LMFDB - Embedded newform 1584.4.a.bc.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1584,4,Mod(1,1584)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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			1.1
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)

$f(q)$	$=$	$q-14.8564 q^{5} -3.07180 q^{7} +O(q^{10})$	$q-14.8564 q^{5} -3.07180 q^{7} -11.0000 q^{11} +5.35898 q^{13} +41.2154 q^{17} -139.923 q^{19} -111.354 q^{23} +95.7128 q^{25} +24.9948 q^{29} -31.4974 q^{31} +45.6359 q^{35} +13.1436 q^{37} -261.072 q^{41} +57.7128 q^{43} -343.846 q^{47} -333.564 q^{49} +342.995 q^{53} +163.420 q^{55} +88.3693 q^{59} +738.697 q^{61} -79.6152 q^{65} -342.359 q^{67} -207.364 q^{71} -1010.60 q^{73} +33.7898 q^{77} -1294.23 q^{79} +441.846 q^{83} -612.313 q^{85} +1489.11 q^{89} -16.4617 q^{91} +2078.75 q^{95} +1346.42 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 20 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 - 20 * q^7	$ 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}+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

Display 100 coefficients 10 coefficients

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
$5$	−14.8564	−1.32880	−0.664399	−	0.747378i	$-0.731312\pi$
$5$	−14.8564	−1.32880	−0.664399	+	0.747378i	$0.731312\pi$
$6$	0	0
$7$	−3.07180	−0.165861	−0.0829307	−	0.996555i	$-0.526428\pi$
$7$	−3.07180	−0.165861	−0.0829307	+	0.996555i	$0.526428\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	5.35898	0.114332	0.0571659	−	0.998365i	$-0.481794\pi$
$13$	5.35898	0.114332	0.0571659	+	0.998365i	$0.481794\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	41.2154	0.588012	0.294006	−	0.955804i	$-0.405011\pi$
$17$	41.2154	0.588012	0.294006	+	0.955804i	$0.405011\pi$
$18$	0	0
$19$	−139.923	−1.68950	−0.844751	−	0.535159i	$-0.820252\pi$
$19$	−139.923	−1.68950	−0.844751	+	0.535159i	$0.820252\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−111.354	−1.00952	−0.504758	−	0.863261i	$-0.668418\pi$
$23$	−111.354	−1.00952	−0.504758	+	0.863261i	$0.668418\pi$
$24$	0	0
$25$	95.7128	0.765703
$26$	0	0
$27$	0	0
$28$	0	0
$29$	24.9948	0.160049	0.0800246	−	0.996793i	$-0.474500\pi$
$29$	24.9948	0.160049	0.0800246	+	0.996793i	$0.474500\pi$
$30$	0	0
$31$	−31.4974	−0.182487	−0.0912436	−	0.995829i	$-0.529084\pi$
$31$	−31.4974	−0.182487	−0.0912436	+	0.995829i	$0.529084\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	45.6359	0.220396
$36$	0	0
$37$	13.1436	0.0583998	0.0291999	−	0.999574i	$-0.490704\pi$
$37$	13.1436	0.0583998	0.0291999	+	0.999574i	$0.490704\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−261.072	−0.994453	−0.497226	−	0.867621i	$-0.665648\pi$
$41$	−261.072	−0.994453	−0.497226	+	0.867621i	$0.665648\pi$
$42$	0	0
$43$	57.7128	0.204677	0.102339	−	0.994750i	$-0.467367\pi$
$43$	57.7128	0.204677	0.102339	+	0.994750i	$0.467367\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−343.846	−1.06713	−0.533565	−	0.845759i	$-0.679148\pi$
$47$	−343.846	−1.06713	−0.533565	+	0.845759i	$0.679148\pi$
$48$	0	0
$49$	−333.564	−0.972490
$50$	0	0
$51$	0	0
$52$	0	0
$53$	342.995	0.888943	0.444471	−	0.895793i	$-0.353392\pi$
$53$	342.995	0.888943	0.444471	+	0.895793i	$0.353392\pi$
$54$	0	0
$55$	163.420	0.400647
$56$	0	0
$57$	0	0
$58$	0	0
$59$	88.3693	0.194995	0.0974975	−	0.995236i	$-0.468916\pi$
$59$	88.3693	0.194995	0.0974975	+	0.995236i	$0.468916\pi$
$60$	0	0
$61$	738.697	1.55050	0.775250	−	0.631654i	$-0.217624\pi$
$61$	738.697	1.55050	0.775250	+	0.631654i	$0.217624\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−79.6152	−0.151924
$66$	0	0
$67$	−342.359	−0.624266	−0.312133	−	0.950038i	$-0.601043\pi$
$67$	−342.359	−0.624266	−0.312133	+	0.950038i	$0.601043\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−207.364	−0.346614	−0.173307	−	0.984868i	$-0.555445\pi$
$71$	−207.364	−0.346614	−0.173307	+	0.984868i	$0.555445\pi$
$72$	0	0
$73$	−1010.60	−1.62030	−0.810149	−	0.586224i	$-0.800614\pi$
$73$	−1010.60	−1.62030	−0.810149	+	0.586224i	$0.800614\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	33.7898	0.0500091
$78$	0	0
$79$	−1294.23	−1.84319	−0.921593	−	0.388157i	$-0.873112\pi$
$79$	−1294.23	−1.84319	−0.921593	+	0.388157i	$0.873112\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	441.846	0.584324	0.292162	−	0.956369i	$-0.405625\pi$
$83$	441.846	0.584324	0.292162	+	0.956369i	$0.405625\pi$
$84$	0	0
$85$	−612.313	−0.781349
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1489.11	1.77355	0.886773	−	0.462205i	$-0.152942\pi$
$89$	1489.11	1.77355	0.886773	+	0.462205i	$0.152942\pi$
$90$	0	0
$91$	−16.4617	−0.0189633
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2078.75	2.24501
$96$	0	0
$97$	1346.42	1.40936	0.704679	−	0.709526i	$-0.251091\pi$
$97$	1346.42	1.40936	0.704679	+	0.709526i	$0.251091\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	161.461	0.159069	0.0795347	−	0.996832i	$-0.474657\pi$
$101$	161.461	0.159069	0.0795347	+	0.996832i	$0.474657\pi$
$102$	0	0
$103$	34.7592	0.0332517	0.0166259	−	0.999862i	$-0.494708\pi$
$103$	34.7592	0.0332517	0.0166259	+	0.999862i	$0.494708\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	832.179	0.751867	0.375934	−	0.926647i	$-0.377322\pi$
$107$	832.179	0.751867	0.375934	+	0.926647i	$0.377322\pi$
$108$	0	0
$109$	1044.26	0.917629	0.458815	−	0.888532i	$-0.348274\pi$
$109$	1044.26	0.917629	0.458815	+	0.888532i	$0.348274\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−295.082	−0.245654	−0.122827	−	0.992428i	$-0.539196\pi$
$113$	−295.082	−0.245654	−0.122827	+	0.992428i	$0.539196\pi$
$114$	0	0
$115$	1654.32	1.34144
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−126.605	−0.0975285
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	435.102	0.311334
$126$	0	0
$127$	1317.60	0.920618	0.460309	−	0.887759i	$-0.347739\pi$
$127$	1317.60	0.920618	0.460309	+	0.887759i	$0.347739\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1600.71	−1.06759	−0.533797	−	0.845612i	$-0.679235\pi$
$131$	−1600.71	−1.06759	−0.533797	+	0.845612i	$0.679235\pi$
$132$	0	0
$133$	429.815	0.280223
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1611.68	−1.00507	−0.502536	−	0.864556i	$-0.667600\pi$
$137$	−1611.68	−1.00507	−0.502536	+	0.864556i	$0.667600\pi$
$138$	0	0
$139$	31.8619	0.0194424	0.00972120	−	0.999953i	$-0.496906\pi$
$139$	31.8619	0.0194424	0.00972120	+	0.999953i	$0.496906\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−58.9488	−0.0344724
$144$	0	0
$145$	−371.334	−0.212673
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2428.34	1.33515	0.667576	−	0.744542i	$-0.267332\pi$
$149$	2428.34	1.33515	0.667576	+	0.744542i	$0.267332\pi$
$150$	0	0
$151$	2576.68	1.38866	0.694328	−	0.719659i	$-0.255702\pi$
$151$	2576.68	1.38866	0.694328	+	0.719659i	$0.255702\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	467.939	0.242489
$156$	0	0
$157$	2475.94	1.25861	0.629305	−	0.777158i	$-0.283340\pi$
$157$	2475.94	1.25861	0.629305	+	0.777158i	$0.283340\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	342.056	0.167440
$162$	0	0
$163$	2725.11	1.30949	0.654745	−	0.755850i	$-0.272776\pi$
$163$	2725.11	1.30949	0.654745	+	0.755850i	$0.272776\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2737.30	1.26837	0.634187	−	0.773180i	$-0.281335\pi$
$167$	2737.30	1.26837	0.634187	+	0.773180i	$0.281335\pi$
$168$	0	0
$169$	−2168.28	−0.986928
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2307.42	−1.01404	−0.507022	−	0.861933i	$-0.669254\pi$
$173$	−2307.42	−1.01404	−0.507022	+	0.861933i	$0.669254\pi$
$174$	0	0
$175$	−294.010	−0.127001
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1312.15	−0.547905	−0.273953	−	0.961743i	$-0.588331\pi$
$179$	−1312.15	−0.547905	−0.273953	+	0.961743i	$0.588331\pi$
$180$	0	0
$181$	−803.174	−0.329831	−0.164916	−	0.986308i	$-0.552735\pi$
$181$	−803.174	−0.329831	−0.164916	+	0.986308i	$0.552735\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−195.267	−0.0776015
$186$	0	0
$187$	−453.369	−0.177292
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1718.25	0.650932	0.325466	−	0.945554i	$-0.394479\pi$
$191$	1718.25	0.650932	0.325466	+	0.945554i	$0.394479\pi$
$192$	0	0
$193$	1340.18	0.499837	0.249919	−	0.968267i	$-0.419596\pi$
$193$	1340.18	0.499837	0.249919	+	0.968267i	$0.419596\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3518.33	1.27244	0.636220	−	0.771508i	$-0.280497\pi$
$197$	3518.33	1.27244	0.636220	+	0.771508i	$0.280497\pi$
$198$	0	0
$199$	−823.692	−0.293417	−0.146709	−	0.989180i	$-0.546868\pi$
$199$	−823.692	−0.293417	−0.146709	+	0.989180i	$0.546868\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−76.7791	−0.0265460
$204$	0	0
$205$	3878.59	1.32143
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1539.15	0.509404
$210$	0	0
$211$	107.343	0.0350228	0.0175114	−	0.999847i	$-0.494426\pi$
$211$	107.343	0.0350228	0.0175114	+	0.999847i	$0.494426\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−857.405	−0.271975
$216$	0	0
$217$	96.7537	0.0302676
$218$	0	0
$219$	0	0
$220$	0	0
$221$	220.873	0.0672285
$222$	0	0
$223$	3933.68	1.18125	0.590625	−	0.806946i	$-0.298881\pi$
$223$	3933.68	1.18125	0.590625	+	0.806946i	$0.298881\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1771.90	−0.518085	−0.259042	−	0.965866i	$-0.583407\pi$
$227$	−1771.90	−0.518085	−0.259042	+	0.965866i	$0.583407\pi$
$228$	0	0
$229$	1915.37	0.552713	0.276356	−	0.961055i	$-0.410873\pi$
$229$	1915.37	0.552713	0.276356	+	0.961055i	$0.410873\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4396.32	−1.23610	−0.618052	−	0.786137i	$-0.712078\pi$
$233$	−4396.32	−1.23610	−0.618052	+	0.786137i	$0.712078\pi$
$234$	0	0
$235$	5108.32	1.41800
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−4084.49	−1.10546	−0.552728	−	0.833362i	$-0.686413\pi$
$239$	−4084.49	−1.10546	−0.552728	+	0.833362i	$0.686413\pi$
$240$	0	0
$241$	3908.58	1.04471	0.522353	−	0.852730i	$-0.325054\pi$
$241$	3908.58	1.04471	0.522353	+	0.852730i	$0.325054\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4955.56	1.29224
$246$	0	0
$247$	−749.845	−0.193164
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1094.89	0.275335	0.137667	−	0.990479i	$-0.456040\pi$
$251$	1094.89	0.275335	0.137667	+	0.990479i	$0.456040\pi$
$252$	0	0
$253$	1224.89	0.304381
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−783.179	−0.190091	−0.0950454	−	0.995473i	$-0.530300\pi$
$257$	−783.179	−0.190091	−0.0950454	+	0.995473i	$0.530300\pi$
$258$	0	0
$259$	−40.3744	−0.00968628
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6180.06	1.44897	0.724484	−	0.689292i	$-0.242078\pi$
$263$	6180.06	1.44897	0.724484	+	0.689292i	$0.242078\pi$
$264$	0	0
$265$	−5095.67	−1.18122
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−986.965	−0.223704	−0.111852	−	0.993725i	$-0.535678\pi$
$269$	−986.965	−0.223704	−0.111852	+	0.993725i	$0.535678\pi$
$270$	0	0
$271$	−4576.99	−1.02595	−0.512975	−	0.858404i	$-0.671457\pi$
$271$	−4576.99	−1.02595	−0.512975	+	0.858404i	$0.671457\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1052.84	−0.230868
$276$	0	0
$277$	567.836	0.123169	0.0615847	−	0.998102i	$-0.480385\pi$
$277$	567.836	0.123169	0.0615847	+	0.998102i	$0.480385\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−5311.01	−1.12750	−0.563752	−	0.825944i	$-0.690643\pi$
$281$	−5311.01	−1.12750	−0.563752	+	0.825944i	$0.690643\pi$
$282$	0	0
$283$	4728.44	0.993204	0.496602	−	0.867978i	$-0.334581\pi$
$283$	4728.44	0.993204	0.496602	+	0.867978i	$0.334581\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	801.960	0.164941
$288$	0	0
$289$	−3214.29	−0.654242
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2328.92	−0.464358	−0.232179	−	0.972673i	$-0.574585\pi$
$293$	−2328.92	−0.464358	−0.232179	+	0.972673i	$0.574585\pi$
$294$	0	0
$295$	−1312.85	−0.259109
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−596.743	−0.115420
$300$	0	0
$301$	−177.282	−0.0339481
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−10974.4	−2.06030
$306$	0	0
$307$	1678.07	0.311962	0.155981	−	0.987760i	$-0.450146\pi$
$307$	1678.07	0.311962	0.155981	+	0.987760i	$0.450146\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3572.71	0.651413	0.325707	−	0.945471i	$-0.394398\pi$
$311$	3572.71	0.651413	0.325707	+	0.945471i	$0.394398\pi$
$312$	0	0
$313$	7184.36	1.29739	0.648697	−	0.761047i	$-0.275314\pi$
$313$	7184.36	1.29739	0.648697	+	0.761047i	$0.275314\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.7077	0.00278306	0.00139153	−	0.999999i	$-0.499557\pi$
$317$	15.7077	0.00278306	0.00139153	+	0.999999i	$0.499557\pi$
$318$	0	0
$319$	−274.943	−0.0482566
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−5766.98	−0.993447
$324$	0	0
$325$	512.923	0.0875442
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1056.23	0.176996
$330$	0	0
$331$	1318.95	0.219022	0.109511	−	0.993986i	$-0.465072\pi$
$331$	1318.95	0.219022	0.109511	+	0.993986i	$0.465072\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5086.22	0.829523
$336$	0	0
$337$	−239.183	−0.0386621	−0.0193310	−	0.999813i	$-0.506154\pi$
$337$	−239.183	−0.0386621	−0.0193310	+	0.999813i	$0.506154\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	346.472	0.0550220
$342$	0	0
$343$	2078.27	0.327160
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5862.79	−0.907006	−0.453503	−	0.891255i	$-0.649826\pi$
$347$	−5862.79	−0.907006	−0.453503	+	0.891255i	$0.649826\pi$
$348$	0	0
$349$	3491.73	0.535553	0.267776	−	0.963481i	$-0.413711\pi$
$349$	3491.73	0.535553	0.267776	+	0.963481i	$0.413711\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10916.7	1.64600	0.822999	−	0.568043i	$-0.192299\pi$
$353$	10916.7	1.64600	0.822999	+	0.568043i	$0.192299\pi$
$354$	0	0
$355$	3080.69	0.460580
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11500.7	−1.69077	−0.845384	−	0.534160i	$-0.820628\pi$
$359$	−11500.7	−1.69077	−0.845384	+	0.534160i	$0.820628\pi$
$360$	0	0
$361$	12719.5	1.85442
$362$	0	0
$363$	0	0
$364$	0	0
$365$	15013.9	2.15305
$366$	0	0
$367$	−6767.01	−0.962493	−0.481246	−	0.876585i	$-0.659816\pi$
$367$	−6767.01	−0.962493	−0.481246	+	0.876585i	$0.659816\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1053.61	−0.147441
$372$	0	0
$373$	−5310.22	−0.737139	−0.368569	−	0.929600i	$-0.620152\pi$
$373$	−5310.22	−0.737139	−0.368569	+	0.929600i	$0.620152\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	133.947	0.0182987
$378$	0	0
$379$	838.267	0.113612	0.0568059	−	0.998385i	$-0.481908\pi$
$379$	838.267	0.113612	0.0568059	+	0.998385i	$0.481908\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2832.16	−0.377851	−0.188925	−	0.981991i	$-0.560500\pi$
$383$	−2832.16	−0.377851	−0.188925	+	0.981991i	$0.560500\pi$
$384$	0	0
$385$	−501.994	−0.0664520
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−3111.25	−0.405519	−0.202759	−	0.979229i	$-0.564991\pi$
$389$	−3111.25	−0.405519	−0.202759	+	0.979229i	$0.564991\pi$
$390$	0	0
$391$	−4589.49	−0.593608
$392$	0	0
$393$	0	0
$394$	0	0
$395$	19227.5	2.44922
$396$	0	0
$397$	14208.7	1.79626	0.898131	−	0.439728i	$-0.144925\pi$
$397$	14208.7	1.79626	0.898131	+	0.439728i	$0.144925\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6261.68	0.779784	0.389892	−	0.920861i	$-0.372512\pi$
$401$	6261.68	0.779784	0.389892	+	0.920861i	$0.372512\pi$
$402$	0	0
$403$	−168.794	−0.0208641
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−144.580	−0.0176082
$408$	0	0
$409$	−4192.50	−0.506860	−0.253430	−	0.967354i	$-0.581559\pi$
$409$	−4192.50	−0.506860	−0.253430	+	0.967354i	$0.581559\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−271.453	−0.0323421
$414$	0	0
$415$	−6564.25	−0.776448
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−9287.15	−1.08283	−0.541416	−	0.840755i	$-0.682112\pi$
$419$	−9287.15	−1.08283	−0.541416	+	0.840755i	$0.682112\pi$
$420$	0	0
$421$	13146.0	1.52185	0.760923	−	0.648842i	$-0.224746\pi$
$421$	13146.0	1.52185	0.760923	+	0.648842i	$0.224746\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3944.84	0.450242
$426$	0	0
$427$	−2269.13	−0.257168
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4909.67	0.548701	0.274351	−	0.961630i	$-0.411537\pi$
$431$	4909.67	0.548701	0.274351	+	0.961630i	$0.411537\pi$
$432$	0	0
$433$	−11743.3	−1.30334	−0.651671	−	0.758502i	$-0.725932\pi$
$433$	−11743.3	−1.30334	−0.651671	+	0.758502i	$0.725932\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	15581.0	1.70558
$438$	0	0
$439$	11824.2	1.28551	0.642754	−	0.766073i	$-0.277792\pi$
$439$	11824.2	1.28551	0.642754	+	0.766073i	$0.277792\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	10102.1	1.08344	0.541722	−	0.840558i	$-0.317772\pi$
$443$	10102.1	1.08344	0.541722	+	0.840558i	$0.317772\pi$
$444$	0	0
$445$	−22122.9	−2.35668
$446$	0	0
$447$	0	0
$448$	0	0
$449$	345.254	0.0362885	0.0181443	−	0.999835i	$-0.494224\pi$
$449$	345.254	0.0362885	0.0181443	+	0.999835i	$0.494224\pi$
$450$	0	0
$451$	2871.79	0.299839
$452$	0	0
$453$	0	0
$454$	0	0
$455$	244.562	0.0251983
$456$	0	0
$457$	−10567.1	−1.08164	−0.540821	−	0.841138i	$-0.681886\pi$
$457$	−10567.1	−1.08164	−0.540821	+	0.841138i	$0.681886\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−4733.96	−0.478270	−0.239135	−	0.970986i	$-0.576864\pi$
$461$	−4733.96	−0.478270	−0.239135	+	0.970986i	$0.576864\pi$
$462$	0	0
$463$	−3431.20	−0.344409	−0.172204	−	0.985061i	$-0.555089\pi$
$463$	−3431.20	−0.344409	−0.172204	+	0.985061i	$0.555089\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5116.96	0.507034	0.253517	−	0.967331i	$-0.418413\pi$
$467$	5116.96	0.507034	0.253517	+	0.967331i	$0.418413\pi$
$468$	0	0
$469$	1051.66	0.103542
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−634.841	−0.0617125
$474$	0	0
$475$	−13392.4	−1.29366
$476$	0	0
$477$	0	0
$478$	0	0
$479$	11566.9	1.10335	0.551675	−	0.834059i	$-0.313989\pi$
$479$	11566.9	1.10335	0.551675	+	0.834059i	$0.313989\pi$
$480$	0	0
$481$	70.4363	0.00667696
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−20002.9	−1.87275
$486$	0	0
$487$	18326.5	1.70525	0.852623	−	0.522527i	$-0.175010\pi$
$487$	18326.5	1.70525	0.852623	+	0.522527i	$0.175010\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−7617.58	−0.700156	−0.350078	−	0.936721i	$-0.613845\pi$
$491$	−7617.58	−0.700156	−0.350078	+	0.936721i	$0.613845\pi$
$492$	0	0
$493$	1030.17	0.0941108
$494$	0	0
$495$	0	0
$496$	0	0
$497$	636.980	0.0574899
$498$	0	0
$499$	−12909.1	−1.15810	−0.579050	−	0.815292i	$-0.696576\pi$
$499$	−12909.1	−1.15810	−0.579050	+	0.815292i	$0.696576\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	10165.7	0.901121	0.450561	−	0.892746i	$-0.351224\pi$
$503$	10165.7	0.901121	0.450561	+	0.892746i	$0.351224\pi$
$504$	0	0
$505$	−2398.74	−0.211371
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−6449.93	−0.561666	−0.280833	−	0.959757i	$-0.590611\pi$
$509$	−6449.93	−0.561666	−0.280833	+	0.959757i	$0.590611\pi$
$510$	0	0
$511$	3104.36	0.268745
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−516.397	−0.0441848
$516$	0	0
$517$	3782.31	0.321752
$518$	0	0
$519$	0	0
$520$	0	0
$521$	19327.4	1.62524	0.812620	−	0.582794i	$-0.198041\pi$
$521$	19327.4	1.62524	0.812620	+	0.582794i	$0.198041\pi$
$522$	0	0
$523$	−6259.09	−0.523310	−0.261655	−	0.965161i	$-0.584268\pi$
$523$	−6259.09	−0.523310	−0.261655	+	0.965161i	$0.584268\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1298.18	−0.107305
$528$	0	0
$529$	232.675	0.0191235
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1399.08	−0.113698
$534$	0	0
$535$	−12363.2	−0.999079
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3669.20	0.293217
$540$	0	0
$541$	−14008.2	−1.11323	−0.556616	−	0.830770i	$-0.687900\pi$
$541$	−14008.2	−1.11323	−0.556616	+	0.830770i	$0.687900\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−15513.9	−1.21934
$546$	0	0
$547$	4949.45	0.386879	0.193440	−	0.981112i	$-0.438036\pi$
$547$	4949.45	0.386879	0.193440	+	0.981112i	$0.438036\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3497.35	−0.270404
$552$	0	0
$553$	3975.60	0.305714
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3801.58	0.289188	0.144594	−	0.989491i	$-0.453812\pi$
$557$	3801.58	0.289188	0.144594	+	0.989491i	$0.453812\pi$
$558$	0	0
$559$	309.282	0.0234011
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−9900.11	−0.741101	−0.370551	−	0.928812i	$-0.620831\pi$
$563$	−9900.11	−0.741101	−0.370551	+	0.928812i	$0.620831\pi$
$564$	0	0
$565$	4383.85	0.326425
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5329.16	−0.392636	−0.196318	−	0.980540i	$-0.562898\pi$
$569$	−5329.16	−0.392636	−0.196318	+	0.980540i	$0.562898\pi$
$570$	0	0
$571$	16962.6	1.24319	0.621597	−	0.783337i	$-0.286484\pi$
$571$	16962.6	1.24319	0.621597	+	0.783337i	$0.286484\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−10658.0	−0.772989
$576$	0	0
$577$	−15487.0	−1.11738	−0.558692	−	0.829375i	$-0.688697\pi$
$577$	−15487.0	−1.11738	−0.558692	+	0.829375i	$0.688697\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1357.26	−0.0969169
$582$	0	0
$583$	−3772.94	−0.268026
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11084.2	0.779373	0.389686	−	0.920948i	$-0.372583\pi$
$587$	11084.2	0.779373	0.389686	+	0.920948i	$0.372583\pi$
$588$	0	0
$589$	4407.22	0.308313
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−4349.68	−0.301214	−0.150607	−	0.988594i	$-0.548123\pi$
$593$	−4349.68	−0.301214	−0.150607	+	0.988594i	$0.548123\pi$
$594$	0	0
$595$	1880.90	0.129596
$596$	0	0
$597$	0	0
$598$	0	0
$599$	13183.9	0.899299	0.449650	−	0.893205i	$-0.351549\pi$
$599$	13183.9	0.899299	0.449650	+	0.893205i	$0.351549\pi$
$600$	0	0
$601$	−18765.0	−1.27361	−0.636806	−	0.771024i	$-0.719745\pi$
$601$	−18765.0	−1.27361	−0.636806	+	0.771024i	$0.719745\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1797.63	−0.120800
$606$	0	0
$607$	−21871.4	−1.46249	−0.731244	−	0.682116i	$-0.761060\pi$
$607$	−21871.4	−1.46249	−0.731244	+	0.682116i	$0.761060\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1842.67	−0.122007
$612$	0	0
$613$	−3527.85	−0.232445	−0.116222	−	0.993223i	$-0.537079\pi$
$613$	−3527.85	−0.232445	−0.116222	+	0.993223i	$0.537079\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	22728.1	1.48298	0.741490	−	0.670963i	$-0.234119\pi$
$617$	22728.1	1.48298	0.741490	+	0.670963i	$0.234119\pi$
$618$	0	0
$619$	21443.3	1.39237	0.696187	−	0.717861i	$-0.254879\pi$
$619$	21443.3	1.39237	0.696187	+	0.717861i	$0.254879\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4574.25	−0.294163
$624$	0	0
$625$	−18428.2	−1.17940
$626$	0	0
$627$	0	0
$628$	0	0
$629$	541.718	0.0343398
$630$	0	0
$631$	−21532.0	−1.35844	−0.679219	−	0.733936i	$-0.737681\pi$
$631$	−21532.0	−1.35844	−0.679219	+	0.733936i	$0.737681\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−19574.9	−1.22332
$636$	0	0
$637$	−1787.56	−0.111187
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−20148.3	−1.24151	−0.620756	−	0.784004i	$-0.713174\pi$
$641$	−20148.3	−1.24151	−0.620756	+	0.784004i	$0.713174\pi$
$642$	0	0
$643$	−28869.7	−1.77062	−0.885310	−	0.465000i	$-0.846054\pi$
$643$	−28869.7	−1.77062	−0.885310	+	0.465000i	$0.846054\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1590.02	−0.0966155	−0.0483077	−	0.998833i	$-0.515383\pi$
$647$	−1590.02	−0.0966155	−0.0483077	+	0.998833i	$0.515383\pi$
$648$	0	0
$649$	−972.062	−0.0587932
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−20028.1	−1.20024	−0.600122	−	0.799909i	$-0.704881\pi$
$653$	−20028.1	−1.20024	−0.600122	+	0.799909i	$0.704881\pi$
$654$	0	0
$655$	23780.8	1.41862
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−10520.7	−0.621897	−0.310948	−	0.950427i	$-0.600647\pi$
$659$	−10520.7	−0.621897	−0.310948	+	0.950427i	$0.600647\pi$
$660$	0	0
$661$	3295.83	0.193938	0.0969690	−	0.995287i	$-0.469085\pi$
$661$	3295.83	0.193938	0.0969690	+	0.995287i	$0.469085\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−6385.51	−0.372360
$666$	0	0
$667$	−2783.27	−0.161572
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−8125.67	−0.467493
$672$	0	0
$673$	−1187.64	−0.0680239	−0.0340119	−	0.999421i	$-0.510828\pi$
$673$	−1187.64	−0.0680239	−0.0340119	+	0.999421i	$0.510828\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−13221.4	−0.750574	−0.375287	−	0.926909i	$-0.622456\pi$
$677$	−13221.4	−0.750574	−0.375287	+	0.926909i	$0.622456\pi$
$678$	0	0
$679$	−4135.91	−0.233758
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−13831.4	−0.774882	−0.387441	−	0.921894i	$-0.626641\pi$
$683$	−13831.4	−0.774882	−0.387441	+	0.921894i	$0.626641\pi$
$684$	0	0
$685$	23943.7	1.33554
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1838.10	0.101635
$690$	0	0
$691$	9817.07	0.540462	0.270231	−	0.962796i	$-0.412900\pi$
$691$	9817.07	0.540462	0.270231	+	0.962796i	$0.412900\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−473.354	−0.0258350
$696$	0	0
$697$	−10760.2	−0.584750
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−29949.8	−1.61368	−0.806838	−	0.590773i	$-0.798823\pi$
$701$	−29949.8	−1.61368	−0.806838	+	0.590773i	$0.798823\pi$
$702$	0	0
$703$	−1839.09	−0.0986667
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−495.976	−0.0263835
$708$	0	0
$709$	11307.5	0.598959	0.299479	−	0.954103i	$-0.403187\pi$
$709$	11307.5	0.598959	0.299479	+	0.954103i	$0.403187\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3507.36	0.184224
$714$	0	0
$715$	875.768	0.0458068
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−32623.4	−1.69214	−0.846070	−	0.533071i	$-0.821038\pi$
$719$	−32623.4	−1.69214	−0.846070	+	0.533071i	$0.821038\pi$
$720$	0	0
$721$	−106.773	−0.00551518
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2392.33	0.122550
$726$	0	0
$727$	502.545	0.0256373	0.0128187	−	0.999918i	$-0.495920\pi$
$727$	502.545	0.0256373	0.0128187	+	0.999918i	$0.495920\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2378.66	0.120353
$732$	0	0
$733$	8631.37	0.434935	0.217467	−	0.976068i	$-0.430220\pi$
$733$	8631.37	0.434935	0.217467	+	0.976068i	$0.430220\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3765.95	0.188223
$738$	0	0
$739$	18357.5	0.913792	0.456896	−	0.889520i	$-0.348961\pi$
$739$	18357.5	0.913792	0.456896	+	0.889520i	$0.348961\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	11182.6	0.552155	0.276078	−	0.961135i	$-0.410965\pi$
$743$	11182.6	0.552155	0.276078	+	0.961135i	$0.410965\pi$
$744$	0	0
$745$	−36076.4	−1.77415
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2556.29	−0.124706
$750$	0	0
$751$	−16733.4	−0.813063	−0.406531	−	0.913637i	$-0.633262\pi$
$751$	−16733.4	−0.813063	−0.406531	+	0.913637i	$0.633262\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−38280.2	−1.84524
$756$	0	0
$757$	−24402.4	−1.17163	−0.585813	−	0.810446i	$-0.699225\pi$
$757$	−24402.4	−1.17163	−0.585813	+	0.810446i	$0.699225\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−8469.33	−0.403434	−0.201717	−	0.979444i	$-0.564652\pi$
$761$	−8469.33	−0.403434	−0.201717	+	0.979444i	$0.564652\pi$
$762$	0	0
$763$	−3207.74	−0.152199
$764$	0	0
$765$	0	0
$766$	0	0
$767$	473.570	0.0222941
$768$	0	0
$769$	32834.7	1.53973	0.769864	−	0.638208i	$-0.220324\pi$
$769$	32834.7	1.53973	0.769864	+	0.638208i	$0.220324\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	35571.4	1.65513	0.827564	−	0.561371i	$-0.189726\pi$
$773$	35571.4	1.65513	0.827564	+	0.561371i	$0.189726\pi$
$774$	0	0
$775$	−3014.71	−0.139731
$776$	0	0
$777$	0	0
$778$	0	0
$779$	36530.0	1.68013
$780$	0	0
$781$	2281.01	0.104508
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−36783.6	−1.67244
$786$	0	0
$787$	−15729.6	−0.712452	−0.356226	−	0.934400i	$-0.615937\pi$
$787$	−15729.6	−0.712452	−0.356226	+	0.934400i	$0.615937\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	906.431	0.0407446
$792$	0	0
$793$	3958.67	0.177272
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−7888.07	−0.350577	−0.175288	−	0.984517i	$-0.556086\pi$
$797$	−7888.07	−0.350577	−0.175288	+	0.984517i	$0.556086\pi$
$798$	0	0
$799$	−14171.8	−0.627485
$800$	0	0
$801$	0	0
$802$	0	0
$803$	11116.6	0.488538
$804$	0	0
$805$	−5081.73	−0.222494
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−5896.97	−0.256275	−0.128138	−	0.991756i	$-0.540900\pi$
$809$	−5896.97	−0.256275	−0.128138	+	0.991756i	$0.540900\pi$
$810$	0	0
$811$	−14197.9	−0.614744	−0.307372	−	0.951589i	$-0.599450\pi$
$811$	−14197.9	−0.614744	−0.307372	+	0.951589i	$0.599450\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−40485.3	−1.74005
$816$	0	0
$817$	−8075.35	−0.345803
$818$	0	0
$819$	0	0
$820$	0	0
$821$	19841.7	0.843459	0.421729	−	0.906722i	$-0.361423\pi$
$821$	19841.7	0.843459	0.421729	+	0.906722i	$0.361423\pi$
$822$	0	0
$823$	28202.2	1.19449	0.597246	−	0.802058i	$-0.296262\pi$
$823$	28202.2	1.19449	0.597246	+	0.802058i	$0.296262\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	34031.0	1.43092	0.715462	−	0.698651i	$-0.246216\pi$
$827$	34031.0	1.43092	0.715462	+	0.698651i	$0.246216\pi$
$828$	0	0
$829$	4931.55	0.206610	0.103305	−	0.994650i	$-0.467058\pi$
$829$	4931.55	0.206610	0.103305	+	0.994650i	$0.467058\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−13748.0	−0.571836
$834$	0	0
$835$	−40666.4	−1.68541
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−38189.8	−1.57146	−0.785731	−	0.618568i	$-0.787713\pi$
$839$	−38189.8	−1.57146	−0.785731	+	0.618568i	$0.787713\pi$
$840$	0	0
$841$	−23764.3	−0.974384
$842$	0	0
$843$	0	0
$844$	0	0
$845$	32212.9	1.31143
$846$	0	0
$847$	−371.687	−0.0150783
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1463.59	−0.0589556
$852$	0	0
$853$	42966.8	1.72469	0.862343	−	0.506325i	$-0.168997\pi$
$853$	42966.8	1.72469	0.862343	+	0.506325i	$0.168997\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	17281.5	0.688828	0.344414	−	0.938818i	$-0.388078\pi$
$857$	17281.5	0.688828	0.344414	+	0.938818i	$0.388078\pi$
$858$	0	0
$859$	−9316.75	−0.370062	−0.185031	−	0.982733i	$-0.559239\pi$
$859$	−9316.75	−0.370062	−0.185031	+	0.982733i	$0.559239\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−9647.65	−0.380544	−0.190272	−	0.981731i	$-0.560937\pi$
$863$	−9647.65	−0.380544	−0.190272	+	0.981731i	$0.560937\pi$
$864$	0	0
$865$	34279.9	1.34746
$866$	0	0
$867$	0	0
$868$	0	0
$869$	14236.5	0.555742
$870$	0	0
$871$	−1834.70	−0.0713735
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1336.55	−0.0516383
$876$	0	0
$877$	19728.7	0.759624	0.379812	−	0.925064i	$-0.375989\pi$
$877$	19728.7	0.759624	0.379812	+	0.925064i	$0.375989\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−19473.9	−0.744712	−0.372356	−	0.928090i	$-0.621450\pi$
$881$	−19473.9	−0.744712	−0.372356	+	0.928090i	$0.621450\pi$
$882$	0	0
$883$	−49092.4	−1.87100	−0.935499	−	0.353329i	$-0.885050\pi$
$883$	−49092.4	−1.87100	−0.935499	+	0.353329i	$0.885050\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	9292.86	0.351774	0.175887	−	0.984410i	$-0.443721\pi$
$887$	9292.86	0.351774	0.175887	+	0.984410i	$0.443721\pi$
$888$	0	0
$889$	−4047.41	−0.152695
$890$	0	0
$891$	0	0
$892$	0	0
$893$	48112.0	1.80292
$894$	0	0
$895$	19493.9	0.728055
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−787.273	−0.0292069
$900$	0	0
$901$	14136.7	0.522709
$902$	0	0
$903$	0	0
$904$	0	0
$905$	11932.3	0.438279
$906$	0	0
$907$	−37688.7	−1.37975	−0.689875	−	0.723928i	$-0.742335\pi$
$907$	−37688.7	−1.37975	−0.689875	+	0.723928i	$0.742335\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	33049.6	1.20196	0.600979	−	0.799265i	$-0.294778\pi$
$911$	33049.6	1.20196	0.600979	+	0.799265i	$0.294778\pi$
$912$	0	0
$913$	−4860.31	−0.176180
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4917.06	0.177073
$918$	0	0
$919$	23148.0	0.830883	0.415442	−	0.909620i	$-0.363627\pi$
$919$	23148.0	0.830883	0.415442	+	0.909620i	$0.363627\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1111.26	−0.0396290
$924$	0	0
$925$	1258.01	0.0447169
$926$	0	0
$927$	0	0
$928$	0	0
$929$	23177.9	0.818561	0.409280	−	0.912409i	$-0.365780\pi$
$929$	23177.9	0.818561	0.409280	+	0.912409i	$0.365780\pi$
$930$	0	0
$931$	46673.3	1.64302
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6735.44	0.235585
$936$	0	0
$937$	−34574.7	−1.20545	−0.602724	−	0.797950i	$-0.705918\pi$
$937$	−34574.7	−1.20545	−0.602724	+	0.797950i	$0.705918\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−41831.2	−1.44916	−0.724578	−	0.689192i	$-0.757966\pi$
$941$	−41831.2	−1.44916	−0.724578	+	0.689192i	$0.757966\pi$
$942$	0	0
$943$	29071.3	1.00392
$944$	0	0
$945$	0	0
$946$	0	0
$947$	27231.2	0.934419	0.467209	−	0.884147i	$-0.345259\pi$
$947$	27231.2	0.934419	0.467209	+	0.884147i	$0.345259\pi$
$948$	0	0
$949$	−5415.79	−0.185252
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−40939.4	−1.39156	−0.695781	−	0.718254i	$-0.744942\pi$
$953$	−40939.4	−1.39156	−0.695781	+	0.718254i	$0.744942\pi$
$954$	0	0
$955$	−25527.0	−0.864956
$956$	0	0
$957$	0	0
$958$	0	0
$959$	4950.74	0.166703
$960$	0	0
$961$	−28798.9	−0.966698
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−19910.3	−0.664182
$966$	0	0
$967$	46173.1	1.53550	0.767750	−	0.640750i	$-0.221376\pi$
$967$	46173.1	1.53550	0.767750	+	0.640750i	$0.221376\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−5153.91	−0.170337	−0.0851683	−	0.996367i	$-0.527143\pi$
$971$	−5153.91	−0.170337	−0.0851683	+	0.996367i	$0.527143\pi$
$972$	0	0
$973$	−97.8734	−0.00322474
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−9692.13	−0.317378	−0.158689	−	0.987329i	$-0.550727\pi$
$977$	−9692.13	−0.317378	−0.158689	+	0.987329i	$0.550727\pi$
$978$	0	0
$979$	−16380.2	−0.534744
$980$	0	0
$981$	0	0
$982$	0	0
$983$	32915.7	1.06800	0.534002	−	0.845483i	$-0.320687\pi$
$983$	32915.7	1.06800	0.534002	+	0.845483i	$0.320687\pi$
$984$	0	0
$985$	−52269.7	−1.69081
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6426.54	−0.206625
$990$	0	0
$991$	−29477.9	−0.944901	−0.472451	−	0.881357i	$-0.656630\pi$
$991$	−29477.9	−0.944901	−0.472451	+	0.881357i	$0.656630\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	12237.1	0.389892
$996$	0	0
$997$	−31944.4	−1.01473	−0.507366	−	0.861731i	$-0.669381\pi$
$997$	−31944.4	−1.01473	−0.507366	+	0.861731i	$0.669381\pi$
$998$	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.1.1		2
3.2	odd	2		176.4.a.i.1.2		2
4.3	odd	2		99.4.a.c.1.1		2
12.11	even	2		11.4.a.a.1.2	✓	2
20.19	odd	2		2475.4.a.q.1.2		2
24.5	odd	2		704.4.a.n.1.1		2
24.11	even	2		704.4.a.p.1.2		2
33.32	even	2		1936.4.a.w.1.2		2
44.43	even	2		1089.4.a.v.1.2		2
60.23	odd	4		275.4.b.c.199.1		4
60.47	odd	4		275.4.b.c.199.4		4
60.59	even	2		275.4.a.b.1.1		2
84.83	odd	2		539.4.a.e.1.2		2
132.35	odd	10		121.4.c.f.81.2		8
132.47	even	10		121.4.c.c.9.2		8
132.59	even	10		121.4.c.c.27.2		8
132.71	even	10		121.4.c.c.3.1		8
132.83	odd	10		121.4.c.f.3.2		8
132.95	odd	10		121.4.c.f.27.1		8
132.107	odd	10		121.4.c.f.9.1		8
132.119	even	10		121.4.c.c.81.1		8
132.131	odd	2		121.4.a.c.1.1		2
156.155	even	2		1859.4.a.a.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.4.a.a.1.2	✓	2	12.11	even	2
99.4.a.c.1.1		2	4.3	odd	2
121.4.a.c.1.1		2	132.131	odd	2
121.4.c.c.3.1		8	132.71	even	10
121.4.c.c.9.2		8	132.47	even	10
121.4.c.c.27.2		8	132.59	even	10
121.4.c.c.81.1		8	132.119	even	10
121.4.c.f.3.2		8	132.83	odd	10
121.4.c.f.9.1		8	132.107	odd	10
121.4.c.f.27.1		8	132.95	odd	10
121.4.c.f.81.2		8	132.35	odd	10
176.4.a.i.1.2		2	3.2	odd	2
275.4.a.b.1.1		2	60.59	even	2
275.4.b.c.199.1		4	60.23	odd	4
275.4.b.c.199.4		4	60.47	odd	4
539.4.a.e.1.2		2	84.83	odd	2
704.4.a.n.1.1		2	24.5	odd	2
704.4.a.p.1.2		2	24.11	even	2
1089.4.a.v.1.2		2	44.43	even	2
1584.4.a.bc.1.1		2	1.1	even	1	trivial
1859.4.a.a.1.1		2	156.155	even	2
1936.4.a.w.1.2		2	33.32	even	2
2475.4.a.q.1.2		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-14.8564 q^{5} -3.07180 q^{7} +O(q^{10})\)	\(q-14.8564 q^{5} -3.07180 q^{7} -11.0000 q^{11} +5.35898 q^{13} +41.2154 q^{17} -139.923 q^{19} -111.354 q^{23} +95.7128 q^{25} +24.9948 q^{29} -31.4974 q^{31} +45.6359 q^{35} +13.1436 q^{37} -261.072 q^{41} +57.7128 q^{43} -343.846 q^{47} -333.564 q^{49} +342.995 q^{53} +163.420 q^{55} +88.3693 q^{59} +738.697 q^{61} -79.6152 q^{65} -342.359 q^{67} -207.364 q^{71} -1010.60 q^{73} +33.7898 q^{77} -1294.23 q^{79} +441.846 q^{83} -612.313 q^{85} +1489.11 q^{89} -16.4617 q^{91} +2078.75 q^{95} +1346.42 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 20 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 - 20 * q^7	\( 2 q - 2 q^{5} - 20 q^{7} - 22 q^{11} + 80 q^{13} + 124 q^{17} - 72 q^{19} - 98 q^{23} + 136 q^{25} - 144 q^{29} + 34 q^{31} - 172 q^{35} + 54 q^{37} - 536 q^{41} + 60 q^{43} - 272 q^{47} - 390 q^{49} + 492 q^{53} + 22 q^{55} + 634 q^{59} + 840 q^{61} + 880 q^{65} - 754 q^{67} - 678 q^{71} - 400 q^{73} + 220 q^{77} - 316 q^{79} + 468 q^{83} + 452 q^{85} + 1842 q^{89} - 1280 q^{91} + 2952 q^{95} + 2194 q^{97}+O(q^{100}) \) 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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