LMFDB - Embedded newform 1152.4.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,4,Mod(1,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1152.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$1152 = 2^{7} \cdot 3^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1152.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:	$67.9702003266$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.4764.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - x^{2} - 12x - 6$ x^3 - x^2 - 12*x - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{4}$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.536923$ of defining polynomial
Character	$\chi$	$=$	1152.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.147691 q^{5} -30.6992 q^{7} +57.1029 q^{11} -58.8076 q^{13} +130.501 q^{17} +52.2168 q^{19} -147.387 q^{23} -124.978 q^{25} +196.855 q^{29} +105.734 q^{31} -4.53399 q^{35} +306.764 q^{37} -1.31975 q^{41} -129.377 q^{43} -270.661 q^{47} +599.439 q^{49} -453.150 q^{53} +8.43360 q^{55} -669.707 q^{59} -456.379 q^{61} -8.68536 q^{65} -542.027 q^{67} +524.921 q^{71} -600.439 q^{73} -1753.01 q^{77} -192.434 q^{79} -920.789 q^{83} +19.2739 q^{85} +892.732 q^{89} +1805.34 q^{91} +7.71196 q^{95} -1165.43 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$3 q - 10 q^{5} - 6 q^{7} + 20 q^{11} - 46 q^{13} + 68 q^{17} + 68 q^{19} - 56 q^{23} + 53 q^{25} - 46 q^{29} + 226 q^{31} - 332 q^{35} - 66 q^{37} + 236 q^{41} + 212 q^{43} - 760 q^{47} + 327 q^{49} - 702 q^{53}+ \cdots - 1222 q^{97}+O(q^{100})$ 3 * q - 10 * q^5 - 6 * q^7 + 20 * q^11 - 46 * q^13 + 68 * q^17 + 68 * q^19 - 56 * q^23 + 53 * q^25 - 46 * q^29 + 226 * q^31 - 332 * q^35 - 66 * q^37 + 236 * q^41 + 212 * q^43 - 760 * q^47 + 327 * q^49 - 702 * q^53 - 152 * q^55 - 600 * q^59 - 122 * q^61 + 1028 * q^65 - 760 * q^67 - 512 * q^71 - 330 * q^73 - 2024 * q^77 + 218 * q^79 - 2340 * q^83 + 392 * q^85 + 2616 * q^89 + 1372 * q^91 - 2680 * q^95 - 1222 * 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
$5$	0.147691	0.0132099	0.00660495	−	0.999978i	$-0.497898\pi$
$5$	0.147691	0.0132099	0.00660495	+	0.999978i	$0.497898\pi$
$6$	0	0
$7$	−30.6992	−1.65760	−0.828800	−	0.559546i	$-0.810976\pi$
$7$	−30.6992	−1.65760	−0.828800	+	0.559546i	$0.810976\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	57.1029	1.56520	0.782599	−	0.622526i	$-0.213893\pi$
$11$	57.1029	1.56520	0.782599	+	0.622526i	$0.213893\pi$
$12$	0	0
$13$	−58.8076	−1.25464	−0.627319	−	0.778763i	$-0.715848\pi$
$13$	−58.8076	−1.25464	−0.627319	+	0.778763i	$0.715848\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	130.501	1.86184	0.930918	−	0.365229i	$-0.119009\pi$
$17$	130.501	1.86184	0.930918	+	0.365229i	$0.119009\pi$
$18$	0	0
$19$	52.2168	0.630492	0.315246	−	0.949010i	$-0.397913\pi$
$19$	52.2168	0.630492	0.315246	+	0.949010i	$0.397913\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−147.387	−1.33619	−0.668096	−	0.744075i	$-0.732890\pi$
$23$	−147.387	−1.33619	−0.668096	+	0.744075i	$0.732890\pi$
$24$	0	0
$25$	−124.978	−0.999825
$26$	0	0
$27$	0	0
$28$	0	0
$29$	196.855	1.26052	0.630259	−	0.776385i	$-0.282949\pi$
$29$	196.855	1.26052	0.630259	+	0.776385i	$0.282949\pi$
$30$	0	0
$31$	105.734	0.612596	0.306298	−	0.951936i	$-0.400910\pi$
$31$	105.734	0.612596	0.306298	+	0.951936i	$0.400910\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.53399	−0.0218967
$36$	0	0
$37$	306.764	1.36302	0.681509	−	0.731810i	$-0.261324\pi$
$37$	306.764	1.36302	0.681509	+	0.731810i	$0.261324\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.31975	−0.00502707	−0.00251353	−	0.999997i	$-0.500800\pi$
$41$	−1.31975	−0.00502707	−0.00251353	+	0.999997i	$0.500800\pi$
$42$	0	0
$43$	−129.377	−0.458831	−0.229416	−	0.973329i	$-0.573681\pi$
$43$	−129.377	−0.458831	−0.229416	+	0.973329i	$0.573681\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−270.661	−0.840000	−0.420000	−	0.907524i	$-0.637970\pi$
$47$	−270.661	−0.840000	−0.420000	+	0.907524i	$0.637970\pi$
$48$	0	0
$49$	599.439	1.74763
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−453.150	−1.17443	−0.587217	−	0.809430i	$-0.699776\pi$
$53$	−453.150	−1.17443	−0.587217	+	0.809430i	$0.699776\pi$
$54$	0	0
$55$	8.43360	0.0206761
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−669.707	−1.47777	−0.738885	−	0.673832i	$-0.764647\pi$
$59$	−669.707	−1.47777	−0.738885	+	0.673832i	$0.764647\pi$
$60$	0	0
$61$	−456.379	−0.957924	−0.478962	−	0.877836i	$-0.658987\pi$
$61$	−456.379	−0.957924	−0.478962	+	0.877836i	$0.658987\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−8.68536	−0.0165736
$66$	0	0
$67$	−542.027	−0.988345	−0.494173	−	0.869364i	$-0.664529\pi$
$67$	−542.027	−0.988345	−0.494173	+	0.869364i	$0.664529\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	524.921	0.877418	0.438709	−	0.898629i	$-0.355436\pi$
$71$	524.921	0.877418	0.438709	+	0.898629i	$0.355436\pi$
$72$	0	0
$73$	−600.439	−0.962685	−0.481343	−	0.876532i	$-0.659851\pi$
$73$	−600.439	−0.962685	−0.481343	+	0.876532i	$0.659851\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1753.01	−2.59447
$78$	0	0
$79$	−192.434	−0.274057	−0.137028	−	0.990567i	$-0.543755\pi$
$79$	−192.434	−0.274057	−0.137028	+	0.990567i	$0.543755\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−920.789	−1.21771	−0.608854	−	0.793283i	$-0.708370\pi$
$83$	−920.789	−1.21771	−0.608854	+	0.793283i	$0.708370\pi$
$84$	0	0
$85$	19.2739	0.0245947
$86$	0	0
$87$	0	0
$88$	0	0
$89$	892.732	1.06325	0.531626	−	0.846979i	$-0.321581\pi$
$89$	892.732	1.06325	0.531626	+	0.846979i	$0.321581\pi$
$90$	0	0
$91$	1805.34	2.07969
$92$	0	0
$93$	0	0
$94$	0	0
$95$	7.71196	0.00832874
$96$	0	0
$97$	−1165.43	−1.21991	−0.609954	−	0.792437i	$-0.708812\pi$
$97$	−1165.43	−1.21991	−0.609954	+	0.792437i	$0.708812\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	18.0902	0.0178222	0.00891111	−	0.999960i	$-0.497163\pi$
$101$	18.0902	0.0178222	0.00891111	+	0.999960i	$0.497163\pi$
$102$	0	0
$103$	−522.212	−0.499564	−0.249782	−	0.968302i	$-0.580359\pi$
$103$	−522.212	−0.499564	−0.249782	+	0.968302i	$0.580359\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−619.745	−0.559935	−0.279968	−	0.960009i	$-0.590324\pi$
$107$	−619.745	−0.559935	−0.279968	+	0.960009i	$0.590324\pi$
$108$	0	0
$109$	571.653	0.502334	0.251167	−	0.967944i	$-0.419186\pi$
$109$	571.653	0.502334	0.251167	+	0.967944i	$0.419186\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	504.082	0.419646	0.209823	−	0.977739i	$-0.432711\pi$
$113$	504.082	0.419646	0.209823	+	0.977739i	$0.432711\pi$
$114$	0	0
$115$	−21.7678	−0.0176510
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4006.28	−3.08618
$120$	0	0
$121$	1929.75	1.44985
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−36.9196	−0.0264175
$126$	0	0
$127$	1431.52	1.00021	0.500107	−	0.865964i	$-0.333294\pi$
$127$	1431.52	1.00021	0.500107	+	0.865964i	$0.333294\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1536.00	−1.02443	−0.512217	−	0.858856i	$-0.671176\pi$
$131$	−1536.00	−1.02443	−0.512217	+	0.858856i	$0.671176\pi$
$132$	0	0
$133$	−1603.01	−1.04510
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2734.14	1.70506	0.852531	−	0.522677i	$-0.175066\pi$
$137$	2734.14	1.70506	0.852531	+	0.522677i	$0.175066\pi$
$138$	0	0
$139$	−2808.47	−1.71375	−0.856877	−	0.515522i	$-0.827598\pi$
$139$	−2808.47	−1.71375	−0.856877	+	0.515522i	$0.827598\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3358.09	−1.96376
$144$	0	0
$145$	29.0737	0.0166513
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1790.08	−0.984222	−0.492111	−	0.870532i	$-0.663775\pi$
$149$	−1790.08	−0.984222	−0.492111	+	0.870532i	$0.663775\pi$
$150$	0	0
$151$	−1848.00	−0.995948	−0.497974	−	0.867192i	$-0.665922\pi$
$151$	−1848.00	−0.995948	−0.497974	+	0.867192i	$0.665922\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	15.6160	0.00809233
$156$	0	0
$157$	2541.93	1.29215	0.646076	−	0.763273i	$-0.276409\pi$
$157$	2541.93	1.29215	0.646076	+	0.763273i	$0.276409\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4524.67	2.21487
$162$	0	0
$163$	−2613.06	−1.25565	−0.627824	−	0.778355i	$-0.716054\pi$
$163$	−2613.06	−1.25565	−0.627824	+	0.778355i	$0.716054\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	687.732	0.318673	0.159336	−	0.987224i	$-0.449065\pi$
$167$	687.732	0.318673	0.159336	+	0.987224i	$0.449065\pi$
$168$	0	0
$169$	1261.33	0.574115
$170$	0	0
$171$	0	0
$172$	0	0
$173$	631.176	0.277384	0.138692	−	0.990336i	$-0.455710\pi$
$173$	631.176	0.277384	0.138692	+	0.990336i	$0.455710\pi$
$174$	0	0
$175$	3836.73	1.65731
$176$	0	0
$177$	0	0
$178$	0	0
$179$	324.959	0.135690	0.0678451	−	0.997696i	$-0.478388\pi$
$179$	324.959	0.135690	0.0678451	+	0.997696i	$0.478388\pi$
$180$	0	0
$181$	−444.868	−0.182689	−0.0913446	−	0.995819i	$-0.529116\pi$
$181$	−444.868	−0.182689	−0.0913446	+	0.995819i	$0.529116\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	45.3063	0.0180053
$186$	0	0
$187$	7452.01	2.91414
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2061.70	−0.781042	−0.390521	−	0.920594i	$-0.627705\pi$
$191$	−2061.70	−0.781042	−0.390521	+	0.920594i	$0.627705\pi$
$192$	0	0
$193$	−1885.67	−0.703283	−0.351642	−	0.936135i	$-0.614376\pi$
$193$	−1885.67	−0.703283	−0.351642	+	0.936135i	$0.614376\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5248.55	−1.89819	−0.949096	−	0.314987i	$-0.898000\pi$
$197$	−5248.55	−1.89819	−0.949096	+	0.314987i	$0.898000\pi$
$198$	0	0
$199$	−3094.00	−1.10215	−0.551076	−	0.834455i	$-0.685782\pi$
$199$	−3094.00	−1.10215	−0.551076	+	0.834455i	$0.685782\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−6043.28	−2.08943
$204$	0	0
$205$	−0.194915	−6.64071e−5 0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2981.73	0.986846
$210$	0	0
$211$	−65.6705	−0.0214263	−0.0107131	−	0.999943i	$-0.503410\pi$
$211$	−65.6705	−0.0214263	−0.0107131	+	0.999943i	$0.503410\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−19.1078	−0.00606111
$216$	0	0
$217$	−3245.96	−1.01544
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−7674.46	−2.33593
$222$	0	0
$223$	−377.711	−0.113423	−0.0567117	−	0.998391i	$-0.518062\pi$
$223$	−377.711	−0.113423	−0.0567117	+	0.998391i	$0.518062\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3627.59	−1.06067	−0.530334	−	0.847789i	$-0.677933\pi$
$227$	−3627.59	−1.06067	−0.530334	+	0.847789i	$0.677933\pi$
$228$	0	0
$229$	−329.399	−0.0950537	−0.0475268	−	0.998870i	$-0.515134\pi$
$229$	−329.399	−0.0950537	−0.0475268	+	0.998870i	$0.515134\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2715.15	−0.763413	−0.381707	−	0.924284i	$-0.624664\pi$
$233$	−2715.15	−0.763413	−0.381707	+	0.924284i	$0.624664\pi$
$234$	0	0
$235$	−39.9743	−0.0110963
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1400.56	0.379056	0.189528	−	0.981875i	$-0.439304\pi$
$239$	1400.56	0.379056	0.189528	+	0.981875i	$0.439304\pi$
$240$	0	0
$241$	147.831	0.0395129	0.0197565	−	0.999805i	$-0.493711\pi$
$241$	147.831	0.0395129	0.0197565	+	0.999805i	$0.493711\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	88.5318	0.0230861
$246$	0	0
$247$	−3070.74	−0.791039
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2880.42	−0.724346	−0.362173	−	0.932111i	$-0.617965\pi$
$251$	−2880.42	−0.724346	−0.362173	+	0.932111i	$0.617965\pi$
$252$	0	0
$253$	−8416.26	−2.09141
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6466.90	−1.56963	−0.784813	−	0.619732i	$-0.787241\pi$
$257$	−6466.90	−1.56963	−0.784813	+	0.619732i	$0.787241\pi$
$258$	0	0
$259$	−9417.40	−2.25934
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−5171.68	−1.21255	−0.606273	−	0.795256i	$-0.707336\pi$
$263$	−5171.68	−1.21255	−0.606273	+	0.795256i	$0.707336\pi$
$264$	0	0
$265$	−66.9263	−0.0155141
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2480.11	0.562136	0.281068	−	0.959688i	$-0.409311\pi$
$269$	2480.11	0.562136	0.281068	+	0.959688i	$0.409311\pi$
$270$	0	0
$271$	2061.23	0.462032	0.231016	−	0.972950i	$-0.425795\pi$
$271$	2061.23	0.462032	0.231016	+	0.972950i	$0.425795\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−7136.62	−1.56493
$276$	0	0
$277$	6241.31	1.35380	0.676902	−	0.736073i	$-0.263322\pi$
$277$	6241.31	1.35380	0.676902	+	0.736073i	$0.263322\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6839.63	1.45202	0.726011	−	0.687683i	$-0.241372\pi$
$281$	6839.63	1.45202	0.726011	+	0.687683i	$0.241372\pi$
$282$	0	0
$283$	8037.94	1.68836	0.844180	−	0.536060i	$-0.180088\pi$
$283$	8037.94	1.68836	0.844180	+	0.536060i	$0.180088\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	40.5151	0.00833286
$288$	0	0
$289$	12117.6	2.46643
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1762.17	0.351354	0.175677	−	0.984448i	$-0.443789\pi$
$293$	1762.17	0.351354	0.175677	+	0.984448i	$0.443789\pi$
$294$	0	0
$295$	−98.9098	−0.0195212
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8667.50	1.67644
$300$	0	0
$301$	3971.75	0.760558
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−67.4031	−0.0126541
$306$	0	0
$307$	−4383.94	−0.814998	−0.407499	−	0.913206i	$-0.633599\pi$
$307$	−4383.94	−0.814998	−0.407499	+	0.913206i	$0.633599\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2791.10	0.508903	0.254451	−	0.967086i	$-0.418105\pi$
$311$	2791.10	0.508903	0.254451	+	0.967086i	$0.418105\pi$
$312$	0	0
$313$	−3379.98	−0.610377	−0.305189	−	0.952292i	$-0.598719\pi$
$313$	−3379.98	−0.610377	−0.305189	+	0.952292i	$0.598719\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1672.82	−0.296388	−0.148194	−	0.988958i	$-0.547346\pi$
$317$	−1672.82	−0.296388	−0.148194	+	0.988958i	$0.547346\pi$
$318$	0	0
$319$	11241.0	1.97296
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6814.36	1.17387
$324$	0	0
$325$	7349.66	1.25442
$326$	0	0
$327$	0	0
$328$	0	0
$329$	8309.08	1.39238
$330$	0	0
$331$	2389.90	0.396860	0.198430	−	0.980115i	$-0.436416\pi$
$331$	2389.90	0.396860	0.198430	+	0.980115i	$0.436416\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−80.0526	−0.0130559
$336$	0	0
$337$	−6669.96	−1.07815	−0.539074	−	0.842258i	$-0.681226\pi$
$337$	−6669.96	−1.07815	−0.539074	+	0.842258i	$0.681226\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6037.75	0.958834
$342$	0	0
$343$	−7872.45	−1.23928
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1502.43	0.232435	0.116217	−	0.993224i	$-0.462923\pi$
$347$	1502.43	0.232435	0.116217	+	0.993224i	$0.462923\pi$
$348$	0	0
$349$	−1238.77	−0.190000	−0.0950000	−	0.995477i	$-0.530285\pi$
$349$	−1238.77	−0.190000	−0.0950000	+	0.995477i	$0.530285\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6421.72	0.968255	0.484127	−	0.874998i	$-0.339137\pi$
$353$	6421.72	0.968255	0.484127	+	0.874998i	$0.339137\pi$
$354$	0	0
$355$	77.5262	0.0115906
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5298.01	−0.778881	−0.389440	−	0.921052i	$-0.627332\pi$
$359$	−5298.01	−0.778881	−0.389440	+	0.921052i	$0.627332\pi$
$360$	0	0
$361$	−4132.41	−0.602479
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−88.6795	−0.0127170
$366$	0	0
$367$	9310.15	1.32421	0.662106	−	0.749410i	$-0.269663\pi$
$367$	9310.15	1.32421	0.662106	+	0.749410i	$0.269663\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	13911.3	1.94674
$372$	0	0
$373$	−11651.8	−1.61745	−0.808726	−	0.588186i	$-0.799842\pi$
$373$	−11651.8	−1.61745	−0.808726	+	0.588186i	$0.799842\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−11576.6	−1.58149
$378$	0	0
$379$	1424.32	0.193040	0.0965201	−	0.995331i	$-0.469229\pi$
$379$	1424.32	0.193040	0.0965201	+	0.995331i	$0.469229\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−6244.09	−0.833050	−0.416525	−	0.909124i	$-0.636752\pi$
$383$	−6244.09	−0.833050	−0.416525	+	0.909124i	$0.636752\pi$
$384$	0	0
$385$	−258.904	−0.0342727
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6434.93	0.838725	0.419362	−	0.907819i	$-0.362254\pi$
$389$	6434.93	0.838725	0.419362	+	0.907819i	$0.362254\pi$
$390$	0	0
$391$	−19234.2	−2.48777
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−28.4207	−0.00362026
$396$	0	0
$397$	−559.727	−0.0707604	−0.0353802	−	0.999374i	$-0.511264\pi$
$397$	−559.727	−0.0707604	−0.0353802	+	0.999374i	$0.511264\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2930.51	−0.364945	−0.182472	−	0.983211i	$-0.558410\pi$
$401$	−2930.51	−0.364945	−0.182472	+	0.983211i	$0.558410\pi$
$402$	0	0
$403$	−6217.98	−0.768585
$404$	0	0
$405$	0	0
$406$	0	0
$407$	17517.1	2.13339
$408$	0	0
$409$	7242.66	0.875614	0.437807	−	0.899069i	$-0.355755\pi$
$409$	7242.66	0.875614	0.437807	+	0.899069i	$0.355755\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	20559.4	2.44955
$414$	0	0
$415$	−135.992	−0.0160858
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1575.55	0.183701	0.0918504	−	0.995773i	$-0.470722\pi$
$419$	1575.55	0.183701	0.0918504	+	0.995773i	$0.470722\pi$
$420$	0	0
$421$	−12220.0	−1.41465	−0.707326	−	0.706888i	$-0.750098\pi$
$421$	−12220.0	−1.41465	−0.707326	+	0.706888i	$0.750098\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−16309.8	−1.86151
$426$	0	0
$427$	14010.5	1.58785
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12669.6	1.41594	0.707971	−	0.706241i	$-0.249611\pi$
$431$	12669.6	1.41594	0.707971	+	0.706241i	$0.249611\pi$
$432$	0	0
$433$	−3704.81	−0.411182	−0.205591	−	0.978638i	$-0.565912\pi$
$433$	−3704.81	−0.411182	−0.205591	+	0.978638i	$0.565912\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7696.10	−0.842459
$438$	0	0
$439$	−787.710	−0.0856387	−0.0428193	−	0.999083i	$-0.513634\pi$
$439$	−787.710	−0.0856387	−0.0428193	+	0.999083i	$0.513634\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5536.26	−0.593760	−0.296880	−	0.954915i	$-0.595946\pi$
$443$	−5536.26	−0.593760	−0.296880	+	0.954915i	$0.595946\pi$
$444$	0	0
$445$	131.849	0.0140454
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−8968.84	−0.942685	−0.471343	−	0.881950i	$-0.656230\pi$
$449$	−8968.84	−0.942685	−0.471343	+	0.881950i	$0.656230\pi$
$450$	0	0
$451$	−75.3614	−0.00786836
$452$	0	0
$453$	0	0
$454$	0	0
$455$	266.633	0.0274724
$456$	0	0
$457$	11755.1	1.20324	0.601619	−	0.798783i	$-0.294523\pi$
$457$	11755.1	1.20324	0.601619	+	0.798783i	$0.294523\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−3595.89	−0.363291	−0.181646	−	0.983364i	$-0.558142\pi$
$461$	−3595.89	−0.363291	−0.181646	+	0.983364i	$0.558142\pi$
$462$	0	0
$463$	−5568.32	−0.558924	−0.279462	−	0.960157i	$-0.590156\pi$
$463$	−5568.32	−0.558924	−0.279462	+	0.960157i	$0.590156\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9741.02	−0.965226	−0.482613	−	0.875834i	$-0.660312\pi$
$467$	−9741.02	−0.965226	−0.482613	+	0.875834i	$0.660312\pi$
$468$	0	0
$469$	16639.8	1.63828
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−7387.78	−0.718162
$474$	0	0
$475$	−6525.96	−0.630382
$476$	0	0
$477$	0	0
$478$	0	0
$479$	11840.6	1.12946	0.564730	−	0.825276i	$-0.308980\pi$
$479$	11840.6	1.12946	0.564730	+	0.825276i	$0.308980\pi$
$480$	0	0
$481$	−18040.0	−1.71009
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−172.123	−0.0161148
$486$	0	0
$487$	16044.0	1.49286	0.746432	−	0.665461i	$-0.231765\pi$
$487$	16044.0	1.49286	0.746432	+	0.665461i	$0.231765\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	11236.6	1.03279	0.516395	−	0.856351i	$-0.327274\pi$
$491$	11236.6	1.03279	0.516395	+	0.856351i	$0.327274\pi$
$492$	0	0
$493$	25689.8	2.34688
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−16114.6	−1.45441
$498$	0	0
$499$	−18926.3	−1.69791	−0.848957	−	0.528461i	$-0.822769\pi$
$499$	−18926.3	−1.69791	−0.848957	+	0.528461i	$0.822769\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	12581.8	1.11530	0.557649	−	0.830077i	$-0.311704\pi$
$503$	12581.8	1.11530	0.557649	+	0.830077i	$0.311704\pi$
$504$	0	0
$505$	2.67177	0.000235430 0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	11042.6	0.961597	0.480799	−	0.876831i	$-0.340347\pi$
$509$	11042.6	0.961597	0.480799	+	0.876831i	$0.340347\pi$
$510$	0	0
$511$	18433.0	1.59575
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−77.1260	−0.00659918
$516$	0	0
$517$	−15455.6	−1.31477
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−6096.70	−0.512671	−0.256335	−	0.966588i	$-0.582515\pi$
$521$	−6096.70	−0.512671	−0.256335	+	0.966588i	$0.582515\pi$
$522$	0	0
$523$	21980.7	1.83776	0.918881	−	0.394536i	$-0.129095\pi$
$523$	21980.7	1.83776	0.918881	+	0.394536i	$0.129095\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	13798.5	1.14055
$528$	0	0
$529$	9556.05	0.785407
$530$	0	0
$531$	0	0
$532$	0	0
$533$	77.6111	0.00630715
$534$	0	0
$535$	−91.5309	−0.00739668
$536$	0	0
$537$	0	0
$538$	0	0
$539$	34229.7	2.73540
$540$	0	0
$541$	−10519.1	−0.835951	−0.417976	−	0.908458i	$-0.637260\pi$
$541$	−10519.1	−0.835951	−0.417976	+	0.908458i	$0.637260\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	84.4281	0.00663578
$546$	0	0
$547$	−4811.42	−0.376090	−0.188045	−	0.982160i	$-0.560215\pi$
$547$	−4811.42	−0.376090	−0.188045	+	0.982160i	$0.560215\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	10279.1	0.794747
$552$	0	0
$553$	5907.55	0.454276
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−6482.69	−0.493143	−0.246571	−	0.969125i	$-0.579304\pi$
$557$	−6482.69	−0.493143	−0.246571	+	0.969125i	$0.579304\pi$
$558$	0	0
$559$	7608.32	0.575666
$560$	0	0
$561$	0	0
$562$	0	0
$563$	23090.2	1.72849	0.864243	−	0.503075i	$-0.167798\pi$
$563$	23090.2	1.72849	0.864243	+	0.503075i	$0.167798\pi$
$564$	0	0
$565$	74.4484	0.00554348
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1781.49	0.131255	0.0656274	−	0.997844i	$-0.479095\pi$
$569$	1781.49	0.131255	0.0656274	+	0.997844i	$0.479095\pi$
$570$	0	0
$571$	−15493.9	−1.13555	−0.567776	−	0.823183i	$-0.692196\pi$
$571$	−15493.9	−1.13555	−0.567776	+	0.823183i	$0.692196\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	18420.2	1.33596
$576$	0	0
$577$	3233.43	0.233292	0.116646	−	0.993174i	$-0.462786\pi$
$577$	3233.43	0.233292	0.116646	+	0.993174i	$0.462786\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	28267.4	2.01847
$582$	0	0
$583$	−25876.2	−1.83822
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−18285.3	−1.28571	−0.642857	−	0.765986i	$-0.722251\pi$
$587$	−18285.3	−1.28571	−0.642857	+	0.765986i	$0.722251\pi$
$588$	0	0
$589$	5521.11	0.386237
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−5783.50	−0.400506	−0.200253	−	0.979744i	$-0.564176\pi$
$593$	−5783.50	−0.400506	−0.200253	+	0.979744i	$0.564176\pi$
$594$	0	0
$595$	−591.692	−0.0407681
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−27718.5	−1.89073	−0.945363	−	0.326018i	$-0.894293\pi$
$599$	−27718.5	−1.89073	−0.945363	+	0.326018i	$0.894293\pi$
$600$	0	0
$601$	−2905.47	−0.197199	−0.0985995	−	0.995127i	$-0.531436\pi$
$601$	−2905.47	−0.197199	−0.0985995	+	0.995127i	$0.531436\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	285.006	0.0191523
$606$	0	0
$607$	−27350.5	−1.82887	−0.914433	−	0.404738i	$-0.867363\pi$
$607$	−27350.5	−1.82887	−0.914433	+	0.404738i	$0.867363\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	15916.9	1.05390
$612$	0	0
$613$	−575.905	−0.0379455	−0.0189727	−	0.999820i	$-0.506040\pi$
$613$	−575.905	−0.0379455	−0.0189727	+	0.999820i	$0.506040\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13186.7	0.860417	0.430208	−	0.902730i	$-0.358440\pi$
$617$	13186.7	0.860417	0.430208	+	0.902730i	$0.358440\pi$
$618$	0	0
$619$	18598.4	1.20764	0.603822	−	0.797120i	$-0.293644\pi$
$619$	18598.4	1.20764	0.603822	+	0.797120i	$0.293644\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−27406.1	−1.76244
$624$	0	0
$625$	15616.8	0.999477
$626$	0	0
$627$	0	0
$628$	0	0
$629$	40033.1	2.53772
$630$	0	0
$631$	−6196.64	−0.390942	−0.195471	−	0.980709i	$-0.562624\pi$
$631$	−6196.64	−0.390942	−0.195471	+	0.980709i	$0.562624\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	211.423	0.0132127
$636$	0	0
$637$	−35251.5	−2.19265
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−13568.8	−0.836091	−0.418045	−	0.908426i	$-0.637285\pi$
$641$	−13568.8	−0.836091	−0.418045	+	0.908426i	$0.637285\pi$
$642$	0	0
$643$	20311.3	1.24573	0.622863	−	0.782331i	$-0.285970\pi$
$643$	20311.3	1.24573	0.622863	+	0.782331i	$0.285970\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	28961.7	1.75982	0.879908	−	0.475145i	$-0.157604\pi$
$647$	28961.7	1.75982	0.879908	+	0.475145i	$0.157604\pi$
$648$	0	0
$649$	−38242.2	−2.31300
$650$	0	0
$651$	0	0
$652$	0	0
$653$	23670.5	1.41853	0.709264	−	0.704943i	$-0.249028\pi$
$653$	23670.5	1.41853	0.709264	+	0.704943i	$0.249028\pi$
$654$	0	0
$655$	−226.854	−0.0135327
$656$	0	0
$657$	0	0
$658$	0	0
$659$	945.123	0.0558676	0.0279338	−	0.999610i	$-0.491107\pi$
$659$	945.123	0.0558676	0.0279338	+	0.999610i	$0.491107\pi$
$660$	0	0
$661$	15682.5	0.922813	0.461407	−	0.887189i	$-0.347345\pi$
$661$	15682.5	0.922813	0.461407	+	0.887189i	$0.347345\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−236.751	−0.0138057
$666$	0	0
$667$	−29013.9	−1.68429
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−26060.6	−1.49934
$672$	0	0
$673$	−12152.8	−0.696070	−0.348035	−	0.937482i	$-0.613151\pi$
$673$	−12152.8	−0.696070	−0.348035	+	0.937482i	$0.613151\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−10521.1	−0.597282	−0.298641	−	0.954366i	$-0.596533\pi$
$677$	−10521.1	−0.597282	−0.298641	+	0.954366i	$0.596533\pi$
$678$	0	0
$679$	35777.6	2.02212
$680$	0	0
$681$	0	0
$682$	0	0
$683$	18474.8	1.03502	0.517509	−	0.855678i	$-0.326859\pi$
$683$	18474.8	1.03502	0.517509	+	0.855678i	$0.326859\pi$
$684$	0	0
$685$	403.809	0.0225237
$686$	0	0
$687$	0	0
$688$	0	0
$689$	26648.7	1.47349
$690$	0	0
$691$	20100.2	1.10658	0.553290	−	0.832989i	$-0.313372\pi$
$691$	20100.2	1.10658	0.553290	+	0.832989i	$0.313372\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−414.787	−0.0226385
$696$	0	0
$697$	−172.229	−0.00935958
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−12867.8	−0.693307	−0.346654	−	0.937993i	$-0.612682\pi$
$701$	−12867.8	−0.693307	−0.346654	+	0.937993i	$0.612682\pi$
$702$	0	0
$703$	16018.2	0.859373
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−555.355	−0.0295421
$708$	0	0
$709$	17082.1	0.904838	0.452419	−	0.891806i	$-0.350561\pi$
$709$	17082.1	0.904838	0.452419	+	0.891806i	$0.350561\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−15583.9	−0.818545
$714$	0	0
$715$	−495.959	−0.0259410
$716$	0	0
$717$	0	0
$718$	0	0
$719$	10251.4	0.531728	0.265864	−	0.964011i	$-0.414343\pi$
$719$	10251.4	0.531728	0.265864	+	0.964011i	$0.414343\pi$
$720$	0	0
$721$	16031.5	0.828076
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−24602.6	−1.26030
$726$	0	0
$727$	−6464.45	−0.329784	−0.164892	−	0.986312i	$-0.552728\pi$
$727$	−6464.45	−0.329784	−0.164892	+	0.986312i	$0.552728\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−16883.8	−0.854268
$732$	0	0
$733$	−9971.20	−0.502449	−0.251224	−	0.967929i	$-0.580833\pi$
$733$	−9971.20	−0.502449	−0.251224	+	0.967929i	$0.580833\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−30951.3	−1.54696
$738$	0	0
$739$	10085.1	0.502011	0.251006	−	0.967986i	$-0.419239\pi$
$739$	10085.1	0.502011	0.251006	+	0.967986i	$0.419239\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	6949.91	0.343159	0.171580	−	0.985170i	$-0.445113\pi$
$743$	6949.91	0.343159	0.171580	+	0.985170i	$0.445113\pi$
$744$	0	0
$745$	−264.379	−0.0130015
$746$	0	0
$747$	0	0
$748$	0	0
$749$	19025.7	0.928148
$750$	0	0
$751$	15770.3	0.766265	0.383132	−	0.923693i	$-0.374845\pi$
$751$	15770.3	0.766265	0.383132	+	0.923693i	$0.374845\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−272.933	−0.0131564
$756$	0	0
$757$	13727.2	0.659082	0.329541	−	0.944141i	$-0.393106\pi$
$757$	13727.2	0.659082	0.329541	+	0.944141i	$0.393106\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	6731.89	0.320671	0.160336	−	0.987063i	$-0.448742\pi$
$761$	6731.89	0.320671	0.160336	+	0.987063i	$0.448742\pi$
$762$	0	0
$763$	−17549.3	−0.832669
$764$	0	0
$765$	0	0
$766$	0	0
$767$	39383.8	1.85406
$768$	0	0
$769$	18919.2	0.887181	0.443591	−	0.896229i	$-0.353704\pi$
$769$	18919.2	0.887181	0.443591	+	0.896229i	$0.353704\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−36138.2	−1.68150	−0.840751	−	0.541422i	$-0.817886\pi$
$773$	−36138.2	−1.68150	−0.840751	+	0.541422i	$0.817886\pi$
$774$	0	0
$775$	−13214.5	−0.612489
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−68.9129	−0.00316953
$780$	0	0
$781$	29974.5	1.37333
$782$	0	0
$783$	0	0
$784$	0	0
$785$	375.420	0.0170692
$786$	0	0
$787$	−26273.6	−1.19003	−0.595015	−	0.803714i	$-0.702854\pi$
$787$	−26273.6	−1.19003	−0.595015	+	0.803714i	$0.702854\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−15474.9	−0.695605
$792$	0	0
$793$	26838.5	1.20185
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−35843.9	−1.59304	−0.796521	−	0.604610i	$-0.793329\pi$
$797$	−35843.9	−1.59304	−0.796521	+	0.604610i	$0.793329\pi$
$798$	0	0
$799$	−35321.6	−1.56394
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−34286.8	−1.50679
$804$	0	0
$805$	668.254	0.0292582
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−3033.64	−0.131838	−0.0659191	−	0.997825i	$-0.520998\pi$
$809$	−3033.64	−0.131838	−0.0659191	+	0.997825i	$0.520998\pi$
$810$	0	0
$811$	−23885.8	−1.03421	−0.517105	−	0.855922i	$-0.672990\pi$
$811$	−23885.8	−1.03421	−0.517105	+	0.855922i	$0.672990\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−385.926	−0.0165870
$816$	0	0
$817$	−6755.63	−0.289289
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−26993.1	−1.14746	−0.573730	−	0.819045i	$-0.694504\pi$
$821$	−26993.1	−1.14746	−0.573730	+	0.819045i	$0.694504\pi$
$822$	0	0
$823$	44105.5	1.86807	0.934034	−	0.357183i	$-0.116263\pi$
$823$	44105.5	1.86807	0.934034	+	0.357183i	$0.116263\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−21548.3	−0.906057	−0.453028	−	0.891496i	$-0.649656\pi$
$827$	−21548.3	−0.906057	−0.453028	+	0.891496i	$0.649656\pi$
$828$	0	0
$829$	−36677.9	−1.53664	−0.768322	−	0.640064i	$-0.778908\pi$
$829$	−36677.9	−1.53664	−0.768322	+	0.640064i	$0.778908\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	78227.5	3.25381
$834$	0	0
$835$	101.572	0.00420963
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−7116.32	−0.292828	−0.146414	−	0.989223i	$-0.546773\pi$
$839$	−7116.32	−0.292828	−0.146414	+	0.989223i	$0.546773\pi$
$840$	0	0
$841$	14362.8	0.588906
$842$	0	0
$843$	0	0
$844$	0	0
$845$	186.287	0.00758399
$846$	0	0
$847$	−59241.6	−2.40327
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−45213.1	−1.82125
$852$	0	0
$853$	−22898.8	−0.919157	−0.459579	−	0.888137i	$-0.652000\pi$
$853$	−22898.8	−0.919157	−0.459579	+	0.888137i	$0.652000\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−20903.5	−0.833196	−0.416598	−	0.909091i	$-0.636778\pi$
$857$	−20903.5	−0.833196	−0.416598	+	0.909091i	$0.636778\pi$
$858$	0	0
$859$	20766.2	0.824833	0.412417	−	0.910995i	$-0.364685\pi$
$859$	20766.2	0.824833	0.412417	+	0.910995i	$0.364685\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	5881.39	0.231987	0.115994	−	0.993250i	$-0.462995\pi$
$863$	5881.39	0.231987	0.115994	+	0.993250i	$0.462995\pi$
$864$	0	0
$865$	93.2191	0.00366421
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−10988.5	−0.428953
$870$	0	0
$871$	31875.3	1.24001
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1133.40	0.0437896
$876$	0	0
$877$	49735.2	1.91498	0.957490	−	0.288466i	$-0.0931454\pi$
$877$	49735.2	1.91498	0.957490	+	0.288466i	$0.0931454\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−26000.8	−0.994312	−0.497156	−	0.867661i	$-0.665622\pi$
$881$	−26000.8	−0.994312	−0.497156	+	0.867661i	$0.665622\pi$
$882$	0	0
$883$	29578.1	1.12728	0.563638	−	0.826022i	$-0.309401\pi$
$883$	29578.1	1.12728	0.563638	+	0.826022i	$0.309401\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−34850.5	−1.31924	−0.659619	−	0.751600i	$-0.729282\pi$
$887$	−34850.5	−1.31924	−0.659619	+	0.751600i	$0.729282\pi$
$888$	0	0
$889$	−43946.6	−1.65795
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−14133.1	−0.529614
$894$	0	0
$895$	47.9935	0.00179245
$896$	0	0
$897$	0	0
$898$	0	0
$899$	20814.3	0.772188
$900$	0	0
$901$	−59136.7	−2.18660
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−65.7030	−0.00241331
$906$	0	0
$907$	−4319.35	−0.158127	−0.0790637	−	0.996870i	$-0.525193\pi$
$907$	−4319.35	−0.158127	−0.0790637	+	0.996870i	$0.525193\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−37252.8	−1.35482	−0.677410	−	0.735605i	$-0.736898\pi$
$911$	−37252.8	−1.35482	−0.677410	+	0.735605i	$0.736898\pi$
$912$	0	0
$913$	−52579.7	−1.90595
$914$	0	0
$915$	0	0
$916$	0	0
$917$	47153.9	1.69810
$918$	0	0
$919$	17996.5	0.645975	0.322987	−	0.946403i	$-0.395313\pi$
$919$	17996.5	0.645975	0.322987	+	0.946403i	$0.395313\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−30869.3	−1.10084
$924$	0	0
$925$	−38338.8	−1.36278
$926$	0	0
$927$	0	0
$928$	0	0
$929$	5536.38	0.195525	0.0977624	−	0.995210i	$-0.468831\pi$
$929$	5536.38	0.195525	0.0977624	+	0.995210i	$0.468831\pi$
$930$	0	0
$931$	31300.8	1.10187
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1100.60	0.0384955
$936$	0	0
$937$	26959.0	0.939926	0.469963	−	0.882686i	$-0.344267\pi$
$937$	26959.0	0.939926	0.469963	+	0.882686i	$0.344267\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−20079.7	−0.695622	−0.347811	−	0.937565i	$-0.613075\pi$
$941$	−20079.7	−0.695622	−0.347811	+	0.937565i	$0.613075\pi$
$942$	0	0
$943$	194.514	0.00671713
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−13308.3	−0.456664	−0.228332	−	0.973583i	$-0.573327\pi$
$947$	−13308.3	−0.456664	−0.228332	+	0.973583i	$0.573327\pi$
$948$	0	0
$949$	35310.3	1.20782
$950$	0	0
$951$	0	0
$952$	0	0
$953$	43058.3	1.46358	0.731792	−	0.681528i	$-0.238684\pi$
$953$	43058.3	1.46358	0.731792	+	0.681528i	$0.238684\pi$
$954$	0	0
$955$	−304.494	−0.0103175
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−83935.9	−2.82631
$960$	0	0
$961$	−18611.2	−0.624727
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−278.497	−0.00929030
$966$	0	0
$967$	−5593.18	−0.186002	−0.0930012	−	0.995666i	$-0.529646\pi$
$967$	−5593.18	−0.186002	−0.0930012	+	0.995666i	$0.529646\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−9902.15	−0.327266	−0.163633	−	0.986521i	$-0.552321\pi$
$971$	−9902.15	−0.327266	−0.163633	+	0.986521i	$0.552321\pi$
$972$	0	0
$973$	86217.8	2.84072
$974$	0	0
$975$	0	0
$976$	0	0
$977$	17066.5	0.558860	0.279430	−	0.960166i	$-0.409855\pi$
$977$	17066.5	0.558860	0.279430	+	0.960166i	$0.409855\pi$
$978$	0	0
$979$	50977.6	1.66420
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−15726.6	−0.510275	−0.255137	−	0.966905i	$-0.582121\pi$
$983$	−15726.6	−0.510275	−0.255137	+	0.966905i	$0.582121\pi$
$984$	0	0
$985$	−775.165	−0.0250749
$986$	0	0
$987$	0	0
$988$	0	0
$989$	19068.5	0.613086
$990$	0	0
$991$	−40492.7	−1.29797	−0.648987	−	0.760799i	$-0.724807\pi$
$991$	−40492.7	−1.29797	−0.648987	+	0.760799i	$0.724807\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−456.957	−0.0145593
$996$	0	0
$997$	−28445.5	−0.903588	−0.451794	−	0.892122i	$-0.649216\pi$
$997$	−28445.5	−0.903588	−0.451794	+	0.892122i	$0.649216\pi$
$998$	0	0
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.4.a.y.1.2	✓	3
3.2	odd	2		1152.4.a.bc.1.2	yes	3
4.3	odd	2		1152.4.a.ba.1.2	yes	3
8.3	odd	2		1152.4.a.bf.1.2	yes	3
8.5	even	2		1152.4.a.bd.1.2	yes	3
12.11	even	2		1152.4.a.be.1.2	yes	3
24.5	odd	2		1152.4.a.z.1.2	yes	3
24.11	even	2		1152.4.a.bb.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.4.a.y.1.2	✓	3	1.1	even	1	trivial
1152.4.a.z.1.2	yes	3	24.5	odd	2
1152.4.a.ba.1.2	yes	3	4.3	odd	2
1152.4.a.bb.1.2	yes	3	24.11	even	2
1152.4.a.bc.1.2	yes	3	3.2	odd	2
1152.4.a.bd.1.2	yes	3	8.5	even	2
1152.4.a.be.1.2	yes	3	12.11	even	2
1152.4.a.bf.1.2	yes	3	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1152.4.a.y.1.2

Level	$1152$
Weight	$4$
Character	1152.1
Self dual	yes
Analytic conductor	$67.970$
Analytic rank	$1$
Dimension	$3$
CM	no
Inner twists	$1$