LMFDB - Embedded newform 2160.4.a.bp.1.2

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 = [3,0,0,0,15,0,0,0,0,0,27] 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:	$0$
Dimension:	$3$
Coefficient field:	3.3.1765.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 11x + 16 $ x^3 - x^2 - 11*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 1080)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.89055$ 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)

$f(q)$	$=$	$q+5.00000 q^{5} +5.60585 q^{7} +53.5556 q^{11} +29.6866 q^{13} +109.192 q^{17} +67.6070 q^{19} -87.4550 q^{23} +25.0000 q^{25} -141.980 q^{29} +219.193 q^{31} +28.0293 q^{35} +436.534 q^{37} -68.7309 q^{41} -16.9311 q^{43} +566.322 q^{47} -311.574 q^{49} -95.0703 q^{53} +267.778 q^{55} +306.600 q^{59} -744.914 q^{61} +148.433 q^{65} -521.489 q^{67} -678.577 q^{71} -879.486 q^{73} +300.225 q^{77} +698.634 q^{79} -1191.92 q^{83} +545.959 q^{85} +840.998 q^{89} +166.419 q^{91} +338.035 q^{95} -1045.59 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 15 q^{5} + 27 q^{11} - 3 q^{13} - 15 q^{17} + 78 q^{19} + 105 q^{23} + 75 q^{25} - 117 q^{29} + 207 q^{31} - 120 q^{37} - 300 q^{41} + 483 q^{43} + 303 q^{47} - 15 q^{49} + 492 q^{53} + 135 q^{55}+ \cdots - 1392 q^{97}+O(q^{100}) $ 3 * q + 15 * q^5 + 27 * q^11 - 3 * q^13 - 15 * q^17 + 78 * q^19 + 105 * q^23 + 75 * q^25 - 117 * q^29 + 207 * q^31 - 120 * q^37 - 300 * q^41 + 483 * q^43 + 303 * q^47 - 15 * q^49 + 492 * q^53 + 135 * q^55 + 240 * q^59 - 444 * q^61 - 15 * q^65 + 522 * q^67 - 168 * q^71 - 876 * q^73 + 150 * q^77 + 2103 * q^79 - 42 * q^83 - 75 * q^85 + 2268 * q^89 + 1596 * q^91 + 390 * q^95 - 1392 * 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
$7$	5.60585	0.302688	0.151344	−	0.988481i	$-0.451640\pi$
$7$	5.60585	0.302688	0.151344	+	0.988481i	$0.451640\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	53.5556	1.46796	0.733982	−	0.679168i	$-0.237659\pi$
$11$	53.5556	1.46796	0.733982	+	0.679168i	$0.237659\pi$
$12$	0	0
$13$	29.6866	0.633352	0.316676	−	0.948534i	$-0.397433\pi$
$13$	29.6866	0.633352	0.316676	+	0.948534i	$0.397433\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	109.192	1.55782	0.778909	−	0.627137i	$-0.215773\pi$
$17$	109.192	1.55782	0.778909	+	0.627137i	$0.215773\pi$
$18$	0	0
$19$	67.6070	0.816322	0.408161	−	0.912910i	$-0.366170\pi$
$19$	67.6070	0.816322	0.408161	+	0.912910i	$0.366170\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−87.4550	−0.792854	−0.396427	−	0.918066i	$-0.629750\pi$
$23$	−87.4550	−0.792854	−0.396427	+	0.918066i	$0.629750\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−141.980	−0.909140	−0.454570	−	0.890711i	$-0.650207\pi$
$29$	−141.980	−0.909140	−0.454570	+	0.890711i	$0.650207\pi$
$30$	0	0
$31$	219.193	1.26994	0.634971	−	0.772536i	$-0.281012\pi$
$31$	219.193	1.26994	0.634971	+	0.772536i	$0.281012\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	28.0293	0.135366
$36$	0	0
$37$	436.534	1.93961	0.969806	−	0.243876i	$-0.0784190\pi$
$37$	436.534	1.93961	0.969806	+	0.243876i	$0.0784190\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−68.7309	−0.261804	−0.130902	−	0.991395i	$-0.541787\pi$
$41$	−68.7309	−0.261804	−0.130902	+	0.991395i	$0.541787\pi$
$42$	0	0
$43$	−16.9311	−0.0600456	−0.0300228	−	0.999549i	$-0.509558\pi$
$43$	−16.9311	−0.0600456	−0.0300228	+	0.999549i	$0.509558\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	566.322	1.75759	0.878793	−	0.477203i	$-0.158349\pi$
$47$	566.322	1.75759	0.878793	+	0.477203i	$0.158349\pi$
$48$	0	0
$49$	−311.574	−0.908380
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−95.0703	−0.246395	−0.123197	−	0.992382i	$-0.539315\pi$
$53$	−95.0703	−0.246395	−0.123197	+	0.992382i	$0.539315\pi$
$54$	0	0
$55$	267.778	0.656494
$56$	0	0
$57$	0	0
$58$	0	0
$59$	306.600	0.676541	0.338271	−	0.941049i	$-0.390158\pi$
$59$	306.600	0.676541	0.338271	+	0.941049i	$0.390158\pi$
$60$	0	0
$61$	−744.914	−1.56355	−0.781774	−	0.623562i	$-0.785685\pi$
$61$	−744.914	−1.56355	−0.781774	+	0.623562i	$0.785685\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	148.433	0.283243
$66$	0	0
$67$	−521.489	−0.950896	−0.475448	−	0.879744i	$-0.657714\pi$
$67$	−521.489	−0.950896	−0.475448	+	0.879744i	$0.657714\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−678.577	−1.13426	−0.567129	−	0.823629i	$-0.691946\pi$
$71$	−678.577	−1.13426	−0.567129	+	0.823629i	$0.691946\pi$
$72$	0	0
$73$	−879.486	−1.41008	−0.705041	−	0.709166i	$-0.749071\pi$
$73$	−879.486	−1.41008	−0.705041	+	0.709166i	$0.749071\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	300.225	0.444335
$78$	0	0
$79$	698.634	0.994968	0.497484	−	0.867473i	$-0.334257\pi$
$79$	698.634	0.994968	0.497484	+	0.867473i	$0.334257\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1191.92	−1.57627	−0.788134	−	0.615504i	$-0.788952\pi$
$83$	−1191.92	−1.57627	−0.788134	+	0.615504i	$0.788952\pi$
$84$	0	0
$85$	545.959	0.696678
$86$	0	0
$87$	0	0
$88$	0	0
$89$	840.998	1.00164	0.500818	−	0.865553i	$-0.333033\pi$
$89$	840.998	1.00164	0.500818	+	0.865553i	$0.333033\pi$
$90$	0	0
$91$	166.419	0.191708
$92$	0	0
$93$	0	0
$94$	0	0
$95$	338.035	0.365070
$96$	0	0
$97$	−1045.59	−1.09447	−0.547234	−	0.836980i	$-0.684319\pi$
$97$	−1045.59	−1.09447	−0.547234	+	0.836980i	$0.684319\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	754.304	0.743129	0.371565	−	0.928407i	$-0.378821\pi$
$101$	754.304	0.743129	0.371565	+	0.928407i	$0.378821\pi$
$102$	0	0
$103$	1807.06	1.72869	0.864346	−	0.502898i	$-0.167733\pi$
$103$	1807.06	1.72869	0.864346	+	0.502898i	$0.167733\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	657.832	0.594346	0.297173	−	0.954824i	$-0.403956\pi$
$107$	657.832	0.594346	0.297173	+	0.954824i	$0.403956\pi$
$108$	0	0
$109$	−981.839	−0.862781	−0.431390	−	0.902165i	$-0.641977\pi$
$109$	−981.839	−0.862781	−0.431390	+	0.902165i	$0.641977\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−850.655	−0.708167	−0.354084	−	0.935214i	$-0.615207\pi$
$113$	−850.655	−0.708167	−0.354084	+	0.935214i	$0.615207\pi$
$114$	0	0
$115$	−437.275	−0.354575
$116$	0	0
$117$	0	0
$118$	0	0
$119$	612.113	0.471532
$120$	0	0
$121$	1537.20	1.15492
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	226.985	0.158596	0.0792978	−	0.996851i	$-0.474732\pi$
$127$	226.985	0.158596	0.0792978	+	0.996851i	$0.474732\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1217.09	0.811735	0.405867	−	0.913932i	$-0.366969\pi$
$131$	1217.09	0.811735	0.405867	+	0.913932i	$0.366969\pi$
$132$	0	0
$133$	378.995	0.247090
$134$	0	0
$135$	0	0
$136$	0	0
$137$	509.696	0.317856	0.158928	−	0.987290i	$-0.449196\pi$
$137$	509.696	0.317856	0.158928	+	0.987290i	$0.449196\pi$
$138$	0	0
$139$	845.621	0.516005	0.258002	−	0.966144i	$-0.416936\pi$
$139$	845.621	0.516005	0.258002	+	0.966144i	$0.416936\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1589.88	0.929738
$144$	0	0
$145$	−709.901	−0.406580
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1036.04	0.569638	0.284819	−	0.958581i	$-0.408066\pi$
$149$	1036.04	0.569638	0.284819	+	0.958581i	$0.408066\pi$
$150$	0	0
$151$	1837.56	0.990321	0.495161	−	0.868801i	$-0.335109\pi$
$151$	1837.56	0.990321	0.495161	+	0.868801i	$0.335109\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1095.97	0.567936
$156$	0	0
$157$	−3080.72	−1.56604	−0.783020	−	0.621996i	$-0.786322\pi$
$157$	−3080.72	−1.56604	−0.783020	+	0.621996i	$0.786322\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−490.260	−0.239987
$162$	0	0
$163$	606.209	0.291300	0.145650	−	0.989336i	$-0.453473\pi$
$163$	606.209	0.291300	0.145650	+	0.989336i	$0.453473\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−250.692	−0.116162	−0.0580812	−	0.998312i	$-0.518498\pi$
$167$	−250.692	−0.116162	−0.0580812	+	0.998312i	$0.518498\pi$
$168$	0	0
$169$	−1315.71	−0.598866
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1264.31	−0.555629	−0.277815	−	0.960635i	$-0.589610\pi$
$173$	−1264.31	−0.555629	−0.277815	+	0.960635i	$0.589610\pi$
$174$	0	0
$175$	140.146	0.0605375
$176$	0	0
$177$	0	0
$178$	0	0
$179$	558.174	0.233072	0.116536	−	0.993186i	$-0.462821\pi$
$179$	558.174	0.233072	0.116536	+	0.993186i	$0.462821\pi$
$180$	0	0
$181$	−4377.85	−1.79781	−0.898904	−	0.438146i	$-0.855635\pi$
$181$	−4377.85	−1.79781	−0.898904	+	0.438146i	$0.855635\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2182.67	0.867421
$186$	0	0
$187$	5847.83	2.28682
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1983.68	−0.751489	−0.375744	−	0.926723i	$-0.622613\pi$
$191$	−1983.68	−0.751489	−0.375744	+	0.926723i	$0.622613\pi$
$192$	0	0
$193$	1092.75	0.407555	0.203778	−	0.979017i	$-0.434678\pi$
$193$	1092.75	0.407555	0.203778	+	0.979017i	$0.434678\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−917.082	−0.331672	−0.165836	−	0.986153i	$-0.553032\pi$
$197$	−917.082	−0.331672	−0.165836	+	0.986153i	$0.553032\pi$
$198$	0	0
$199$	−1672.55	−0.595797	−0.297899	−	0.954598i	$-0.596286\pi$
$199$	−1672.55	−0.595797	−0.297899	+	0.954598i	$0.596286\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−795.920	−0.275185
$204$	0	0
$205$	−343.655	−0.117082
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3620.73	1.19833
$210$	0	0
$211$	−4890.34	−1.59557	−0.797785	−	0.602942i	$-0.793995\pi$
$211$	−4890.34	−1.59557	−0.797785	+	0.602942i	$0.793995\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−84.6553	−0.0268532
$216$	0	0
$217$	1228.76	0.384396
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3241.53	0.986647
$222$	0	0
$223$	−25.4162	−0.00763227	−0.00381614	−	0.999993i	$-0.501215\pi$
$223$	−25.4162	−0.00763227	−0.00381614	+	0.999993i	$0.501215\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6185.05	1.80844	0.904220	−	0.427067i	$-0.140453\pi$
$227$	6185.05	1.80844	0.904220	+	0.427067i	$0.140453\pi$
$228$	0	0
$229$	−633.968	−0.182942	−0.0914712	−	0.995808i	$-0.529157\pi$
$229$	−633.968	−0.182942	−0.0914712	+	0.995808i	$0.529157\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5585.10	1.57035	0.785176	−	0.619273i	$-0.212573\pi$
$233$	5585.10	1.57035	0.785176	+	0.619273i	$0.212573\pi$
$234$	0	0
$235$	2831.61	0.786016
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2686.83	0.727181	0.363591	−	0.931559i	$-0.381551\pi$
$239$	2686.83	0.727181	0.363591	+	0.931559i	$0.381551\pi$
$240$	0	0
$241$	−1601.60	−0.428084	−0.214042	−	0.976824i	$-0.568663\pi$
$241$	−1601.60	−0.428084	−0.214042	+	0.976824i	$0.568663\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1557.87	−0.406240
$246$	0	0
$247$	2007.02	0.517019
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4955.28	1.24611	0.623057	−	0.782177i	$-0.285891\pi$
$251$	4955.28	1.24611	0.623057	+	0.782177i	$0.285891\pi$
$252$	0	0
$253$	−4683.70	−1.16388
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6161.72	−1.49556	−0.747778	−	0.663949i	$-0.768879\pi$
$257$	−6161.72	−1.49556	−0.747778	+	0.663949i	$0.768879\pi$
$258$	0	0
$259$	2447.14	0.587097
$260$	0	0
$261$	0	0
$262$	0	0
$263$	768.688	0.180226	0.0901128	−	0.995932i	$-0.471277\pi$
$263$	768.688	0.180226	0.0901128	+	0.995932i	$0.471277\pi$
$264$	0	0
$265$	−475.351	−0.110191
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7989.19	1.81082	0.905408	−	0.424542i	$-0.139565\pi$
$269$	7989.19	1.81082	0.905408	+	0.424542i	$0.139565\pi$
$270$	0	0
$271$	3875.39	0.868683	0.434341	−	0.900748i	$-0.356981\pi$
$271$	3875.39	0.868683	0.434341	+	0.900748i	$0.356981\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1338.89	0.293593
$276$	0	0
$277$	1121.71	0.243311	0.121656	−	0.992572i	$-0.461180\pi$
$277$	1121.71	0.243311	0.121656	+	0.992572i	$0.461180\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−5681.52	−1.20616	−0.603081	−	0.797680i	$-0.706060\pi$
$281$	−5681.52	−1.20616	−0.603081	+	0.797680i	$0.706060\pi$
$282$	0	0
$283$	−6366.11	−1.33719	−0.668597	−	0.743625i	$-0.733105\pi$
$283$	−6366.11	−1.33719	−0.668597	+	0.743625i	$0.733105\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−385.296	−0.0792449
$288$	0	0
$289$	7009.86	1.42680
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5316.63	1.06007	0.530035	−	0.847976i	$-0.322179\pi$
$293$	5316.63	1.06007	0.530035	+	0.847976i	$0.322179\pi$
$294$	0	0
$295$	1533.00	0.302558
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2596.24	−0.502155
$300$	0	0
$301$	−94.9130	−0.0181751
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3724.57	−0.699240
$306$	0	0
$307$	−2310.21	−0.429480	−0.214740	−	0.976671i	$-0.568890\pi$
$307$	−2310.21	−0.429480	−0.214740	+	0.976671i	$0.568890\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2874.98	0.524197	0.262099	−	0.965041i	$-0.415585\pi$
$311$	2874.98	0.524197	0.262099	+	0.965041i	$0.415585\pi$
$312$	0	0
$313$	−7587.20	−1.37014	−0.685070	−	0.728478i	$-0.740228\pi$
$313$	−7587.20	−1.37014	−0.685070	+	0.728478i	$0.740228\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7699.56	1.36420	0.682099	−	0.731260i	$-0.261067\pi$
$317$	7699.56	1.36420	0.682099	+	0.731260i	$0.261067\pi$
$318$	0	0
$319$	−7603.83	−1.33459
$320$	0	0
$321$	0	0
$322$	0	0
$323$	7382.14	1.27168
$324$	0	0
$325$	742.164	0.126670
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3174.72	0.531999
$330$	0	0
$331$	5469.00	0.908168	0.454084	−	0.890959i	$-0.349967\pi$
$331$	5469.00	0.908168	0.454084	+	0.890959i	$0.349967\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2607.45	−0.425254
$336$	0	0
$337$	6073.73	0.981772	0.490886	−	0.871224i	$-0.336673\pi$
$337$	6073.73	0.981772	0.490886	+	0.871224i	$0.336673\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	11739.0	1.86423
$342$	0	0
$343$	−3669.45	−0.577643
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6463.73	−0.999975	−0.499988	−	0.866033i	$-0.666662\pi$
$347$	−6463.73	−0.999975	−0.499988	+	0.866033i	$0.666662\pi$
$348$	0	0
$349$	−1630.31	−0.250053	−0.125026	−	0.992153i	$-0.539902\pi$
$349$	−1630.31	−0.250053	−0.125026	+	0.992153i	$0.539902\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8136.15	1.22675	0.613376	−	0.789791i	$-0.289811\pi$
$353$	8136.15	1.22675	0.613376	+	0.789791i	$0.289811\pi$
$354$	0	0
$355$	−3392.88	−0.507255
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8847.53	1.30071	0.650355	−	0.759631i	$-0.274620\pi$
$359$	8847.53	1.30071	0.650355	+	0.759631i	$0.274620\pi$
$360$	0	0
$361$	−2288.29	−0.333619
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4397.43	−0.630608
$366$	0	0
$367$	−6938.76	−0.986922	−0.493461	−	0.869768i	$-0.664268\pi$
$367$	−6938.76	−0.986922	−0.493461	+	0.869768i	$0.664268\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−532.950	−0.0745806
$372$	0	0
$373$	7950.83	1.10369	0.551847	−	0.833945i	$-0.313923\pi$
$373$	7950.83	1.10369	0.551847	+	0.833945i	$0.313923\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4214.90	−0.575805
$378$	0	0
$379$	11947.9	1.61932	0.809659	−	0.586901i	$-0.199652\pi$
$379$	11947.9	1.61932	0.809659	+	0.586901i	$0.199652\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1499.09	−0.199999	−0.0999997	−	0.994987i	$-0.531884\pi$
$383$	−1499.09	−0.199999	−0.0999997	+	0.994987i	$0.531884\pi$
$384$	0	0
$385$	1501.12	0.198713
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−708.852	−0.0923913	−0.0461956	−	0.998932i	$-0.514710\pi$
$389$	−708.852	−0.0923913	−0.0461956	+	0.998932i	$0.514710\pi$
$390$	0	0
$391$	−9549.38	−1.23512
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3493.17	0.444963
$396$	0	0
$397$	2541.17	0.321253	0.160626	−	0.987015i	$-0.448649\pi$
$397$	2541.17	0.321253	0.160626	+	0.987015i	$0.448649\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2135.53	0.265944	0.132972	−	0.991120i	$-0.457548\pi$
$401$	2135.53	0.265944	0.132972	+	0.991120i	$0.457548\pi$
$402$	0	0
$403$	6507.09	0.804321
$404$	0	0
$405$	0	0
$406$	0	0
$407$	23378.8	2.84728
$408$	0	0
$409$	7921.49	0.957684	0.478842	−	0.877901i	$-0.341057\pi$
$409$	7921.49	0.957684	0.478842	+	0.877901i	$0.341057\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1718.76	0.204781
$414$	0	0
$415$	−5959.60	−0.704928
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−13067.7	−1.52362	−0.761811	−	0.647799i	$-0.775690\pi$
$419$	−13067.7	−1.52362	−0.761811	+	0.647799i	$0.775690\pi$
$420$	0	0
$421$	14532.5	1.68235	0.841175	−	0.540762i	$-0.181864\pi$
$421$	14532.5	1.68235	0.841175	+	0.540762i	$0.181864\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2729.80	0.311564
$426$	0	0
$427$	−4175.88	−0.473267
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1363.94	0.152433	0.0762164	−	0.997091i	$-0.475716\pi$
$431$	1363.94	0.152433	0.0762164	+	0.997091i	$0.475716\pi$
$432$	0	0
$433$	10621.2	1.17880	0.589401	−	0.807841i	$-0.299364\pi$
$433$	10621.2	1.17880	0.589401	+	0.807841i	$0.299364\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5912.57	−0.647224
$438$	0	0
$439$	9505.12	1.03338	0.516691	−	0.856172i	$-0.327164\pi$
$439$	9505.12	1.03338	0.516691	+	0.856172i	$0.327164\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	377.897	0.0405292	0.0202646	−	0.999795i	$-0.493549\pi$
$443$	377.897	0.0405292	0.0202646	+	0.999795i	$0.493549\pi$
$444$	0	0
$445$	4204.99	0.447945
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1424.29	0.149703	0.0748513	−	0.997195i	$-0.476152\pi$
$449$	1424.29	0.149703	0.0748513	+	0.997195i	$0.476152\pi$
$450$	0	0
$451$	−3680.93	−0.384319
$452$	0	0
$453$	0	0
$454$	0	0
$455$	832.093	0.0857343
$456$	0	0
$457$	−10979.1	−1.12381	−0.561903	−	0.827203i	$-0.689931\pi$
$457$	−10979.1	−1.12381	−0.561903	+	0.827203i	$0.689931\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	10729.6	1.08401	0.542003	−	0.840377i	$-0.317666\pi$
$461$	10729.6	1.08401	0.542003	+	0.840377i	$0.317666\pi$
$462$	0	0
$463$	−1314.85	−0.131979	−0.0659895	−	0.997820i	$-0.521020\pi$
$463$	−1314.85	−0.131979	−0.0659895	+	0.997820i	$0.521020\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7772.33	0.770151	0.385075	−	0.922885i	$-0.374175\pi$
$467$	7772.33	0.770151	0.385075	+	0.922885i	$0.374175\pi$
$468$	0	0
$469$	−2923.39	−0.287824
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−906.753	−0.0881449
$474$	0	0
$475$	1690.18	0.163264
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−17266.5	−1.64702	−0.823512	−	0.567299i	$-0.807988\pi$
$479$	−17266.5	−1.64702	−0.823512	+	0.567299i	$0.807988\pi$
$480$	0	0
$481$	12959.2	1.22846
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5227.94	−0.489461
$486$	0	0
$487$	123.244	0.0114676	0.00573380	−	0.999984i	$-0.498175\pi$
$487$	123.244	0.0114676	0.00573380	+	0.999984i	$0.498175\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	13322.6	1.22453	0.612263	−	0.790654i	$-0.290259\pi$
$491$	13322.6	1.22453	0.612263	+	0.790654i	$0.290259\pi$
$492$	0	0
$493$	−15503.1	−1.41627
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3804.00	−0.343326
$498$	0	0
$499$	−5106.10	−0.458077	−0.229039	−	0.973417i	$-0.573558\pi$
$499$	−5106.10	−0.458077	−0.229039	+	0.973417i	$0.573558\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−856.862	−0.0759554	−0.0379777	−	0.999279i	$-0.512092\pi$
$503$	−856.862	−0.0759554	−0.0379777	+	0.999279i	$0.512092\pi$
$504$	0	0
$505$	3771.52	0.332338
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−568.382	−0.0494953	−0.0247476	−	0.999694i	$-0.507878\pi$
$509$	−568.382	−0.0494953	−0.0247476	+	0.999694i	$0.507878\pi$
$510$	0	0
$511$	−4930.27	−0.426814
$512$	0	0
$513$	0	0
$514$	0	0
$515$	9035.32	0.773095
$516$	0	0
$517$	30329.7	2.58007
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−3341.28	−0.280968	−0.140484	−	0.990083i	$-0.544866\pi$
$521$	−3341.28	−0.280968	−0.140484	+	0.990083i	$0.544866\pi$
$522$	0	0
$523$	−20342.2	−1.70077	−0.850383	−	0.526165i	$-0.823630\pi$
$523$	−20342.2	−1.70077	−0.850383	+	0.526165i	$0.823630\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	23934.1	1.97834
$528$	0	0
$529$	−4518.62	−0.371383
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2040.39	−0.165814
$534$	0	0
$535$	3289.16	0.265800
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−16686.5	−1.33347
$540$	0	0
$541$	5490.90	0.436363	0.218181	−	0.975908i	$-0.429988\pi$
$541$	5490.90	0.436363	0.218181	+	0.975908i	$0.429988\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−4909.19	−0.385847
$546$	0	0
$547$	−13539.2	−1.05831	−0.529153	−	0.848527i	$-0.677490\pi$
$547$	−13539.2	−1.05831	−0.529153	+	0.848527i	$0.677490\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−9598.86	−0.742151
$552$	0	0
$553$	3916.44	0.301164
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9817.92	0.746856	0.373428	−	0.927659i	$-0.378182\pi$
$557$	9817.92	0.746856	0.373428	+	0.927659i	$0.378182\pi$
$558$	0	0
$559$	−502.625	−0.0380300
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−8971.81	−0.671610	−0.335805	−	0.941931i	$-0.609008\pi$
$563$	−8971.81	−0.671610	−0.335805	+	0.941931i	$0.609008\pi$
$564$	0	0
$565$	−4253.28	−0.316702
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−13876.7	−1.02239	−0.511195	−	0.859464i	$-0.670797\pi$
$569$	−13876.7	−1.02239	−0.511195	+	0.859464i	$0.670797\pi$
$570$	0	0
$571$	3513.33	0.257492	0.128746	−	0.991678i	$-0.458905\pi$
$571$	3513.33	0.257492	0.128746	+	0.991678i	$0.458905\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2186.38	−0.158571
$576$	0	0
$577$	−17636.6	−1.27248	−0.636241	−	0.771491i	$-0.719511\pi$
$577$	−17636.6	−1.27248	−0.636241	+	0.771491i	$0.719511\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−6681.73	−0.477117
$582$	0	0
$583$	−5091.54	−0.361698
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1129.54	−0.0794229	−0.0397114	−	0.999211i	$-0.512644\pi$
$587$	−1129.54	−0.0794229	−0.0397114	+	0.999211i	$0.512644\pi$
$588$	0	0
$589$	14819.0	1.03668
$590$	0	0
$591$	0	0
$592$	0	0
$593$	25431.4	1.76111	0.880557	−	0.473940i	$-0.157169\pi$
$593$	25431.4	1.76111	0.880557	+	0.473940i	$0.157169\pi$
$594$	0	0
$595$	3060.57	0.210876
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−24719.7	−1.68618	−0.843089	−	0.537775i	$-0.819265\pi$
$599$	−24719.7	−1.68618	−0.843089	+	0.537775i	$0.819265\pi$
$600$	0	0
$601$	−24379.3	−1.65466	−0.827331	−	0.561714i	$-0.810142\pi$
$601$	−24379.3	−1.65466	−0.827331	+	0.561714i	$0.810142\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	7686.00	0.516496
$606$	0	0
$607$	−16258.8	−1.08719	−0.543596	−	0.839347i	$-0.682937\pi$
$607$	−16258.8	−1.08719	−0.543596	+	0.839347i	$0.682937\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	16812.2	1.11317
$612$	0	0
$613$	−13069.4	−0.861124	−0.430562	−	0.902561i	$-0.641685\pi$
$613$	−13069.4	−0.861124	−0.430562	+	0.902561i	$0.641685\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−20592.5	−1.34363	−0.671816	−	0.740718i	$-0.734485\pi$
$617$	−20592.5	−1.34363	−0.671816	+	0.740718i	$0.734485\pi$
$618$	0	0
$619$	−24687.2	−1.60301	−0.801504	−	0.597989i	$-0.795967\pi$
$619$	−24687.2	−1.60301	−0.801504	+	0.597989i	$0.795967\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4714.51	0.303183
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	47665.9	3.02157
$630$	0	0
$631$	−18847.2	−1.18906	−0.594529	−	0.804074i	$-0.702662\pi$
$631$	−18847.2	−1.18906	−0.594529	+	0.804074i	$0.702662\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1134.92	0.0709261
$636$	0	0
$637$	−9249.57	−0.575324
$638$	0	0
$639$	0	0
$640$	0	0
$641$	3418.42	0.210639	0.105319	−	0.994438i	$-0.466414\pi$
$641$	3418.42	0.210639	0.105319	+	0.994438i	$0.466414\pi$
$642$	0	0
$643$	−20369.8	−1.24931	−0.624656	−	0.780900i	$-0.714761\pi$
$643$	−20369.8	−1.24931	−0.624656	+	0.780900i	$0.714761\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15267.3	0.927699	0.463849	−	0.885914i	$-0.346468\pi$
$647$	15267.3	0.927699	0.463849	+	0.885914i	$0.346468\pi$
$648$	0	0
$649$	16420.1	0.993139
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10839.6	0.649595	0.324797	−	0.945784i	$-0.394704\pi$
$653$	10839.6	0.649595	0.324797	+	0.945784i	$0.394704\pi$
$654$	0	0
$655$	6085.43	0.363019
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1589.00	−0.0939282	−0.0469641	−	0.998897i	$-0.514955\pi$
$659$	−1589.00	−0.0939282	−0.0469641	+	0.998897i	$0.514955\pi$
$660$	0	0
$661$	−15061.5	−0.886272	−0.443136	−	0.896454i	$-0.646134\pi$
$661$	−15061.5	−0.886272	−0.443136	+	0.896454i	$0.646134\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1894.98	0.110502
$666$	0	0
$667$	12416.9	0.720815
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−39894.3	−2.29523
$672$	0	0
$673$	18477.3	1.05832	0.529159	−	0.848523i	$-0.322508\pi$
$673$	18477.3	1.05832	0.529159	+	0.848523i	$0.322508\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−28782.1	−1.63395	−0.816975	−	0.576673i	$-0.804351\pi$
$677$	−28782.1	−1.63395	−0.816975	+	0.576673i	$0.804351\pi$
$678$	0	0
$679$	−5861.42	−0.331282
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1542.58	−0.0864205	−0.0432103	−	0.999066i	$-0.513759\pi$
$683$	−1542.58	−0.0864205	−0.0432103	+	0.999066i	$0.513759\pi$
$684$	0	0
$685$	2548.48	0.142149
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2822.31	−0.156054
$690$	0	0
$691$	8037.73	0.442503	0.221252	−	0.975217i	$-0.428986\pi$
$691$	8037.73	0.442503	0.221252	+	0.975217i	$0.428986\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4228.11	0.230764
$696$	0	0
$697$	−7504.86	−0.407843
$698$	0	0
$699$	0	0
$700$	0	0
$701$	3686.72	0.198639	0.0993193	−	0.995056i	$-0.468333\pi$
$701$	3686.72	0.198639	0.0993193	+	0.995056i	$0.468333\pi$
$702$	0	0
$703$	29512.7	1.58335
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4228.52	0.224936
$708$	0	0
$709$	20887.3	1.10640	0.553200	−	0.833049i	$-0.313407\pi$
$709$	20887.3	1.10640	0.553200	+	0.833049i	$0.313407\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−19169.5	−1.00688
$714$	0	0
$715$	7949.41	0.415791
$716$	0	0
$717$	0	0
$718$	0	0
$719$	30381.8	1.57587	0.787935	−	0.615758i	$-0.211150\pi$
$719$	30381.8	1.57587	0.787935	+	0.615758i	$0.211150\pi$
$720$	0	0
$721$	10130.1	0.523254
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−3549.50	−0.181828
$726$	0	0
$727$	36256.3	1.84962	0.924810	−	0.380430i	$-0.124224\pi$
$727$	36256.3	1.84962	0.924810	+	0.380430i	$0.124224\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1848.73	−0.0935402
$732$	0	0
$733$	206.568	0.0104089	0.00520447	−	0.999986i	$-0.498343\pi$
$733$	206.568	0.0104089	0.00520447	+	0.999986i	$0.498343\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−27928.7	−1.39588
$738$	0	0
$739$	28859.8	1.43657	0.718285	−	0.695749i	$-0.244927\pi$
$739$	28859.8	1.43657	0.718285	+	0.695749i	$0.244927\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−9452.34	−0.466720	−0.233360	−	0.972390i	$-0.574972\pi$
$743$	−9452.34	−0.466720	−0.233360	+	0.972390i	$0.574972\pi$
$744$	0	0
$745$	5180.22	0.254750
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3687.71	0.179901
$750$	0	0
$751$	13833.5	0.672160	0.336080	−	0.941833i	$-0.390899\pi$
$751$	13833.5	0.672160	0.336080	+	0.941833i	$0.390899\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	9187.80	0.442885
$756$	0	0
$757$	−4267.28	−0.204884	−0.102442	−	0.994739i	$-0.532666\pi$
$757$	−4267.28	−0.204884	−0.102442	+	0.994739i	$0.532666\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−21502.5	−1.02426	−0.512131	−	0.858907i	$-0.671144\pi$
$761$	−21502.5	−1.02426	−0.512131	+	0.858907i	$0.671144\pi$
$762$	0	0
$763$	−5504.04	−0.261153
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9101.91	0.428489
$768$	0	0
$769$	17936.2	0.841086	0.420543	−	0.907273i	$-0.361840\pi$
$769$	17936.2	0.841086	0.420543	+	0.907273i	$0.361840\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−13672.3	−0.636167	−0.318083	−	0.948063i	$-0.603039\pi$
$773$	−13672.3	−0.636167	−0.318083	+	0.948063i	$0.603039\pi$
$774$	0	0
$775$	5479.83	0.253989
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4646.69	−0.213716
$780$	0	0
$781$	−36341.6	−1.66505
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−15403.6	−0.700354
$786$	0	0
$787$	−14681.1	−0.664963	−0.332482	−	0.943110i	$-0.607886\pi$
$787$	−14681.1	−0.664963	−0.332482	+	0.943110i	$0.607886\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−4768.65	−0.214353
$792$	0	0
$793$	−22113.9	−0.990276
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−11761.8	−0.522740	−0.261370	−	0.965239i	$-0.584174\pi$
$797$	−11761.8	−0.522740	−0.261370	+	0.965239i	$0.584174\pi$
$798$	0	0
$799$	61837.7	2.73800
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−47101.4	−2.06995
$804$	0	0
$805$	−2451.30	−0.107325
$806$	0	0
$807$	0	0
$808$	0	0
$809$	18821.1	0.817941	0.408971	−	0.912548i	$-0.365888\pi$
$809$	18821.1	0.817941	0.408971	+	0.912548i	$0.365888\pi$
$810$	0	0
$811$	−13804.6	−0.597713	−0.298856	−	0.954298i	$-0.596605\pi$
$811$	−13804.6	−0.597713	−0.298856	+	0.954298i	$0.596605\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3031.05	0.130274
$816$	0	0
$817$	−1144.66	−0.0490166
$818$	0	0
$819$	0	0
$820$	0	0
$821$	21364.1	0.908176	0.454088	−	0.890957i	$-0.349965\pi$
$821$	21364.1	0.908176	0.454088	+	0.890957i	$0.349965\pi$
$822$	0	0
$823$	−30739.7	−1.30197	−0.650983	−	0.759092i	$-0.725643\pi$
$823$	−30739.7	−1.30197	−0.650983	+	0.759092i	$0.725643\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3820.31	0.160635	0.0803176	−	0.996769i	$-0.474407\pi$
$827$	3820.31	0.160635	0.0803176	+	0.996769i	$0.474407\pi$
$828$	0	0
$829$	22902.8	0.959526	0.479763	−	0.877398i	$-0.340723\pi$
$829$	22902.8	0.959526	0.479763	+	0.877398i	$0.340723\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−34021.4	−1.41509
$834$	0	0
$835$	−1253.46	−0.0519494
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−32660.2	−1.34393	−0.671963	−	0.740585i	$-0.734549\pi$
$839$	−32660.2	−1.34393	−0.671963	+	0.740585i	$0.734549\pi$
$840$	0	0
$841$	−4230.63	−0.173465
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−6578.54	−0.267821
$846$	0	0
$847$	8617.31	0.349580
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−38177.1	−1.53783
$852$	0	0
$853$	3474.35	0.139460	0.0697301	−	0.997566i	$-0.477786\pi$
$853$	3474.35	0.139460	0.0697301	+	0.997566i	$0.477786\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−36755.3	−1.46504	−0.732519	−	0.680746i	$-0.761656\pi$
$857$	−36755.3	−1.46504	−0.732519	+	0.680746i	$0.761656\pi$
$858$	0	0
$859$	20322.2	0.807201	0.403600	−	0.914935i	$-0.367759\pi$
$859$	20322.2	0.807201	0.403600	+	0.914935i	$0.367759\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−8541.33	−0.336907	−0.168453	−	0.985710i	$-0.553877\pi$
$863$	−8541.33	−0.336907	−0.168453	+	0.985710i	$0.553877\pi$
$864$	0	0
$865$	−6321.56	−0.248485
$866$	0	0
$867$	0	0
$868$	0	0
$869$	37415.7	1.46058
$870$	0	0
$871$	−15481.2	−0.602252
$872$	0	0
$873$	0	0
$874$	0	0
$875$	700.732	0.0270732
$876$	0	0
$877$	−30963.2	−1.19219	−0.596096	−	0.802913i	$-0.703282\pi$
$877$	−30963.2	−1.19219	−0.596096	+	0.802913i	$0.703282\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−25296.4	−0.967376	−0.483688	−	0.875241i	$-0.660703\pi$
$881$	−25296.4	−0.967376	−0.483688	+	0.875241i	$0.660703\pi$
$882$	0	0
$883$	20261.1	0.772187	0.386094	−	0.922460i	$-0.373824\pi$
$883$	20261.1	0.772187	0.386094	+	0.922460i	$0.373824\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	6869.30	0.260032	0.130016	−	0.991512i	$-0.458497\pi$
$887$	6869.30	0.260032	0.130016	+	0.991512i	$0.458497\pi$
$888$	0	0
$889$	1272.44	0.0480049
$890$	0	0
$891$	0	0
$892$	0	0
$893$	38287.3	1.43476
$894$	0	0
$895$	2790.87	0.104233
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−31121.1	−1.15456
$900$	0	0
$901$	−10380.9	−0.383838
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−21889.3	−0.804004
$906$	0	0
$907$	−11567.2	−0.423465	−0.211733	−	0.977328i	$-0.567911\pi$
$907$	−11567.2	−0.423465	−0.211733	+	0.977328i	$0.567911\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−27971.1	−1.01726	−0.508631	−	0.860985i	$-0.669848\pi$
$911$	−27971.1	−1.01726	−0.508631	+	0.860985i	$0.669848\pi$
$912$	0	0
$913$	−63834.0	−2.31391
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6822.80	0.245702
$918$	0	0
$919$	254.867	0.00914828	0.00457414	−	0.999990i	$-0.498544\pi$
$919$	254.867	0.00914828	0.00457414	+	0.999990i	$0.498544\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−20144.6	−0.718384
$924$	0	0
$925$	10913.3	0.387923
$926$	0	0
$927$	0	0
$928$	0	0
$929$	8347.83	0.294815	0.147408	−	0.989076i	$-0.452907\pi$
$929$	8347.83	0.294815	0.147408	+	0.989076i	$0.452907\pi$
$930$	0	0
$931$	−21064.6	−0.741531
$932$	0	0
$933$	0	0
$934$	0	0
$935$	29239.2	1.02270
$936$	0	0
$937$	4128.98	0.143957	0.0719786	−	0.997406i	$-0.477069\pi$
$937$	4128.98	0.143957	0.0719786	+	0.997406i	$0.477069\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−43760.0	−1.51598	−0.757988	−	0.652268i	$-0.773818\pi$
$941$	−43760.0	−1.51598	−0.757988	+	0.652268i	$0.773818\pi$
$942$	0	0
$943$	6010.87	0.207572
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8035.72	0.275740	0.137870	−	0.990450i	$-0.455974\pi$
$947$	8035.72	0.275740	0.137870	+	0.990450i	$0.455974\pi$
$948$	0	0
$949$	−26108.9	−0.893078
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−10733.9	−0.364854	−0.182427	−	0.983219i	$-0.558395\pi$
$953$	−10733.9	−0.364854	−0.182427	+	0.983219i	$0.558395\pi$
$954$	0	0
$955$	−9918.42	−0.336076
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2857.28	0.0962110
$960$	0	0
$961$	18254.6	0.612755
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5463.77	0.182264
$966$	0	0
$967$	5800.85	0.192909	0.0964543	−	0.995337i	$-0.469250\pi$
$967$	5800.85	0.192909	0.0964543	+	0.995337i	$0.469250\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	47046.8	1.55490	0.777448	−	0.628947i	$-0.216514\pi$
$971$	47046.8	1.55490	0.777448	+	0.628947i	$0.216514\pi$
$972$	0	0
$973$	4740.43	0.156188
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−32043.6	−1.04930	−0.524649	−	0.851318i	$-0.675804\pi$
$977$	−32043.6	−1.04930	−0.524649	+	0.851318i	$0.675804\pi$
$978$	0	0
$979$	45040.1	1.47037
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−25178.5	−0.816957	−0.408478	−	0.912768i	$-0.633940\pi$
$983$	−25178.5	−0.816957	−0.408478	+	0.912768i	$0.633940\pi$
$984$	0	0
$985$	−4585.41	−0.148328
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1480.71	0.0476074
$990$	0	0
$991$	−18011.8	−0.577360	−0.288680	−	0.957426i	$-0.593216\pi$
$991$	−18011.8	−0.577360	−0.288680	+	0.957426i	$0.593216\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−8362.73	−0.266449
$996$	0	0
$997$	−31687.1	−1.00656	−0.503281	−	0.864123i	$-0.667874\pi$
$997$	−31687.1	−1.00656	−0.503281	+	0.864123i	$0.667874\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.a.bp.1.2		3
3.2	odd	2		2160.4.a.bh.1.2		3
4.3	odd	2		1080.4.a.l.1.2	yes	3
12.11	even	2		1080.4.a.f.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.a.f.1.2	✓	3	12.11	even	2
1080.4.a.l.1.2	yes	3	4.3	odd	2
2160.4.a.bh.1.2		3	3.2	odd	2
2160.4.a.bp.1.2		3	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+5.00000 q^{5} +5.60585 q^{7} +53.5556 q^{11} +29.6866 q^{13} +109.192 q^{17} +67.6070 q^{19} -87.4550 q^{23} +25.0000 q^{25} -141.980 q^{29} +219.193 q^{31} +28.0293 q^{35} +436.534 q^{37} -68.7309 q^{41} -16.9311 q^{43} +566.322 q^{47} -311.574 q^{49} -95.0703 q^{53} +267.778 q^{55} +306.600 q^{59} -744.914 q^{61} +148.433 q^{65} -521.489 q^{67} -678.577 q^{71} -879.486 q^{73} +300.225 q^{77} +698.634 q^{79} -1191.92 q^{83} +545.959 q^{85} +840.998 q^{89} +166.419 q^{91} +338.035 q^{95} -1045.59 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 15 q^{5} + 27 q^{11} - 3 q^{13} - 15 q^{17} + 78 q^{19} + 105 q^{23} + 75 q^{25} - 117 q^{29} + 207 q^{31} - 120 q^{37} - 300 q^{41} + 483 q^{43} + 303 q^{47} - 15 q^{49} + 492 q^{53} + 135 q^{55}+ \cdots - 1392 q^{97}+O(q^{100}) \) 3 * q + 15 * q^5 + 27 * q^11 - 3 * q^13 - 15 * q^17 + 78 * q^19 + 105 * q^23 + 75 * q^25 - 117 * q^29 + 207 * q^31 - 120 * q^37 - 300 * q^41 + 483 * q^43 + 303 * q^47 - 15 * q^49 + 492 * q^53 + 135 * q^55 + 240 * q^59 - 444 * q^61 - 15 * q^65 + 522 * q^67 - 168 * q^71 - 876 * q^73 + 150 * q^77 + 2103 * q^79 - 42 * q^83 - 75 * q^85 + 2268 * q^89 + 1596 * q^91 + 390 * q^95 - 1392 * q^97

Introduction

L-functions

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