475.93	0.128526	0.0642628	−	0.997933i	$-0.479530\pi$
$509$	1475.93	0.128526	0.0642628	+	0.997933i	$0.479530\pi$
$510$	0	0
$511$	−10336.4	−0.894821
$512$	0	0
$513$	0	0
$514$	0	0
$515$	22180.4	1.89784
$516$	0	0
$517$	−790.307	−0.0672295
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−7609.43	−0.639875	−0.319938	−	0.947439i	$-0.603662\pi$
$521$	−7609.43	−0.639875	−0.319938	+	0.947439i	$0.603662\pi$
$522$	0	0
$523$	−12452.9	−1.04116	−0.520581	−	0.853812i	$-0.674285\pi$
$523$	−12452.9	−1.04116	−0.520581	+	0.853812i	$0.674285\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5422.18	0.448186
$528$	0	0
$529$	−11988.7	−0.985344
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−20520.9	−1.66765
$534$	0	0
$535$	−6224.81	−0.503031
$536$	0	0
$537$	0	0
$538$	0	0
$539$	620.795	0.0496095
$540$	0	0
$541$	9312.17	0.740039	0.370020	−	0.929024i	$-0.379351\pi$
$541$	9312.17	0.740039	0.370020	+	0.929024i	$0.379351\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−826.105	−0.0649292
$546$	0	0
$547$	11018.6	0.861278	0.430639	−	0.902524i	$-0.358288\pi$
$547$	11018.6	0.861278	0.430639	+	0.902524i	$0.358288\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−11478.6	−0.887490
$552$	0	0
$553$	−16559.6	−1.27339
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12018.4	0.914250	0.457125	−	0.889403i	$-0.348879\pi$
$557$	12018.4	0.914250	0.457125	+	0.889403i	$0.348879\pi$
$558$	0	0
$559$	170.718	0.0129170
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−8763.89	−0.656046	−0.328023	−	0.944670i	$-0.606382\pi$
$563$	−8763.89	−0.656046	−0.328023	+	0.944670i	$0.606382\pi$
$564$	0	0
$565$	25778.1	1.91946
$566$	0	0
$567$	0	0
$568$	0	0
$569$	10273.2	0.756895	0.378447	−	0.925623i	$-0.376458\pi$
$569$	10273.2	0.756895	0.378447	+	0.925623i	$0.376458\pi$
$570$	0	0
$571$	−2602.62	−0.190747	−0.0953734	−	0.995442i	$-0.530404\pi$
$571$	−2602.62	−0.190747	−0.0953734	+	0.995442i	$0.530404\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	537.988	0.0390185
$576$	0	0
$577$	−19727.0	−1.42331	−0.711653	−	0.702532i	$-0.752053\pi$
$577$	−19727.0	−1.42331	−0.711653	+	0.702532i	$0.752053\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−442.739	−0.0316143
$582$	0	0
$583$	−1639.06	−0.116437
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−10116.2	−0.711309	−0.355654	−	0.934618i	$-0.615742\pi$
$587$	−10116.2	−0.711309	−0.355654	+	0.934618i	$0.615742\pi$
$588$	0	0
$589$	4448.78	0.311221
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−3130.32	−0.216774	−0.108387	−	0.994109i	$-0.534569\pi$
$593$	−3130.32	−0.216774	−0.108387	+	0.994109i	$0.534569\pi$
$594$	0	0
$595$	−18016.9	−1.24138
$596$	0	0
$597$	0	0
$598$	0	0
$599$	10080.1	0.687581	0.343790	−	0.939046i	$-0.388289\pi$
$599$	10080.1	0.687581	0.343790	+	0.939046i	$0.388289\pi$
$600$	0	0
$601$	4777.02	0.324224	0.162112	−	0.986772i	$-0.448169\pi$
$601$	4777.02	0.324224	0.162112	+	0.986772i	$0.448169\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1555.63	0.104537
$606$	0	0
$607$	2571.35	0.171941	0.0859703	−	0.996298i	$-0.472601\pi$
$607$	2571.35	0.171941	0.0859703	+	0.996298i	$0.472601\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5362.67	0.355074
$612$	0	0
$613$	12711.9	0.837564	0.418782	−	0.908087i	$-0.362457\pi$
$613$	12711.9	0.837564	0.418782	+	0.908087i	$0.362457\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−16236.1	−1.05939	−0.529693	−	0.848189i	$-0.677693\pi$
$617$	−16236.1	−1.05939	−0.529693	+	0.848189i	$0.677693\pi$
$618$	0	0
$619$	−12657.3	−0.821874	−0.410937	−	0.911664i	$-0.634798\pi$
$619$	−12657.3	−0.821874	−0.410937	+	0.911664i	$0.634798\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5973.75	−0.384162
$624$	0	0
$625$	−19037.8	−1.21842
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3382.28	0.214404
$630$	0	0
$631$	3949.97	0.249201	0.124600	−	0.992207i	$-0.460235\pi$
$631$	3949.97	0.249201	0.124600	+	0.992207i	$0.460235\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1409.13	−0.0880621
$636$	0	0
$637$	−4212.44	−0.262014
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7398.27	0.455872	0.227936	−	0.973676i	$-0.426802\pi$
$641$	7398.27	0.455872	0.227936	+	0.973676i	$0.426802\pi$
$642$	0	0
$643$	12491.7	0.766134	0.383067	−	0.923721i	$-0.374868\pi$
$643$	12491.7	0.766134	0.383067	+	0.923721i	$0.374868\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−10472.0	−0.636315	−0.318158	−	0.948038i	$-0.603064\pi$
$647$	−10472.0	−0.636315	−0.318158	+	0.948038i	$0.603064\pi$
$648$	0	0
$649$	−6001.94	−0.363015
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−6337.94	−0.379820	−0.189910	−	0.981801i	$-0.560820\pi$
$653$	−6337.94	−0.379820	−0.189910	+	0.981801i	$0.560820\pi$
$654$	0	0
$655$	14871.2	0.887121
$656$	0	0
$657$	0	0
$658$	0	0
$659$	15196.7	0.898302	0.449151	−	0.893456i	$-0.351726\pi$
$659$	15196.7	0.898302	0.449151	+	0.893456i	$0.351726\pi$
$660$	0	0
$661$	2298.17	0.135232	0.0676161	−	0.997711i	$-0.478461\pi$
$661$	2298.17	0.135232	0.0676161	+	0.997711i	$0.478461\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−14782.5	−0.862016
$666$	0	0
$667$	−2256.73	−0.131006
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1114.33	−0.0641106
$672$	0	0
$673$	23199.6	1.32880	0.664398	−	0.747379i	$-0.268688\pi$
$673$	23199.6	1.32880	0.664398	+	0.747379i	$0.268688\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2145.38	0.121793	0.0608963	−	0.998144i	$-0.480604\pi$
$677$	2145.38	0.121793	0.0608963	+	0.998144i	$0.480604\pi$
$678$	0	0
$679$	−14348.1	−0.810941
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−29544.6	−1.65519	−0.827593	−	0.561329i	$-0.810290\pi$
$683$	−29544.6	−1.65519	−0.827593	+	0.561329i	$0.810290\pi$
$684$	0	0
$685$	−2549.72	−0.142219
$686$	0	0
$687$	0	0
$688$	0	0
$689$	11121.9	0.614964
$690$	0	0
$691$	−27803.1	−1.53065	−0.765325	−	0.643644i	$-0.777422\pi$
$691$	−27803.1	−1.53065	−0.765325	+	0.643644i	$0.777422\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	37285.4	2.03498
$696$	0	0
$697$	−22759.8	−1.23686
$698$	0	0
$699$	0	0
$700$	0	0
$701$	19697.8	1.06130	0.530652	−	0.847590i	$-0.321947\pi$
$701$	19697.8	1.06130	0.530652	+	0.847590i	$0.321947\pi$
$702$	0	0
$703$	2775.09	0.148883
$704$	0	0
$705$	0	0
$706$	0	0
$707$	21896.0	1.16476
$708$	0	0
$709$	19122.5	1.01292	0.506460	−	0.862263i	$-0.330954\pi$
$709$	19122.5	1.01292	0.506460	+	0.862263i	$0.330954\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	874.641	0.0459405
$714$	0	0
$715$	−10555.8	−0.552117
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1837.44	0.0953060	0.0476530	−	0.998864i	$-0.484826\pi$
$719$	1837.44	0.0953060	0.0476530	+	0.998864i	$0.484826\pi$
$720$	0	0
$721$	−29205.2	−1.50854
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−6808.33	−0.348765
$726$	0	0
$727$	7555.46	0.385442	0.192721	−	0.981254i	$-0.438269\pi$
$727$	7555.46	0.385442	0.192721	+	0.981254i	$0.438269\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	189.344	0.00958021
$732$	0	0
$733$	11984.6	0.603905	0.301952	−	0.953323i	$-0.402362\pi$
$733$	11984.6	0.603905	0.301952	+	0.953323i	$0.402362\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4528.05	0.226313
$738$	0	0
$739$	27142.5	1.35109	0.675543	−	0.737321i	$-0.263909\pi$
$739$	27142.5	1.35109	0.675543	+	0.737321i	$0.263909\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−29222.6	−1.44290	−0.721450	−	0.692467i	$-0.756524\pi$
$743$	−29222.6	−1.44290	−0.721450	+	0.692467i	$0.756524\pi$
$744$	0	0
$745$	−44847.6	−2.20549
$746$	0	0
$747$	0	0
$748$	0	0
$749$	8196.29	0.399847
$750$	0	0
$751$	8859.39	0.430471	0.215236	−	0.976562i	$-0.430948\pi$
$751$	8859.39	0.430471	0.215236	+	0.976562i	$0.430948\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	14956.2	0.720941
$756$	0	0
$757$	35734.4	1.71571	0.857853	−	0.513896i	$-0.171798\pi$
$757$	35734.4	1.71571	0.857853	+	0.513896i	$0.171798\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−34394.7	−1.63838	−0.819189	−	0.573524i	$-0.805576\pi$
$761$	−34394.7	−1.63838	−0.819189	+	0.573524i	$0.805576\pi$
$762$	0	0
$763$	1087.74	0.0516107
$764$	0	0
$765$	0	0
$766$	0	0
$767$	40726.4	1.91727
$768$	0	0
$769$	−11602.7	−0.544091	−0.272045	−	0.962284i	$-0.587700\pi$
$769$	−11602.7	−0.544091	−0.272045	+	0.962284i	$0.587700\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	12680.6	0.590026	0.295013	−	0.955493i	$-0.404676\pi$
$773$	12680.6	0.590026	0.295013	+	0.955493i	$0.404676\pi$
$774$	0	0
$775$	2638.71	0.122303
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−18674.0	−0.858876
$780$	0	0
$781$	5176.99	0.237193
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4397.62	0.199946
$786$	0	0
$787$	4417.61	0.200090	0.100045	−	0.994983i	$-0.468101\pi$
$787$	4417.61	0.200090	0.100045	+	0.994983i	$0.468101\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−33942.4	−1.52573
$792$	0	0
$793$	7561.33	0.338601
$794$	0	0
$795$	0	0
$796$	0	0
$797$	27030.1	1.20132	0.600661	−	0.799504i	$-0.294904\pi$
$797$	27030.1	1.20132	0.600661	+	0.799504i	$0.294904\pi$
$798$	0	0
$799$	5947.75	0.263350
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−6716.60	−0.295173
$804$	0	0
$805$	−2906.27	−0.127246
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−23647.0	−1.02767	−0.513835	−	0.857889i	$-0.671776\pi$
$809$	−23647.0	−1.02767	−0.513835	+	0.857889i	$0.671776\pi$
$810$	0	0
$811$	−33486.1	−1.44988	−0.724941	−	0.688811i	$-0.758133\pi$
$811$	−33486.1	−1.44988	−0.724941	+	0.688811i	$0.758133\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	17933.3	0.770768
$816$	0	0
$817$	155.353	0.00665251
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2605.69	−0.110766	−0.0553832	−	0.998465i	$-0.517638\pi$
$821$	−2605.69	−0.110766	−0.0553832	+	0.998465i	$0.517638\pi$
$822$	0	0
$823$	−31976.2	−1.35434	−0.677169	−	0.735828i	$-0.736793\pi$
$823$	−31976.2	−1.35434	−0.677169	+	0.735828i	$0.736793\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−37759.0	−1.58768	−0.793839	−	0.608128i	$-0.791921\pi$
$827$	−37759.0	−1.58768	−0.793839	+	0.608128i	$0.791921\pi$
$828$	0	0
$829$	−1137.55	−0.0476584	−0.0238292	−	0.999716i	$-0.507586\pi$
$829$	−1137.55	−0.0476584	−0.0238292	+	0.999716i	$0.507586\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−4672.03	−0.194329
$834$	0	0
$835$	6154.40	0.255068
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−37372.2	−1.53782	−0.768911	−	0.639356i	$-0.779201\pi$
$839$	−37372.2	−1.53782	−0.768911	+	0.639356i	$0.779201\pi$
$840$	0	0
$841$	4170.26	0.170989
$842$	0	0
$843$	0	0
$844$	0	0
$845$	43381.1	1.76610
$846$	0	0
$847$	−2048.31	−0.0830943
$848$	0	0
$849$	0	0
$850$	0	0
$851$	545.589	0.0219772
$852$	0	0
$853$	−22490.8	−0.902780	−0.451390	−	0.892327i	$-0.649072\pi$
$853$	−22490.8	−0.902780	−0.451390	+	0.892327i	$0.649072\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−43409.5	−1.73027	−0.865135	−	0.501539i	$-0.832767\pi$
$857$	−43409.5	−1.73027	−0.865135	+	0.501539i	$0.832767\pi$
$858$	0	0
$859$	−29533.2	−1.17306	−0.586532	−	0.809926i	$-0.699507\pi$
$859$	−29533.2	−1.17306	−0.586532	+	0.809926i	$0.699507\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	14351.6	0.566090	0.283045	−	0.959107i	$-0.408655\pi$
$863$	14351.6	0.566090	0.283045	+	0.959107i	$0.408655\pi$
$864$	0	0
$865$	−23251.9	−0.913975
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−10760.5	−0.420051
$870$	0	0
$871$	−30725.3	−1.19528
$872$	0	0
$873$	0	0
$874$	0	0
$875$	18436.5	0.712307
$876$	0	0
$877$	−43248.7	−1.66523	−0.832614	−	0.553854i	$-0.813156\pi$
$877$	−43248.7	−1.66523	−0.832614	+	0.553854i	$0.813156\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−3816.13	−0.145935	−0.0729675	−	0.997334i	$-0.523247\pi$
$881$	−3816.13	−0.145935	−0.0729675	+	0.997334i	$0.523247\pi$
$882$	0	0
$883$	−48787.6	−1.85938	−0.929690	−	0.368343i	$-0.879925\pi$
$883$	−48787.6	−1.85938	−0.929690	+	0.368343i	$0.879925\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	41495.1	1.57077	0.785384	−	0.619009i	$-0.212466\pi$
$887$	41495.1	1.57077	0.785384	+	0.619009i	$0.212466\pi$
$888$	0	0
$889$	1855.41	0.0699984
$890$	0	0
$891$	0	0
$892$	0	0
$893$	4880.01	0.182870
$894$	0	0
$895$	−56951.9	−2.12703
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−11068.7	−0.410637
$900$	0	0
$901$	12335.3	0.456104
$902$	0	0
$903$	0	0
$904$	0	0
$905$	43829.7	1.60989
$906$	0	0
$907$	−21615.3	−0.791316	−0.395658	−	0.918398i	$-0.629483\pi$
$907$	−21615.3	−0.791316	−0.395658	+	0.918398i	$0.629483\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	3646.35	0.132611	0.0663057	−	0.997799i	$-0.478879\pi$
$911$	3646.35	0.132611	0.0663057	+	0.997799i	$0.478879\pi$
$912$	0	0
$913$	−287.693	−0.0104285
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−19581.1	−0.705151
$918$	0	0
$919$	−31280.0	−1.12278	−0.561388	−	0.827553i	$-0.689733\pi$
$919$	−31280.0	−1.12278	−0.561388	+	0.827553i	$0.689733\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−35128.7	−1.25274
$924$	0	0
$925$	1645.99	0.0585079
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−6557.92	−0.231602	−0.115801	−	0.993272i	$-0.536944\pi$
$929$	−6557.92	−0.231602	−0.115801	+	0.993272i	$0.536944\pi$
$930$	0	0
$931$	−3833.30	−0.134942
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−11707.4	−0.409491
$936$	0	0
$937$	−24473.3	−0.853265	−0.426632	−	0.904425i	$-0.640300\pi$
$937$	−24473.3	−0.853265	−0.426632	+	0.904425i	$0.640300\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−15420.8	−0.534224	−0.267112	−	0.963665i	$-0.586069\pi$
$941$	−15420.8	−0.534224	−0.267112	+	0.963665i	$0.586069\pi$
$942$	0	0
$943$	−3671.34	−0.126782
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−33141.2	−1.13722	−0.568608	−	0.822609i	$-0.692518\pi$
$947$	−33141.2	−1.13722	−0.568608	+	0.822609i	$0.692518\pi$
$948$	0	0
$949$	45575.8	1.55896
$950$	0	0
$951$	0	0
$952$	0	0
$953$	20735.4	0.704813	0.352406	−	0.935847i	$-0.385363\pi$
$953$	20735.4	0.704813	0.352406	+	0.935847i	$0.385363\pi$
$954$	0	0
$955$	37589.0	1.27367
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3357.26	0.113046
$960$	0	0
$961$	−25501.1	−0.856000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−31937.7	−1.06540
$966$	0	0
$967$	8178.87	0.271990	0.135995	−	0.990710i	$-0.456577\pi$
$967$	8178.87	0.271990	0.135995	+	0.990710i	$0.456577\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−20576.1	−0.680039	−0.340020	−	0.940418i	$-0.610434\pi$
$971$	−20576.1	−0.680039	−0.340020	+	0.940418i	$0.610434\pi$
$972$	0	0
$973$	−49094.1	−1.61756
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−14541.9	−0.476188	−0.238094	−	0.971242i	$-0.576523\pi$
$977$	−14541.9	−0.476188	−0.238094	+	0.971242i	$0.576523\pi$
$978$	0	0
$979$	−3881.76	−0.126723
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−29285.7	−0.950223	−0.475111	−	0.879926i	$-0.657592\pi$
$983$	−29285.7	−0.950223	−0.475111	+	0.879926i	$0.657592\pi$
$984$	0	0
$985$	65897.7	2.13165
$986$	0	0
$987$	0	0
$988$	0	0
$989$	30.5427	0.000982004 0
$990$	0	0
$991$	38085.9	1.22083	0.610413	−	0.792083i	$-0.291003\pi$
$991$	38085.9	1.22083	0.610413	+	0.792083i	$0.291003\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	98.8940	0.00315090
$996$	0	0
$997$	−26803.6	−0.851434	−0.425717	−	0.904856i	$-0.639978\pi$
$997$	−26803.6	−0.851434	−0.425717	+	0.904856i	$0.639978\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1584.4.a.bc.1.2		2
3.2	odd	2		176.4.a.i.1.1		2
4.3	odd	2		99.4.a.c.1.2		2
12.11	even	2		11.4.a.a.1.1	✓	2
20.19	odd	2		2475.4.a.q.1.1		2
24.5	odd	2		704.4.a.n.1.2		2
24.11	even	2		704.4.a.p.1.1		2
33.32	even	2		1936.4.a.w.1.1		2
44.43	even	2		1089.4.a.v.1.1		2
60.23	odd	4		275.4.b.c.199.3		4
60.47	odd	4		275.4.b.c.199.2		4
60.59	even	2		275.4.a.b.1.2		2
84.83	odd	2		539.4.a.e.1.1		2
132.35	odd	10		121.4.c.f.81.1		8
132.47	even	10		121.4.c.c.9.1		8
132.59	even	10		121.4.c.c.27.1		8
132.71	even	10		121.4.c.c.3.2		8
132.83	odd	10		121.4.c.f.3.1		8
132.95	odd	10		121.4.c.f.27.2		8
132.107	odd	10		121.4.c.f.9.2		8
132.119	even	10		121.4.c.c.81.2		8
132.131	odd	2		121.4.a.c.1.2		2
156.155	even	2		1859.4.a.a.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.4.a.a.1.1	✓	2	12.11	even	2
99.4.a.c.1.2		2	4.3	odd	2
121.4.a.c.1.2		2	132.131	odd	2
121.4.c.c.3.2		8	132.71	even	10
121.4.c.c.9.1		8	132.47	even	10
121.4.c.c.27.1		8	132.59	even	10
121.4.c.c.81.2		8	132.119	even	10
121.4.c.f.3.1		8	132.83	odd	10
121.4.c.f.9.2		8	132.107	odd	10
121.4.c.f.27.2		8	132.95	odd	10
121.4.c.f.81.1		8	132.35	odd	10
176.4.a.i.1.1		2	3.2	odd	2
275.4.a.b.1.2		2	60.59	even	2
275.4.b.c.199.2		4	60.47	odd	4
275.4.b.c.199.3		4	60.23	odd	4
539.4.a.e.1.1		2	84.83	odd	2
704.4.a.n.1.2		2	24.5	odd	2
704.4.a.p.1.1		2	24.11	even	2
1089.4.a.v.1.1		2	44.43	even	2
1584.4.a.bc.1.2		2	1.1	even	1	trivial
1859.4.a.a.1.2		2	156.155	even	2
1936.4.a.w.1.1		2	33.32	even	2
2475.4.a.q.1.1		2	20.19	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:	\( N \)	\(=\)	\( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1584.a (trivial)

\(f(q)\)	\(=\)	\(q+12.8564 q^{5} -16.9282 q^{7} -11.0000 q^{11} +74.6410 q^{13} +82.7846 q^{17} +67.9230 q^{19} +13.3538 q^{23} +40.2872 q^{25} -168.995 q^{29} +65.4974 q^{31} -217.636 q^{35} +40.8564 q^{37} -274.928 q^{41} +2.28719 q^{43} +71.8461 q^{47} -56.4359 q^{49} +149.005 q^{53} -141.420 q^{55} +545.631 q^{59} +101.303 q^{61} +959.615 q^{65} -411.641 q^{67} -470.636 q^{71} +610.600 q^{73} +186.210 q^{77} +978.225 q^{79} +26.1539 q^{83} +1064.31 q^{85} +352.887 q^{89} -1263.54 q^{91} +873.246 q^{95} +847.585 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 20 q^{7} - 22 q^{11} + 80 q^{13} + 124 q^{17} - 72 q^{19} - 98 q^{23} + 136 q^{25} - 144 q^{29} + 34 q^{31} - 172 q^{35} + 54 q^{37} - 536 q^{41} + 60 q^{43} - 272 q^{47} - 390 q^{49} + 492 q^{53}+ \cdots + 2194 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 20 * q^7 - 22 * q^11 + 80 * q^13 + 124 * q^17 - 72 * q^19 - 98 * q^23 + 136 * q^25 - 144 * q^29 + 34 * q^31 - 172 * q^35 + 54 * q^37 - 536 * q^41 + 60 * q^43 - 272 * q^47 - 390 * q^49 + 492 * q^53 + 22 * q^55 + 634 * q^59 + 840 * q^61 + 880 * q^65 - 754 * q^67 - 678 * q^71 - 400 * q^73 + 220 * q^77 - 316 * q^79 + 468 * q^83 + 452 * q^85 + 1842 * q^89 - 1280 * q^91 + 2952 * q^95 + 2194 * q^97

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