LMFDB - Embedded newform 2160.4.a.bk.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2160,4,Mod(1,2160)] 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("2160.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2160.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,-15,0,10,0,0,0,28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$127.444125612$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1257.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 8x + 9 $ x^3 - x^2 - 8*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	no (minimal twist has level 1080)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.72396$ of defining polynomial
Character	$\chi$	$=$	2160.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-5.00000 q^{5} -24.0794 q^{7} +2.95982 q^{11} -22.1759 q^{13} +76.0223 q^{17} +72.1187 q^{19} -176.003 q^{23} +25.0000 q^{25} +42.5134 q^{29} +327.010 q^{31} +120.397 q^{35} +182.089 q^{37} +154.624 q^{41} -173.531 q^{43} -338.059 q^{47} +236.820 q^{49} -26.5490 q^{53} -14.7991 q^{55} +391.670 q^{59} -191.191 q^{61} +110.879 q^{65} +507.583 q^{67} +576.547 q^{71} -390.440 q^{73} -71.2709 q^{77} -1220.43 q^{79} -247.332 q^{83} -380.111 q^{85} +1500.81 q^{89} +533.983 q^{91} -360.594 q^{95} +959.331 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 15 q^{5} + 10 q^{7} + 28 q^{11} - 78 q^{13} - 11 q^{17} + 71 q^{19} - 25 q^{23} + 75 q^{25} + 118 q^{29} + 107 q^{31} - 50 q^{35} - 410 q^{37} + 592 q^{41} - 52 q^{43} - 580 q^{47} + 479 q^{49} + 169 q^{53}+ \cdots - 840 q^{97}+O(q^{100}) $ 3 * q - 15 * q^5 + 10 * q^7 + 28 * q^11 - 78 * q^13 - 11 * q^17 + 71 * q^19 - 25 * q^23 + 75 * q^25 + 118 * q^29 + 107 * q^31 - 50 * q^35 - 410 * q^37 + 592 * q^41 - 52 * q^43 - 580 * q^47 + 479 * q^49 + 169 * q^53 - 140 * q^55 + 234 * q^59 - 673 * q^61 + 390 * q^65 - 386 * q^67 + 16 * q^71 - 892 * q^73 + 1800 * q^77 - 1263 * q^79 - 1815 * q^83 + 55 * q^85 + 1800 * q^89 - 1284 * q^91 - 355 * q^95 - 840 * 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
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−24.0794	−1.30017	−0.650084	−	0.759862i	$-0.725266\pi$
$7$	−24.0794	−1.30017	−0.650084	+	0.759862i	$0.725266\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.95982	0.0811292	0.0405646	−	0.999177i	$-0.487084\pi$
$11$	2.95982	0.0811292	0.0405646	+	0.999177i	$0.487084\pi$
$12$	0	0
$13$	−22.1759	−0.473114	−0.236557	−	0.971618i	$-0.576019\pi$
$13$	−22.1759	−0.473114	−0.236557	+	0.971618i	$0.576019\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	76.0223	1.08459	0.542297	−	0.840187i	$-0.317555\pi$
$17$	76.0223	1.08459	0.542297	+	0.840187i	$0.317555\pi$
$18$	0	0
$19$	72.1187	0.870798	0.435399	−	0.900238i	$-0.356607\pi$
$19$	72.1187	0.870798	0.435399	+	0.900238i	$0.356607\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−176.003	−1.59562	−0.797810	−	0.602909i	$-0.794008\pi$
$23$	−176.003	−1.59562	−0.797810	+	0.602909i	$0.794008\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	42.5134	0.272226	0.136113	−	0.990693i	$-0.456539\pi$
$29$	42.5134	0.272226	0.136113	+	0.990693i	$0.456539\pi$
$30$	0	0
$31$	327.010	1.89460	0.947301	−	0.320345i	$-0.103799\pi$
$31$	327.010	1.89460	0.947301	+	0.320345i	$0.103799\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	120.397	0.581453
$36$	0	0
$37$	182.089	0.809061	0.404530	−	0.914525i	$-0.367435\pi$
$37$	182.089	0.809061	0.404530	+	0.914525i	$0.367435\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	154.624	0.588981	0.294491	−	0.955654i	$-0.404850\pi$
$41$	154.624	0.588981	0.294491	+	0.955654i	$0.404850\pi$
$42$	0	0
$43$	−173.531	−0.615424	−0.307712	−	0.951479i	$-0.599563\pi$
$43$	−173.531	−0.615424	−0.307712	+	0.951479i	$0.599563\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−338.059	−1.04917	−0.524585	−	0.851358i	$-0.675779\pi$
$47$	−338.059	−1.04917	−0.524585	+	0.851358i	$0.675779\pi$
$48$	0	0
$49$	236.820	0.690436
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−26.5490	−0.0688074	−0.0344037	−	0.999408i	$-0.510953\pi$
$53$	−26.5490	−0.0688074	−0.0344037	+	0.999408i	$0.510953\pi$
$54$	0	0
$55$	−14.7991	−0.0362821
$56$	0	0
$57$	0	0
$58$	0	0
$59$	391.670	0.864256	0.432128	−	0.901812i	$-0.357763\pi$
$59$	391.670	0.864256	0.432128	+	0.901812i	$0.357763\pi$
$60$	0	0
$61$	−191.191	−0.401303	−0.200652	−	0.979663i	$-0.564306\pi$
$61$	−191.191	−0.401303	−0.200652	+	0.979663i	$0.564306\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	110.879	0.211583
$66$	0	0
$67$	507.583	0.925540	0.462770	−	0.886478i	$-0.346856\pi$
$67$	507.583	0.925540	0.462770	+	0.886478i	$0.346856\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	576.547	0.963712	0.481856	−	0.876251i	$-0.339963\pi$
$71$	576.547	0.963712	0.481856	+	0.876251i	$0.339963\pi$
$72$	0	0
$73$	−390.440	−0.625994	−0.312997	−	0.949754i	$-0.601333\pi$
$73$	−390.440	−0.625994	−0.312997	+	0.949754i	$0.601333\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−71.2709	−0.105482
$78$	0	0
$79$	−1220.43	−1.73808	−0.869042	−	0.494739i	$-0.835264\pi$
$79$	−1220.43	−1.73808	−0.869042	+	0.494739i	$0.835264\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−247.332	−0.327088	−0.163544	−	0.986536i	$-0.552292\pi$
$83$	−247.332	−0.327088	−0.163544	+	0.986536i	$0.552292\pi$
$84$	0	0
$85$	−380.111	−0.485045
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1500.81	1.78748	0.893740	−	0.448586i	$-0.148072\pi$
$89$	1500.81	1.78748	0.893740	+	0.448586i	$0.148072\pi$
$90$	0	0
$91$	533.983	0.615128
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−360.594	−0.389433
$96$	0	0
$97$	959.331	1.00418	0.502089	−	0.864816i	$-0.332565\pi$
$97$	959.331	1.00418	0.502089	+	0.864816i	$0.332565\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	587.011	0.578314	0.289157	−	0.957282i	$-0.406625\pi$
$101$	587.011	0.578314	0.289157	+	0.957282i	$0.406625\pi$
$102$	0	0
$103$	−1359.95	−1.30097	−0.650483	−	0.759521i	$-0.725433\pi$
$103$	−1359.95	−1.30097	−0.650483	+	0.759521i	$0.725433\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1041.85	−0.941306	−0.470653	−	0.882318i	$-0.655982\pi$
$107$	−1041.85	−0.941306	−0.470653	+	0.882318i	$0.655982\pi$
$108$	0	0
$109$	−457.732	−0.402227	−0.201114	−	0.979568i	$-0.564456\pi$
$109$	−457.732	−0.402227	−0.201114	+	0.979568i	$0.564456\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	467.527	0.389214	0.194607	−	0.980881i	$-0.437657\pi$
$113$	467.527	0.389214	0.194607	+	0.980881i	$0.437657\pi$
$114$	0	0
$115$	880.017	0.713583
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1830.57	−1.41015
$120$	0	0
$121$	−1322.24	−0.993418
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−2368.02	−1.65455	−0.827274	−	0.561799i	$-0.810109\pi$
$127$	−2368.02	−1.65455	−0.827274	+	0.561799i	$0.810109\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2532.97	−1.68937	−0.844683	−	0.535267i	$-0.820211\pi$
$131$	−2532.97	−1.68937	−0.844683	+	0.535267i	$0.820211\pi$
$132$	0	0
$133$	−1736.58	−1.13218
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−424.096	−0.264474	−0.132237	−	0.991218i	$-0.542216\pi$
$137$	−424.096	−0.264474	−0.132237	+	0.991218i	$0.542216\pi$
$138$	0	0
$139$	−1461.14	−0.891596	−0.445798	−	0.895134i	$-0.647080\pi$
$139$	−1461.14	−0.891596	−0.445798	+	0.895134i	$0.647080\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−65.6368	−0.0383834
$144$	0	0
$145$	−212.567	−0.121743
$146$	0	0
$147$	0	0
$148$	0	0
$149$	497.826	0.273715	0.136857	−	0.990591i	$-0.456300\pi$
$149$	497.826	0.273715	0.136857	+	0.990591i	$0.456300\pi$
$150$	0	0
$151$	−708.375	−0.381767	−0.190883	−	0.981613i	$-0.561135\pi$
$151$	−708.375	−0.381767	−0.190883	+	0.981613i	$0.561135\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1635.05	−0.847292
$156$	0	0
$157$	351.998	0.178933	0.0894666	−	0.995990i	$-0.471484\pi$
$157$	351.998	0.178933	0.0894666	+	0.995990i	$0.471484\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4238.06	2.07457
$162$	0	0
$163$	791.353	0.380267	0.190133	−	0.981758i	$-0.439108\pi$
$163$	791.353	0.380267	0.190133	+	0.981758i	$0.439108\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2756.06	−1.27707	−0.638535	−	0.769593i	$-0.720459\pi$
$167$	−2756.06	−1.27707	−0.638535	+	0.769593i	$0.720459\pi$
$168$	0	0
$169$	−1705.23	−0.776163
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3691.62	1.62236	0.811182	−	0.584794i	$-0.198825\pi$
$173$	3691.62	1.62236	0.811182	+	0.584794i	$0.198825\pi$
$174$	0	0
$175$	−601.986	−0.260034
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2036.15	0.850217	0.425109	−	0.905142i	$-0.360236\pi$
$179$	2036.15	0.850217	0.425109	+	0.905142i	$0.360236\pi$
$180$	0	0
$181$	−3268.74	−1.34234	−0.671170	−	0.741304i	$-0.734208\pi$
$181$	−3268.74	−1.34234	−0.671170	+	0.741304i	$0.734208\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−910.445	−0.361823
$186$	0	0
$187$	225.013	0.0879922
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2192.48	0.830589	0.415294	−	0.909687i	$-0.363679\pi$
$191$	2192.48	0.830589	0.415294	+	0.909687i	$0.363679\pi$
$192$	0	0
$193$	3326.76	1.24075	0.620376	−	0.784304i	$-0.286980\pi$
$193$	3326.76	1.24075	0.620376	+	0.784304i	$0.286980\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1818.42	−0.657648	−0.328824	−	0.944391i	$-0.606652\pi$
$197$	−1818.42	−0.657648	−0.328824	+	0.944391i	$0.606652\pi$
$198$	0	0
$199$	1191.04	0.424275	0.212137	−	0.977240i	$-0.431958\pi$
$199$	1191.04	0.424275	0.212137	+	0.977240i	$0.431958\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1023.70	−0.353939
$204$	0	0
$205$	−773.121	−0.263400
$206$	0	0
$207$	0	0
$208$	0	0
$209$	213.459	0.0706471
$210$	0	0
$211$	−2929.49	−0.955803	−0.477902	−	0.878413i	$-0.658602\pi$
$211$	−2929.49	−0.955803	−0.477902	+	0.878413i	$0.658602\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	867.656	0.275226
$216$	0	0
$217$	−7874.21	−2.46330
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1685.86	−0.513137
$222$	0	0
$223$	−1479.80	−0.444370	−0.222185	−	0.975004i	$-0.571319\pi$
$223$	−1479.80	−0.444370	−0.222185	+	0.975004i	$0.571319\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1543.24	−0.451228	−0.225614	−	0.974217i	$-0.572439\pi$
$227$	−1543.24	−0.451228	−0.225614	+	0.974217i	$0.572439\pi$
$228$	0	0
$229$	3163.97	0.913018	0.456509	−	0.889719i	$-0.349100\pi$
$229$	3163.97	0.913018	0.456509	+	0.889719i	$0.349100\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1466.95	−0.412459	−0.206229	−	0.978504i	$-0.566119\pi$
$233$	−1466.95	−0.412459	−0.206229	+	0.978504i	$0.566119\pi$
$234$	0	0
$235$	1690.29	0.469203
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1346.42	0.364405	0.182202	−	0.983261i	$-0.441677\pi$
$239$	1346.42	0.364405	0.182202	+	0.983261i	$0.441677\pi$
$240$	0	0
$241$	−47.7311	−0.0127578	−0.00637890	−	0.999980i	$-0.502030\pi$
$241$	−47.7311	−0.0127578	−0.00637890	+	0.999980i	$0.502030\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1184.10	−0.308772
$246$	0	0
$247$	−1599.30	−0.411987
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5074.18	−1.27601	−0.638007	−	0.770031i	$-0.720241\pi$
$251$	−5074.18	−1.27601	−0.638007	+	0.770031i	$0.720241\pi$
$252$	0	0
$253$	−520.939	−0.129451
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3088.03	0.749518	0.374759	−	0.927122i	$-0.377725\pi$
$257$	3088.03	0.749518	0.374759	+	0.927122i	$0.377725\pi$
$258$	0	0
$259$	−4384.60	−1.05191
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4777.33	−1.12009	−0.560043	−	0.828463i	$-0.689216\pi$
$263$	−4777.33	−1.12009	−0.560043	+	0.828463i	$0.689216\pi$
$264$	0	0
$265$	132.745	0.0307716
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7972.82	1.80710	0.903552	−	0.428478i	$-0.140950\pi$
$269$	7972.82	1.80710	0.903552	+	0.428478i	$0.140950\pi$
$270$	0	0
$271$	−3478.08	−0.779625	−0.389813	−	0.920894i	$-0.627460\pi$
$271$	−3478.08	−0.779625	−0.389813	+	0.920894i	$0.627460\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	73.9956	0.0162258
$276$	0	0
$277$	−7166.62	−1.55451	−0.777257	−	0.629184i	$-0.783389\pi$
$277$	−7166.62	−1.55451	−0.777257	+	0.629184i	$0.783389\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−8157.14	−1.73172	−0.865862	−	0.500284i	$-0.833229\pi$
$281$	−8157.14	−1.73172	−0.865862	+	0.500284i	$0.833229\pi$
$282$	0	0
$283$	−4435.42	−0.931654	−0.465827	−	0.884876i	$-0.654243\pi$
$283$	−4435.42	−0.931654	−0.465827	+	0.884876i	$0.654243\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3723.26	−0.765774
$288$	0	0
$289$	866.384	0.176345
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1811.46	−0.361183	−0.180592	−	0.983558i	$-0.557801\pi$
$293$	−1811.46	−0.361183	−0.180592	+	0.983558i	$0.557801\pi$
$294$	0	0
$295$	−1958.35	−0.386507
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3903.03	0.754911
$300$	0	0
$301$	4178.53	0.800155
$302$	0	0
$303$	0	0
$304$	0	0
$305$	955.955	0.179468
$306$	0	0
$307$	9369.97	1.74193	0.870965	−	0.491345i	$-0.163495\pi$
$307$	9369.97	1.74193	0.870965	+	0.491345i	$0.163495\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	824.869	0.150399	0.0751994	−	0.997169i	$-0.476041\pi$
$311$	824.869	0.150399	0.0751994	+	0.997169i	$0.476041\pi$
$312$	0	0
$313$	3447.76	0.622616	0.311308	−	0.950309i	$-0.399233\pi$
$313$	3447.76	0.622616	0.311308	+	0.950309i	$0.399233\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3822.49	−0.677264	−0.338632	−	0.940919i	$-0.609964\pi$
$317$	−3822.49	−0.677264	−0.338632	+	0.940919i	$0.609964\pi$
$318$	0	0
$319$	125.832	0.0220854
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5482.63	0.944463
$324$	0	0
$325$	−554.397	−0.0946229
$326$	0	0
$327$	0	0
$328$	0	0
$329$	8140.27	1.36410
$330$	0	0
$331$	297.908	0.0494698	0.0247349	−	0.999694i	$-0.492126\pi$
$331$	297.908	0.0494698	0.0247349	+	0.999694i	$0.492126\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2537.92	−0.413914
$336$	0	0
$337$	738.773	0.119417	0.0597085	−	0.998216i	$-0.480983\pi$
$337$	738.773	0.119417	0.0597085	+	0.998216i	$0.480983\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	967.891	0.153707
$342$	0	0
$343$	2556.77	0.402485
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3379.76	0.522868	0.261434	−	0.965221i	$-0.415805\pi$
$347$	3379.76	0.522868	0.261434	+	0.965221i	$0.415805\pi$
$348$	0	0
$349$	−2622.01	−0.402157	−0.201078	−	0.979575i	$-0.564445\pi$
$349$	−2622.01	−0.402157	−0.201078	+	0.979575i	$0.564445\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3052.95	0.460318	0.230159	−	0.973153i	$-0.426075\pi$
$353$	3052.95	0.460318	0.230159	+	0.973153i	$0.426075\pi$
$354$	0	0
$355$	−2882.73	−0.430985
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1613.36	0.237187	0.118593	−	0.992943i	$-0.462162\pi$
$359$	1613.36	0.237187	0.118593	+	0.992943i	$0.462162\pi$
$360$	0	0
$361$	−1657.89	−0.241710
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1952.20	0.279953
$366$	0	0
$367$	−5387.79	−0.766323	−0.383161	−	0.923681i	$-0.625165\pi$
$367$	−5387.79	−0.766323	−0.383161	+	0.923681i	$0.625165\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	639.286	0.0894611
$372$	0	0
$373$	−13742.0	−1.90759	−0.953796	−	0.300455i	$-0.902861\pi$
$373$	−13742.0	−1.90759	−0.953796	+	0.300455i	$0.902861\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−942.773	−0.128794
$378$	0	0
$379$	−6648.79	−0.901123	−0.450561	−	0.892745i	$-0.648776\pi$
$379$	−6648.79	−0.901123	−0.450561	+	0.892745i	$0.648776\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	9086.53	1.21227	0.606136	−	0.795361i	$-0.292719\pi$
$383$	9086.53	1.21227	0.606136	+	0.795361i	$0.292719\pi$
$384$	0	0
$385$	356.355	0.0471728
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−3098.40	−0.403844	−0.201922	−	0.979402i	$-0.564719\pi$
$389$	−3098.40	−0.403844	−0.201922	+	0.979402i	$0.564719\pi$
$390$	0	0
$391$	−13380.2	−1.73060
$392$	0	0
$393$	0	0
$394$	0	0
$395$	6102.13	0.777294
$396$	0	0
$397$	−13686.5	−1.73024	−0.865122	−	0.501562i	$-0.832759\pi$
$397$	−13686.5	−1.73024	−0.865122	+	0.501562i	$0.832759\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3734.83	0.465108	0.232554	−	0.972583i	$-0.425292\pi$
$401$	3734.83	0.465108	0.232554	+	0.972583i	$0.425292\pi$
$402$	0	0
$403$	−7251.73	−0.896363
$404$	0	0
$405$	0	0
$406$	0	0
$407$	538.952	0.0656384
$408$	0	0
$409$	3672.62	0.444009	0.222004	−	0.975046i	$-0.428740\pi$
$409$	3672.62	0.444009	0.222004	+	0.975046i	$0.428740\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−9431.20	−1.12368
$414$	0	0
$415$	1236.66	0.146278
$416$	0	0
$417$	0	0
$418$	0	0
$419$	9111.68	1.06237	0.531187	−	0.847255i	$-0.321746\pi$
$419$	9111.68	1.06237	0.531187	+	0.847255i	$0.321746\pi$
$420$	0	0
$421$	13640.0	1.57903	0.789514	−	0.613733i	$-0.210333\pi$
$421$	13640.0	1.57903	0.789514	+	0.613733i	$0.210333\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1900.56	0.216919
$426$	0	0
$427$	4603.77	0.521762
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9252.97	1.03411	0.517053	−	0.855953i	$-0.327029\pi$
$431$	9252.97	1.03411	0.517053	+	0.855953i	$0.327029\pi$
$432$	0	0
$433$	−8990.85	−0.997858	−0.498929	−	0.866643i	$-0.666273\pi$
$433$	−8990.85	−0.997858	−0.498929	+	0.866643i	$0.666273\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−12693.1	−1.38946
$438$	0	0
$439$	−10409.3	−1.13169	−0.565843	−	0.824513i	$-0.691449\pi$
$439$	−10409.3	−1.13169	−0.565843	+	0.824513i	$0.691449\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3083.63	−0.330717	−0.165359	−	0.986233i	$-0.552878\pi$
$443$	−3083.63	−0.330717	−0.165359	+	0.986233i	$0.552878\pi$
$444$	0	0
$445$	−7504.05	−0.799385
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−16734.4	−1.75890	−0.879448	−	0.475996i	$-0.842088\pi$
$449$	−16734.4	−1.75890	−0.879448	+	0.475996i	$0.842088\pi$
$450$	0	0
$451$	457.660	0.0477836
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2669.92	−0.275094
$456$	0	0
$457$	−19008.4	−1.94568	−0.972841	−	0.231473i	$-0.925646\pi$
$457$	−19008.4	−1.94568	−0.972841	+	0.231473i	$0.925646\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−8624.58	−0.871338	−0.435669	−	0.900107i	$-0.643488\pi$
$461$	−8624.58	−0.871338	−0.435669	+	0.900107i	$0.643488\pi$
$462$	0	0
$463$	−8116.40	−0.814689	−0.407345	−	0.913275i	$-0.633545\pi$
$463$	−8116.40	−0.814689	−0.407345	+	0.913275i	$0.633545\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3645.17	−0.361196	−0.180598	−	0.983557i	$-0.557803\pi$
$467$	−3645.17	−0.361196	−0.180598	+	0.983557i	$0.557803\pi$
$468$	0	0
$469$	−12222.3	−1.20336
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−513.622	−0.0499289
$474$	0	0
$475$	1802.97	0.174160
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6118.27	0.583613	0.291807	−	0.956477i	$-0.405744\pi$
$479$	6118.27	0.583613	0.291807	+	0.956477i	$0.405744\pi$
$480$	0	0
$481$	−4037.99	−0.382778
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4796.66	−0.449082
$486$	0	0
$487$	20400.0	1.89818	0.949091	−	0.315004i	$-0.102006\pi$
$487$	20400.0	1.89818	0.949091	+	0.315004i	$0.102006\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4429.08	−0.407091	−0.203545	−	0.979066i	$-0.565246\pi$
$491$	−4429.08	−0.407091	−0.203545	+	0.979066i	$0.565246\pi$
$492$	0	0
$493$	3231.96	0.295254
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−13882.9	−1.25299
$498$	0	0
$499$	−17665.2	−1.58477	−0.792386	−	0.610020i	$-0.791161\pi$
$499$	−17665.2	−1.58477	−0.792386	+	0.610020i	$0.791161\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1518.17	0.134577	0.0672883	−	0.997734i	$-0.478565\pi$
$503$	1518.17	0.134577	0.0672883	+	0.997734i	$0.478565\pi$
$504$	0	0
$505$	−2935.05	−0.258630
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−8679.43	−0.755813	−0.377906	−	0.925844i	$-0.623356\pi$
$509$	−8679.43	−0.755813	−0.377906	+	0.925844i	$0.623356\pi$
$510$	0	0
$511$	9401.58	0.813897
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6799.73	0.581809
$516$	0	0
$517$	−1000.59	−0.0851182
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5080.54	0.427222	0.213611	−	0.976919i	$-0.431478\pi$
$521$	5080.54	0.427222	0.213611	+	0.976919i	$0.431478\pi$
$522$	0	0
$523$	−13889.3	−1.16125	−0.580626	−	0.814170i	$-0.697192\pi$
$523$	−13889.3	−1.16125	−0.580626	+	0.814170i	$0.697192\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	24860.0	2.05487
$528$	0	0
$529$	18810.2	1.54600
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−3428.93	−0.278656
$534$	0	0
$535$	5209.26	0.420965
$536$	0	0
$537$	0	0
$538$	0	0
$539$	700.944	0.0560145
$540$	0	0
$541$	−20787.4	−1.65198	−0.825989	−	0.563686i	$-0.809383\pi$
$541$	−20787.4	−1.65198	−0.825989	+	0.563686i	$0.809383\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2288.66	0.179882
$546$	0	0
$547$	2015.14	0.157516	0.0787578	−	0.996894i	$-0.474905\pi$
$547$	2015.14	0.157516	0.0787578	+	0.996894i	$0.474905\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3066.01	0.237054
$552$	0	0
$553$	29387.2	2.25980
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16460.8	1.25218	0.626092	−	0.779750i	$-0.284654\pi$
$557$	16460.8	1.25218	0.626092	+	0.779750i	$0.284654\pi$
$558$	0	0
$559$	3848.21	0.291166
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−968.373	−0.0724903	−0.0362452	−	0.999343i	$-0.511540\pi$
$563$	−968.373	−0.0724903	−0.0362452	+	0.999343i	$0.511540\pi$
$564$	0	0
$565$	−2337.63	−0.174062
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−3305.91	−0.243569	−0.121785	−	0.992557i	$-0.538862\pi$
$569$	−3305.91	−0.243569	−0.121785	+	0.992557i	$0.538862\pi$
$570$	0	0
$571$	−22291.2	−1.63373	−0.816863	−	0.576831i	$-0.804289\pi$
$571$	−22291.2	−1.63373	−0.816863	+	0.576831i	$0.804289\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4400.09	−0.319124
$576$	0	0
$577$	12887.3	0.929820	0.464910	−	0.885358i	$-0.346087\pi$
$577$	12887.3	0.929820	0.464910	+	0.885358i	$0.346087\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	5955.63	0.425269
$582$	0	0
$583$	−78.5805	−0.00558228
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8042.37	−0.565493	−0.282746	−	0.959195i	$-0.591245\pi$
$587$	−8042.37	−0.565493	−0.282746	+	0.959195i	$0.591245\pi$
$588$	0	0
$589$	23583.5	1.64982
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−4779.03	−0.330946	−0.165473	−	0.986214i	$-0.552915\pi$
$593$	−4779.03	−0.330946	−0.165473	+	0.986214i	$0.552915\pi$
$594$	0	0
$595$	9152.87	0.630640
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−4890.20	−0.333569	−0.166785	−	0.985993i	$-0.553338\pi$
$599$	−4890.20	−0.333569	−0.166785	+	0.985993i	$0.553338\pi$
$600$	0	0
$601$	−2732.86	−0.185484	−0.0927418	−	0.995690i	$-0.529563\pi$
$601$	−2732.86	−0.185484	−0.0927418	+	0.995690i	$0.529563\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	6611.20	0.444270
$606$	0	0
$607$	18006.9	1.20408	0.602041	−	0.798465i	$-0.294355\pi$
$607$	18006.9	1.20408	0.602041	+	0.798465i	$0.294355\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	7496.76	0.496377
$612$	0	0
$613$	−9779.27	−0.644341	−0.322170	−	0.946682i	$-0.604412\pi$
$613$	−9779.27	−0.644341	−0.322170	+	0.946682i	$0.604412\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18024.5	1.17608	0.588040	−	0.808832i	$-0.299900\pi$
$617$	18024.5	1.17608	0.588040	+	0.808832i	$0.299900\pi$
$618$	0	0
$619$	−2990.51	−0.194182	−0.0970911	−	0.995275i	$-0.530954\pi$
$619$	−2990.51	−0.194182	−0.0970911	+	0.995275i	$0.530954\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−36138.7	−2.32402
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	13842.8	0.877503
$630$	0	0
$631$	28203.6	1.77935	0.889673	−	0.456599i	$-0.150932\pi$
$631$	28203.6	1.77935	0.889673	+	0.456599i	$0.150932\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	11840.1	0.739936
$636$	0	0
$637$	−5251.69	−0.326655
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−27172.9	−1.67436	−0.837180	−	0.546928i	$-0.815797\pi$
$641$	−27172.9	−1.67436	−0.837180	+	0.546928i	$0.815797\pi$
$642$	0	0
$643$	278.480	0.0170796	0.00853979	−	0.999964i	$-0.497282\pi$
$643$	278.480	0.0170796	0.00853979	+	0.999964i	$0.497282\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	22641.1	1.37576	0.687878	−	0.725826i	$-0.258542\pi$
$647$	22641.1	1.37576	0.687878	+	0.725826i	$0.258542\pi$
$648$	0	0
$649$	1159.28	0.0701164
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−23064.5	−1.38221	−0.691104	−	0.722755i	$-0.742875\pi$
$653$	−23064.5	−1.38221	−0.691104	+	0.722755i	$0.742875\pi$
$654$	0	0
$655$	12664.9	0.755507
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−22595.0	−1.33563	−0.667813	−	0.744329i	$-0.732769\pi$
$659$	−22595.0	−1.33563	−0.667813	+	0.744329i	$0.732769\pi$
$660$	0	0
$661$	−14627.9	−0.860756	−0.430378	−	0.902649i	$-0.641620\pi$
$661$	−14627.9	−0.860756	−0.430378	+	0.902649i	$0.641620\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	8682.89	0.506328
$666$	0	0
$667$	−7482.50	−0.434368
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−565.892	−0.0325574
$672$	0	0
$673$	22246.9	1.27423	0.637113	−	0.770770i	$-0.280128\pi$
$673$	22246.9	1.27423	0.637113	+	0.770770i	$0.280128\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	13502.1	0.766513	0.383256	−	0.923642i	$-0.374803\pi$
$677$	13502.1	0.766513	0.383256	+	0.923642i	$0.374803\pi$
$678$	0	0
$679$	−23100.2	−1.30560
$680$	0	0
$681$	0	0
$682$	0	0
$683$	25582.5	1.43322	0.716609	−	0.697475i	$-0.245693\pi$
$683$	25582.5	1.43322	0.716609	+	0.697475i	$0.245693\pi$
$684$	0	0
$685$	2120.48	0.118276
$686$	0	0
$687$	0	0
$688$	0	0
$689$	588.749	0.0325538
$690$	0	0
$691$	12434.6	0.684565	0.342283	−	0.939597i	$-0.388800\pi$
$691$	12434.6	0.684565	0.342283	+	0.939597i	$0.388800\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7305.68	0.398734
$696$	0	0
$697$	11754.9	0.638806
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−13773.1	−0.742088	−0.371044	−	0.928615i	$-0.621000\pi$
$701$	−13773.1	−0.742088	−0.371044	+	0.928615i	$0.621000\pi$
$702$	0	0
$703$	13132.0	0.704529
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−14134.9	−0.751906
$708$	0	0
$709$	−11950.6	−0.633023	−0.316511	−	0.948589i	$-0.602512\pi$
$709$	−11950.6	−0.633023	−0.316511	+	0.948589i	$0.602512\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−57554.8	−3.02306
$714$	0	0
$715$	328.184	0.0171656
$716$	0	0
$717$	0	0
$718$	0	0
$719$	23596.2	1.22391	0.611953	−	0.790894i	$-0.290384\pi$
$719$	23596.2	1.22391	0.611953	+	0.790894i	$0.290384\pi$
$720$	0	0
$721$	32746.7	1.69147
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1062.83	0.0544451
$726$	0	0
$727$	14414.5	0.735358	0.367679	−	0.929953i	$-0.380152\pi$
$727$	14414.5	0.735358	0.367679	+	0.929953i	$0.380152\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−13192.2	−0.667486
$732$	0	0
$733$	28989.8	1.46079	0.730397	−	0.683023i	$-0.239335\pi$
$733$	28989.8	1.46079	0.730397	+	0.683023i	$0.239335\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1502.36	0.0750883
$738$	0	0
$739$	−5035.96	−0.250678	−0.125339	−	0.992114i	$-0.540002\pi$
$739$	−5035.96	−0.250678	−0.125339	+	0.992114i	$0.540002\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−36268.2	−1.79078	−0.895390	−	0.445282i	$-0.853103\pi$
$743$	−36268.2	−1.79078	−0.895390	+	0.445282i	$0.853103\pi$
$744$	0	0
$745$	−2489.13	−0.122409
$746$	0	0
$747$	0	0
$748$	0	0
$749$	25087.2	1.22386
$750$	0	0
$751$	−6876.86	−0.334141	−0.167071	−	0.985945i	$-0.553431\pi$
$751$	−6876.86	−0.334141	−0.167071	+	0.985945i	$0.553431\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	3541.88	0.170731
$756$	0	0
$757$	−32381.0	−1.55470	−0.777350	−	0.629068i	$-0.783437\pi$
$757$	−32381.0	−1.55470	−0.777350	+	0.629068i	$0.783437\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−9253.37	−0.440781	−0.220391	−	0.975412i	$-0.570733\pi$
$761$	−9253.37	−0.440781	−0.220391	+	0.975412i	$0.570733\pi$
$762$	0	0
$763$	11021.9	0.522963
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8685.64	−0.408892
$768$	0	0
$769$	31587.9	1.48126	0.740630	−	0.671913i	$-0.234527\pi$
$769$	31587.9	1.48126	0.740630	+	0.671913i	$0.234527\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	38886.4	1.80937	0.904687	−	0.426077i	$-0.140105\pi$
$773$	38886.4	1.80937	0.904687	+	0.426077i	$0.140105\pi$
$774$	0	0
$775$	8175.24	0.378920
$776$	0	0
$777$	0	0
$778$	0	0
$779$	11151.3	0.512884
$780$	0	0
$781$	1706.48	0.0781851
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1759.99	−0.0800214
$786$	0	0
$787$	8792.97	0.398266	0.199133	−	0.979972i	$-0.436187\pi$
$787$	8792.97	0.398266	0.199133	+	0.979972i	$0.436187\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−11257.8	−0.506044
$792$	0	0
$793$	4239.83	0.189862
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−21136.5	−0.939391	−0.469695	−	0.882829i	$-0.655636\pi$
$797$	−21136.5	−0.939391	−0.469695	+	0.882829i	$0.655636\pi$
$798$	0	0
$799$	−25700.0	−1.13792
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1155.63	−0.0507864
$804$	0	0
$805$	−21190.3	−0.927777
$806$	0	0
$807$	0	0
$808$	0	0
$809$	31974.0	1.38955	0.694776	−	0.719227i	$-0.255504\pi$
$809$	31974.0	1.38955	0.694776	+	0.719227i	$0.255504\pi$
$810$	0	0
$811$	15694.2	0.679528	0.339764	−	0.940511i	$-0.389653\pi$
$811$	15694.2	0.679528	0.339764	+	0.940511i	$0.389653\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−3956.76	−0.170061
$816$	0	0
$817$	−12514.8	−0.535910
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−156.635	−0.00665848	−0.00332924	−	0.999994i	$-0.501060\pi$
$821$	−156.635	−0.00665848	−0.00332924	+	0.999994i	$0.501060\pi$
$822$	0	0
$823$	−13428.1	−0.568743	−0.284372	−	0.958714i	$-0.591785\pi$
$823$	−13428.1	−0.568743	−0.284372	+	0.958714i	$0.591785\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	33717.9	1.41776	0.708880	−	0.705329i	$-0.249201\pi$
$827$	33717.9	1.41776	0.708880	+	0.705329i	$0.249201\pi$
$828$	0	0
$829$	38733.5	1.62276	0.811380	−	0.584518i	$-0.198717\pi$
$829$	38733.5	1.62276	0.811380	+	0.584518i	$0.198717\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	18003.6	0.748843
$834$	0	0
$835$	13780.3	0.571123
$836$	0	0
$837$	0	0
$838$	0	0
$839$	14106.0	0.580443	0.290221	−	0.956960i	$-0.406271\pi$
$839$	14106.0	0.580443	0.290221	+	0.956960i	$0.406271\pi$
$840$	0	0
$841$	−22581.6	−0.925893
$842$	0	0
$843$	0	0
$844$	0	0
$845$	8526.15	0.347111
$846$	0	0
$847$	31838.8	1.29161
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−32048.3	−1.29095
$852$	0	0
$853$	−24614.3	−0.988015	−0.494007	−	0.869458i	$-0.664468\pi$
$853$	−24614.3	−0.988015	−0.494007	+	0.869458i	$0.664468\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	21660.2	0.863360	0.431680	−	0.902027i	$-0.357921\pi$
$857$	21660.2	0.863360	0.431680	+	0.902027i	$0.357921\pi$
$858$	0	0
$859$	26600.0	1.05655	0.528277	−	0.849072i	$-0.322838\pi$
$859$	26600.0	1.05655	0.528277	+	0.849072i	$0.322838\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−47381.0	−1.86891	−0.934454	−	0.356083i	$-0.884112\pi$
$863$	−47381.0	−1.86891	−0.934454	+	0.356083i	$0.884112\pi$
$864$	0	0
$865$	−18458.1	−0.725543
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3612.24	−0.141009
$870$	0	0
$871$	−11256.1	−0.437886
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3009.93	0.116291
$876$	0	0
$877$	−31597.5	−1.21662	−0.608308	−	0.793701i	$-0.708151\pi$
$877$	−31597.5	−1.21662	−0.608308	+	0.793701i	$0.708151\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	33206.9	1.26989	0.634943	−	0.772559i	$-0.281024\pi$
$881$	33206.9	1.26989	0.634943	+	0.772559i	$0.281024\pi$
$882$	0	0
$883$	12479.4	0.475612	0.237806	−	0.971313i	$-0.423572\pi$
$883$	12479.4	0.475612	0.237806	+	0.971313i	$0.423572\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	7331.93	0.277545	0.138772	−	0.990324i	$-0.455684\pi$
$887$	7331.93	0.277545	0.138772	+	0.990324i	$0.455684\pi$
$888$	0	0
$889$	57020.5	2.15119
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−24380.4	−0.913615
$894$	0	0
$895$	−10180.7	−0.380229
$896$	0	0
$897$	0	0
$898$	0	0
$899$	13902.3	0.515759
$900$	0	0
$901$	−2018.32	−0.0746281
$902$	0	0
$903$	0	0
$904$	0	0
$905$	16343.7	0.600312
$906$	0	0
$907$	4489.18	0.164345	0.0821725	−	0.996618i	$-0.473814\pi$
$907$	4489.18	0.164345	0.0821725	+	0.996618i	$0.473814\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−25710.9	−0.935059	−0.467529	−	0.883978i	$-0.654856\pi$
$911$	−25710.9	−0.935059	−0.467529	+	0.883978i	$0.654856\pi$
$912$	0	0
$913$	−732.061	−0.0265363
$914$	0	0
$915$	0	0
$916$	0	0
$917$	60992.6	2.19646
$918$	0	0
$919$	−24637.6	−0.884353	−0.442176	−	0.896928i	$-0.645793\pi$
$919$	−24637.6	−0.884353	−0.442176	+	0.896928i	$0.645793\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−12785.4	−0.455946
$924$	0	0
$925$	4552.23	0.161812
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−6833.00	−0.241317	−0.120658	−	0.992694i	$-0.538501\pi$
$929$	−6833.00	−0.241317	−0.120658	+	0.992694i	$0.538501\pi$
$930$	0	0
$931$	17079.1	0.601231
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1125.06	−0.0393513
$936$	0	0
$937$	−25120.0	−0.875810	−0.437905	−	0.899021i	$-0.644279\pi$
$937$	−25120.0	−0.875810	−0.437905	+	0.899021i	$0.644279\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	54122.3	1.87496	0.937480	−	0.348040i	$-0.113153\pi$
$941$	54122.3	1.87496	0.937480	+	0.348040i	$0.113153\pi$
$942$	0	0
$943$	−27214.4	−0.939790
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−47670.7	−1.63579	−0.817894	−	0.575370i	$-0.804858\pi$
$947$	−47670.7	−1.63579	−0.817894	+	0.575370i	$0.804858\pi$
$948$	0	0
$949$	8658.36	0.296167
$950$	0	0
$951$	0	0
$952$	0	0
$953$	11963.3	0.406642	0.203321	−	0.979112i	$-0.434826\pi$
$953$	11963.3	0.406642	0.203321	+	0.979112i	$0.434826\pi$
$954$	0	0
$955$	−10962.4	−0.371451
$956$	0	0
$957$	0	0
$958$	0	0
$959$	10212.0	0.343861
$960$	0	0
$961$	77144.3	2.58952
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−16633.8	−0.554881
$966$	0	0
$967$	36546.2	1.21535	0.607676	−	0.794185i	$-0.292102\pi$
$967$	36546.2	1.21535	0.607676	+	0.794185i	$0.292102\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	785.854	0.0259725	0.0129862	−	0.999916i	$-0.495866\pi$
$971$	785.854	0.0259725	0.0129862	+	0.999916i	$0.495866\pi$
$972$	0	0
$973$	35183.3	1.15922
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−16159.0	−0.529144	−0.264572	−	0.964366i	$-0.585231\pi$
$977$	−16159.0	−0.529144	−0.264572	+	0.964366i	$0.585231\pi$
$978$	0	0
$979$	4442.14	0.145017
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−19453.9	−0.631213	−0.315607	−	0.948890i	$-0.602208\pi$
$983$	−19453.9	−0.631213	−0.315607	+	0.948890i	$0.602208\pi$
$984$	0	0
$985$	9092.08	0.294109
$986$	0	0
$987$	0	0
$988$	0	0
$989$	30542.1	0.981983
$990$	0	0
$991$	−25557.5	−0.819234	−0.409617	−	0.912258i	$-0.634338\pi$
$991$	−25557.5	−0.819234	−0.409617	+	0.912258i	$0.634338\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−5955.20	−0.189741
$996$	0	0
$997$	−10058.6	−0.319519	−0.159760	−	0.987156i	$-0.551072\pi$
$997$	−10058.6	−0.319519	−0.159760	+	0.987156i	$0.551072\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.a.bk.1.1		3
3.2	odd	2		2160.4.a.bs.1.1		3
4.3	odd	2		1080.4.a.d.1.3	✓	3
12.11	even	2		1080.4.a.j.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.a.d.1.3	✓	3	4.3	odd	2
1080.4.a.j.1.3	yes	3	12.11	even	2
2160.4.a.bk.1.1		3	1.1	even	1	trivial
2160.4.a.bs.1.1		3	3.2	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:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2160.a (trivial)

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} -24.0794 q^{7} +2.95982 q^{11} -22.1759 q^{13} +76.0223 q^{17} +72.1187 q^{19} -176.003 q^{23} +25.0000 q^{25} +42.5134 q^{29} +327.010 q^{31} +120.397 q^{35} +182.089 q^{37} +154.624 q^{41} -173.531 q^{43} -338.059 q^{47} +236.820 q^{49} -26.5490 q^{53} -14.7991 q^{55} +391.670 q^{59} -191.191 q^{61} +110.879 q^{65} +507.583 q^{67} +576.547 q^{71} -390.440 q^{73} -71.2709 q^{77} -1220.43 q^{79} -247.332 q^{83} -380.111 q^{85} +1500.81 q^{89} +533.983 q^{91} -360.594 q^{95} +959.331 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 15 q^{5} + 10 q^{7} + 28 q^{11} - 78 q^{13} - 11 q^{17} + 71 q^{19} - 25 q^{23} + 75 q^{25} + 118 q^{29} + 107 q^{31} - 50 q^{35} - 410 q^{37} + 592 q^{41} - 52 q^{43} - 580 q^{47} + 479 q^{49} + 169 q^{53}+ \cdots - 840 q^{97}+O(q^{100}) \) 3 * q - 15 * q^5 + 10 * q^7 + 28 * q^11 - 78 * q^13 - 11 * q^17 + 71 * q^19 - 25 * q^23 + 75 * q^25 + 118 * q^29 + 107 * q^31 - 50 * q^35 - 410 * q^37 + 592 * q^41 - 52 * q^43 - 580 * q^47 + 479 * q^49 + 169 * q^53 - 140 * q^55 + 234 * q^59 - 673 * q^61 + 390 * q^65 - 386 * q^67 + 16 * q^71 - 892 * q^73 + 1800 * q^77 - 1263 * q^79 - 1815 * q^83 + 55 * q^85 + 1800 * q^89 - 1284 * q^91 - 355 * q^95 - 840 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more