LMFDB - Embedded newform 2160.4.a.bt.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,4,Mod(1,2160)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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

Self dual:	yes
Analytic conductor:	$127.444125612$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.697.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 7x - 5 $ x^3 - 7*x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\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			$2.94883$ 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)

$f(q)$	$=$	$q+5.00000 q^{5} +14.4806 q^{7} +O(q^{10})$	$q+5.00000 q^{5} +14.4806 q^{7} +13.9443 q^{11} -38.8276 q^{13} +126.063 q^{17} -60.5995 q^{19} +50.3522 q^{23} +25.0000 q^{25} +71.3302 q^{29} +28.7087 q^{31} +72.4028 q^{35} +76.6164 q^{37} -396.151 q^{41} +351.146 q^{43} -4.16809 q^{47} -133.313 q^{49} -171.983 q^{53} +69.7213 q^{55} +116.634 q^{59} +523.955 q^{61} -194.138 q^{65} +872.421 q^{67} +802.371 q^{71} -13.5264 q^{73} +201.921 q^{77} -597.959 q^{79} +75.9439 q^{83} +630.316 q^{85} -799.744 q^{89} -562.246 q^{91} -302.998 q^{95} +445.123 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 15 q^{5} + 24 q^{7}+O(q^{10}) $ 3 * q + 15 * q^5 + 24 * q^7	$ 3 q + 15 q^{5} + 24 q^{7} - 6 q^{11} + 48 q^{13} - 27 q^{17} + 195 q^{19} + 27 q^{23} + 75 q^{25} + 60 q^{29} + 279 q^{31} + 120 q^{35} - 138 q^{37} + 66 q^{41} - 222 q^{43} + 264 q^{47} - 237 q^{49} - 507 q^{53} - 30 q^{55} + 960 q^{59} + 543 q^{61} + 240 q^{65} + 1086 q^{67} + 1818 q^{71} + 1362 q^{73} - 1776 q^{77} + 129 q^{79} + 1569 q^{83} - 135 q^{85} - 1770 q^{89} - 1488 q^{91} + 975 q^{95} - 336 q^{97}+O(q^{100}) $ 3 * q + 15 * q^5 + 24 * q^7 - 6 * q^11 + 48 * q^13 - 27 * q^17 + 195 * q^19 + 27 * q^23 + 75 * q^25 + 60 * q^29 + 279 * q^31 + 120 * q^35 - 138 * q^37 + 66 * q^41 - 222 * q^43 + 264 * q^47 - 237 * q^49 - 507 * q^53 - 30 * q^55 + 960 * q^59 + 543 * q^61 + 240 * q^65 + 1086 * q^67 + 1818 * q^71 + 1362 * q^73 - 1776 * q^77 + 129 * q^79 + 1569 * q^83 - 135 * q^85 - 1770 * q^89 - 1488 * q^91 + 975 * q^95 - 336 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	14.4806	0.781877	0.390938	−	0.920417i	$-0.372151\pi$
$7$	14.4806	0.781877	0.390938	+	0.920417i	$0.372151\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	13.9443	0.382214	0.191107	−	0.981569i	$-0.438792\pi$
$11$	13.9443	0.382214	0.191107	+	0.981569i	$0.438792\pi$
$12$	0	0
$13$	−38.8276	−0.828373	−0.414186	−	0.910192i	$-0.635934\pi$
$13$	−38.8276	−0.828373	−0.414186	+	0.910192i	$0.635934\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	126.063	1.79852	0.899259	−	0.437416i	$-0.144106\pi$
$17$	126.063	1.79852	0.899259	+	0.437416i	$0.144106\pi$
$18$	0	0
$19$	−60.5995	−0.731709	−0.365855	−	0.930672i	$-0.619223\pi$
$19$	−60.5995	−0.731709	−0.365855	+	0.930672i	$0.619223\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	50.3522	0.456485	0.228243	−	0.973604i	$-0.426702\pi$
$23$	50.3522	0.456485	0.228243	+	0.973604i	$0.426702\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	71.3302	0.456748	0.228374	−	0.973574i	$-0.426659\pi$
$29$	71.3302	0.456748	0.228374	+	0.973574i	$0.426659\pi$
$30$	0	0
$31$	28.7087	0.166330	0.0831650	−	0.996536i	$-0.473497\pi$
$31$	28.7087	0.166330	0.0831650	+	0.996536i	$0.473497\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	72.4028	0.349666
$36$	0	0
$37$	76.6164	0.340423	0.170212	−	0.985408i	$-0.445555\pi$
$37$	76.6164	0.340423	0.170212	+	0.985408i	$0.445555\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−396.151	−1.50898	−0.754492	−	0.656309i	$-0.772117\pi$
$41$	−396.151	−1.50898	−0.754492	+	0.656309i	$0.772117\pi$
$42$	0	0
$43$	351.146	1.24533	0.622665	−	0.782488i	$-0.286050\pi$
$43$	351.146	1.24533	0.622665	+	0.782488i	$0.286050\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.16809	−0.0129357	−0.00646786	−	0.999979i	$-0.502059\pi$
$47$	−4.16809	−0.0129357	−0.00646786	+	0.999979i	$0.502059\pi$
$48$	0	0
$49$	−133.313	−0.388669
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−171.983	−0.445730	−0.222865	−	0.974849i	$-0.571541\pi$
$53$	−171.983	−0.445730	−0.222865	+	0.974849i	$0.571541\pi$
$54$	0	0
$55$	69.7213	0.170931
$56$	0	0
$57$	0	0
$58$	0	0
$59$	116.634	0.257363	0.128682	−	0.991686i	$-0.458926\pi$
$59$	116.634	0.257363	0.128682	+	0.991686i	$0.458926\pi$
$60$	0	0
$61$	523.955	1.09976	0.549882	−	0.835242i	$-0.314673\pi$
$61$	523.955	1.09976	0.549882	+	0.835242i	$0.314673\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−194.138	−0.370460
$66$	0	0
$67$	872.421	1.59079	0.795397	−	0.606088i	$-0.207262\pi$
$67$	872.421	1.59079	0.795397	+	0.606088i	$0.207262\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	802.371	1.34118	0.670591	−	0.741827i	$-0.266040\pi$
$71$	802.371	1.34118	0.670591	+	0.741827i	$0.266040\pi$
$72$	0	0
$73$	−13.5264	−0.0216869	−0.0108435	−	0.999941i	$-0.503452\pi$
$73$	−13.5264	−0.0216869	−0.0108435	+	0.999941i	$0.503452\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	201.921	0.298844
$78$	0	0
$79$	−597.959	−0.851590	−0.425795	−	0.904820i	$-0.640006\pi$
$79$	−597.959	−0.851590	−0.425795	+	0.904820i	$0.640006\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	75.9439	0.100433	0.0502164	−	0.998738i	$-0.484009\pi$
$83$	75.9439	0.100433	0.0502164	+	0.998738i	$0.484009\pi$
$84$	0	0
$85$	630.316	0.804322
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−799.744	−0.952502	−0.476251	−	0.879309i	$-0.658005\pi$
$89$	−799.744	−0.952502	−0.476251	+	0.879309i	$0.658005\pi$
$90$	0	0
$91$	−562.246	−0.647685
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−302.998	−0.327230
$96$	0	0
$97$	445.123	0.465932	0.232966	−	0.972485i	$-0.425157\pi$
$97$	445.123	0.465932	0.232966	+	0.972485i	$0.425157\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	202.087	0.199093	0.0995466	−	0.995033i	$-0.468261\pi$
$101$	202.087	0.199093	0.0995466	+	0.995033i	$0.468261\pi$
$102$	0	0
$103$	−782.676	−0.748732	−0.374366	−	0.927281i	$-0.622140\pi$
$103$	−782.676	−0.748732	−0.374366	+	0.927281i	$0.622140\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1886.42	1.70437	0.852184	−	0.523242i	$-0.175278\pi$
$107$	1886.42	1.70437	0.852184	+	0.523242i	$0.175278\pi$
$108$	0	0
$109$	604.673	0.531351	0.265675	−	0.964063i	$-0.414405\pi$
$109$	604.673	0.531351	0.265675	+	0.964063i	$0.414405\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1585.09	−1.31958	−0.659792	−	0.751448i	$-0.729356\pi$
$113$	−1585.09	−1.31958	−0.659792	+	0.751448i	$0.729356\pi$
$114$	0	0
$115$	251.761	0.204146
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1825.47	1.40622
$120$	0	0
$121$	−1136.56	−0.853913
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	1464.36	1.02316	0.511579	−	0.859236i	$-0.329061\pi$
$127$	1464.36	1.02316	0.511579	+	0.859236i	$0.329061\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−420.667	−0.280564	−0.140282	−	0.990112i	$-0.544801\pi$
$131$	−420.667	−0.280564	−0.140282	+	0.990112i	$0.544801\pi$
$132$	0	0
$133$	−877.515	−0.572107
$134$	0	0
$135$	0	0
$136$	0	0
$137$	154.631	0.0964305	0.0482152	−	0.998837i	$-0.484647\pi$
$137$	154.631	0.0964305	0.0482152	+	0.998837i	$0.484647\pi$
$138$	0	0
$139$	1838.83	1.12207	0.561033	−	0.827793i	$-0.310404\pi$
$139$	1838.83	1.12207	0.561033	+	0.827793i	$0.310404\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−541.422	−0.316615
$144$	0	0
$145$	356.651	0.204264
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1647.61	−0.905887	−0.452943	−	0.891539i	$-0.649626\pi$
$149$	−1647.61	−0.905887	−0.452943	+	0.891539i	$0.649626\pi$
$150$	0	0
$151$	−1023.34	−0.551509	−0.275754	−	0.961228i	$-0.588928\pi$
$151$	−1023.34	−0.551509	−0.275754	+	0.961228i	$0.588928\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	143.543	0.0743851
$156$	0	0
$157$	−880.210	−0.447442	−0.223721	−	0.974653i	$-0.571820\pi$
$157$	−880.210	−0.447442	−0.223721	+	0.974653i	$0.571820\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	729.128	0.356915
$162$	0	0
$163$	−1346.77	−0.647159	−0.323580	−	0.946201i	$-0.604886\pi$
$163$	−1346.77	−0.647159	−0.323580	+	0.946201i	$0.604886\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2579.87	1.19543	0.597714	−	0.801709i	$-0.296076\pi$
$167$	2579.87	1.19543	0.597714	+	0.801709i	$0.296076\pi$
$168$	0	0
$169$	−689.415	−0.313799
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2877.71	−1.26467	−0.632337	−	0.774694i	$-0.717904\pi$
$173$	−2877.71	−1.26467	−0.632337	+	0.774694i	$0.717904\pi$
$174$	0	0
$175$	362.014	0.156375
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1906.79	−0.796204	−0.398102	−	0.917341i	$-0.630331\pi$
$179$	−1906.79	−0.796204	−0.398102	+	0.917341i	$0.630331\pi$
$180$	0	0
$181$	4841.10	1.98805	0.994023	−	0.109173i	$-0.0348201\pi$
$181$	4841.10	1.98805	0.994023	+	0.109173i	$0.0348201\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	383.082	0.152242
$186$	0	0
$187$	1757.86	0.687418
$188$	0	0
$189$	0	0
$190$	0	0
$191$	104.064	0.0394230	0.0197115	−	0.999806i	$-0.493725\pi$
$191$	104.064	0.0394230	0.0197115	+	0.999806i	$0.493725\pi$
$192$	0	0
$193$	3903.96	1.45603	0.728013	−	0.685563i	$-0.240444\pi$
$193$	3903.96	1.45603	0.728013	+	0.685563i	$0.240444\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4844.92	−1.75222	−0.876108	−	0.482115i	$-0.839869\pi$
$197$	−4844.92	−1.75222	−0.876108	+	0.482115i	$0.839869\pi$
$198$	0	0
$199$	947.744	0.337607	0.168803	−	0.985650i	$-0.446010\pi$
$199$	947.744	0.337607	0.168803	+	0.985650i	$0.446010\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1032.90	0.357120
$204$	0	0
$205$	−1980.75	−0.674838
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−845.015	−0.279669
$210$	0	0
$211$	3869.01	1.26234	0.631169	−	0.775645i	$-0.282575\pi$
$211$	3869.01	1.26234	0.631169	+	0.775645i	$0.282575\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1755.73	0.556929
$216$	0	0
$217$	415.718	0.130050
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4894.73	−1.48984
$222$	0	0
$223$	3123.95	0.938094	0.469047	−	0.883173i	$-0.344597\pi$
$223$	3123.95	0.938094	0.469047	+	0.883173i	$0.344597\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1436.53	0.420025	0.210013	−	0.977699i	$-0.432649\pi$
$227$	1436.53	0.420025	0.210013	+	0.977699i	$0.432649\pi$
$228$	0	0
$229$	3375.00	0.973913	0.486956	−	0.873426i	$-0.338107\pi$
$229$	3375.00	0.973913	0.486956	+	0.873426i	$0.338107\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	396.743	0.111552	0.0557758	−	0.998443i	$-0.482237\pi$
$233$	396.743	0.111552	0.0557758	+	0.998443i	$0.482237\pi$
$234$	0	0
$235$	−20.8405	−0.00578503
$236$	0	0
$237$	0	0
$238$	0	0
$239$	394.888	0.106875	0.0534376	−	0.998571i	$-0.482982\pi$
$239$	394.888	0.106875	0.0534376	+	0.998571i	$0.482982\pi$
$240$	0	0
$241$	−1003.83	−0.268309	−0.134154	−	0.990960i	$-0.542832\pi$
$241$	−1003.83	−0.268309	−0.134154	+	0.990960i	$0.542832\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−666.567	−0.173818
$246$	0	0
$247$	2352.93	0.606128
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6616.38	1.66383	0.831916	−	0.554901i	$-0.187244\pi$
$251$	6616.38	1.66383	0.831916	+	0.554901i	$0.187244\pi$
$252$	0	0
$253$	702.124	0.174475
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5198.46	1.26175	0.630877	−	0.775883i	$-0.282695\pi$
$257$	5198.46	1.26175	0.630877	+	0.775883i	$0.282695\pi$
$258$	0	0
$259$	1109.45	0.266169
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2229.79	0.522793	0.261396	−	0.965232i	$-0.415817\pi$
$263$	2229.79	0.522793	0.261396	+	0.965232i	$0.415817\pi$
$264$	0	0
$265$	−859.916	−0.199337
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1227.05	−0.278121	−0.139060	−	0.990284i	$-0.544408\pi$
$269$	−1227.05	−0.278121	−0.139060	+	0.990284i	$0.544408\pi$
$270$	0	0
$271$	271.993	0.0609683	0.0304842	−	0.999535i	$-0.490295\pi$
$271$	271.993	0.0609683	0.0304842	+	0.999535i	$0.490295\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	348.606	0.0764427
$276$	0	0
$277$	1373.62	0.297952	0.148976	−	0.988841i	$-0.452402\pi$
$277$	1373.62	0.297952	0.148976	+	0.988841i	$0.452402\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−583.170	−0.123804	−0.0619022	−	0.998082i	$-0.519717\pi$
$281$	−583.170	−0.123804	−0.0619022	+	0.998082i	$0.519717\pi$
$282$	0	0
$283$	7637.81	1.60431	0.802157	−	0.597113i	$-0.203686\pi$
$283$	7637.81	1.60431	0.802157	+	0.597113i	$0.203686\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5736.49	−1.17984
$288$	0	0
$289$	10978.9	2.23467
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1859.74	0.370810	0.185405	−	0.982662i	$-0.440640\pi$
$293$	1859.74	0.370810	0.185405	+	0.982662i	$0.440640\pi$
$294$	0	0
$295$	583.169	0.115096
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1955.06	−0.378140
$300$	0	0
$301$	5084.79	0.973695
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2619.78	0.491830
$306$	0	0
$307$	−3932.33	−0.731042	−0.365521	−	0.930803i	$-0.619109\pi$
$307$	−3932.33	−0.731042	−0.365521	+	0.930803i	$0.619109\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4756.90	−0.867328	−0.433664	−	0.901075i	$-0.642780\pi$
$311$	−4756.90	−0.867328	−0.433664	+	0.901075i	$0.642780\pi$
$312$	0	0
$313$	9621.84	1.73757	0.868784	−	0.495192i	$-0.164902\pi$
$313$	9621.84	1.73757	0.868784	+	0.495192i	$0.164902\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5132.07	0.909293	0.454647	−	0.890672i	$-0.349766\pi$
$317$	5132.07	0.909293	0.454647	+	0.890672i	$0.349766\pi$
$318$	0	0
$319$	994.646	0.174575
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−7639.37	−1.31599
$324$	0	0
$325$	−970.691	−0.165675
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−60.3564	−0.0101141
$330$	0	0
$331$	7436.78	1.23493	0.617466	−	0.786598i	$-0.288159\pi$
$331$	7436.78	1.23493	0.617466	+	0.786598i	$0.288159\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4362.11	0.711425
$336$	0	0
$337$	−10303.6	−1.66549	−0.832745	−	0.553656i	$-0.813232\pi$
$337$	−10303.6	−1.66549	−0.832745	+	0.553656i	$0.813232\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	400.321	0.0635736
$342$	0	0
$343$	−6897.28	−1.08577
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3021.04	0.467371	0.233685	−	0.972312i	$-0.424921\pi$
$347$	3021.04	0.467371	0.233685	+	0.972312i	$0.424921\pi$
$348$	0	0
$349$	10116.8	1.55169	0.775845	−	0.630923i	$-0.217324\pi$
$349$	10116.8	1.55169	0.775845	+	0.630923i	$0.217324\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2056.72	−0.310108	−0.155054	−	0.987906i	$-0.549555\pi$
$353$	−2056.72	−0.310108	−0.155054	+	0.987906i	$0.549555\pi$
$354$	0	0
$355$	4011.86	0.599795
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2034.25	−0.299062	−0.149531	−	0.988757i	$-0.547776\pi$
$359$	−2034.25	−0.299062	−0.149531	+	0.988757i	$0.547776\pi$
$360$	0	0
$361$	−3186.70	−0.464601
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−67.6319	−0.00969868
$366$	0	0
$367$	5846.96	0.831631	0.415816	−	0.909449i	$-0.363496\pi$
$367$	5846.96	0.831631	0.415816	+	0.909449i	$0.363496\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2490.41	−0.348506
$372$	0	0
$373$	−6479.76	−0.899489	−0.449744	−	0.893157i	$-0.648485\pi$
$373$	−6479.76	−0.899489	−0.449744	+	0.893157i	$0.648485\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2769.58	−0.378357
$378$	0	0
$379$	−4930.57	−0.668249	−0.334124	−	0.942529i	$-0.608441\pi$
$379$	−4930.57	−0.668249	−0.334124	+	0.942529i	$0.608441\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−4843.57	−0.646200	−0.323100	−	0.946365i	$-0.604725\pi$
$383$	−4843.57	−0.646200	−0.323100	+	0.946365i	$0.604725\pi$
$384$	0	0
$385$	1009.60	0.133647
$386$	0	0
$387$	0	0
$388$	0	0
$389$	11498.9	1.49875	0.749377	−	0.662144i	$-0.230353\pi$
$389$	11498.9	1.49875	0.749377	+	0.662144i	$0.230353\pi$
$390$	0	0
$391$	6347.56	0.820997
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2989.79	−0.380843
$396$	0	0
$397$	8087.11	1.02237	0.511185	−	0.859471i	$-0.329207\pi$
$397$	8087.11	1.02237	0.511185	+	0.859471i	$0.329207\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4956.24	0.617214	0.308607	−	0.951190i	$-0.400137\pi$
$401$	4956.24	0.617214	0.308607	+	0.951190i	$0.400137\pi$
$402$	0	0
$403$	−1114.69	−0.137783
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1068.36	0.130114
$408$	0	0
$409$	−4001.74	−0.483798	−0.241899	−	0.970301i	$-0.577770\pi$
$409$	−4001.74	−0.483798	−0.241899	+	0.970301i	$0.577770\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1688.92	0.201226
$414$	0	0
$415$	379.720	0.0449149
$416$	0	0
$417$	0	0
$418$	0	0
$419$	6684.47	0.779374	0.389687	−	0.920947i	$-0.372583\pi$
$419$	6684.47	0.779374	0.389687	+	0.920947i	$0.372583\pi$
$420$	0	0
$421$	−3614.61	−0.418445	−0.209223	−	0.977868i	$-0.567093\pi$
$421$	−3614.61	−0.418445	−0.209223	+	0.977868i	$0.567093\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3151.58	0.359704
$426$	0	0
$427$	7587.17	0.859880
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−11100.0	−1.24053	−0.620263	−	0.784394i	$-0.712974\pi$
$431$	−11100.0	−1.24053	−0.620263	+	0.784394i	$0.712974\pi$
$432$	0	0
$433$	11212.5	1.24443	0.622216	−	0.782846i	$-0.286233\pi$
$433$	11212.5	1.24443	0.622216	+	0.782846i	$0.286233\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3051.32	−0.334015
$438$	0	0
$439$	−7039.25	−0.765297	−0.382648	−	0.923894i	$-0.624988\pi$
$439$	−7039.25	−0.765297	−0.382648	+	0.923894i	$0.624988\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	11542.1	1.23788	0.618942	−	0.785437i	$-0.287562\pi$
$443$	11542.1	1.23788	0.618942	+	0.785437i	$0.287562\pi$
$444$	0	0
$445$	−3998.72	−0.425972
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2433.58	−0.255785	−0.127893	−	0.991788i	$-0.540821\pi$
$449$	−2433.58	−0.255785	−0.127893	+	0.991788i	$0.540821\pi$
$450$	0	0
$451$	−5524.03	−0.576754
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2811.23	−0.289654
$456$	0	0
$457$	−11241.9	−1.15071	−0.575354	−	0.817904i	$-0.695136\pi$
$457$	−11241.9	−1.15071	−0.575354	+	0.817904i	$0.695136\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9356.41	0.945275	0.472637	−	0.881257i	$-0.343302\pi$
$461$	9356.41	0.945275	0.472637	+	0.881257i	$0.343302\pi$
$462$	0	0
$463$	10726.8	1.07671	0.538356	−	0.842717i	$-0.319045\pi$
$463$	10726.8	1.07671	0.538356	+	0.842717i	$0.319045\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10271.5	−1.01780	−0.508898	−	0.860827i	$-0.669947\pi$
$467$	−10271.5	−1.01780	−0.508898	+	0.860827i	$0.669947\pi$
$468$	0	0
$469$	12633.2	1.24381
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4896.46	0.475982
$474$	0	0
$475$	−1514.99	−0.146342
$476$	0	0
$477$	0	0
$478$	0	0
$479$	7530.03	0.718280	0.359140	−	0.933284i	$-0.383070\pi$
$479$	7530.03	0.718280	0.359140	+	0.933284i	$0.383070\pi$
$480$	0	0
$481$	−2974.83	−0.281997
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2225.61	0.208371
$486$	0	0
$487$	17882.0	1.66388	0.831942	−	0.554862i	$-0.187229\pi$
$487$	17882.0	1.66388	0.831942	+	0.554862i	$0.187229\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−13309.9	−1.22336	−0.611679	−	0.791106i	$-0.709506\pi$
$491$	−13309.9	−1.22336	−0.611679	+	0.791106i	$0.709506\pi$
$492$	0	0
$493$	8992.11	0.821469
$494$	0	0
$495$	0	0
$496$	0	0
$497$	11618.8	1.04864
$498$	0	0
$499$	−9876.59	−0.886046	−0.443023	−	0.896510i	$-0.646094\pi$
$499$	−9876.59	−0.886046	−0.443023	+	0.896510i	$0.646094\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1983.95	−0.175864	−0.0879321	−	0.996126i	$-0.528026\pi$
$503$	−1983.95	−0.175864	−0.0879321	+	0.996126i	$0.528026\pi$
$504$	0	0
$505$	1010.44	0.0890372
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9423.36	0.820595	0.410298	−	0.911952i	$-0.365425\pi$
$509$	9423.36	0.820595	0.410298	+	0.911952i	$0.365425\pi$
$510$	0	0
$511$	−195.870	−0.0169565
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3913.38	−0.334843
$516$	0	0
$517$	−58.1210	−0.00494421
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−12938.3	−1.08798	−0.543988	−	0.839093i	$-0.683086\pi$
$521$	−12938.3	−1.08798	−0.543988	+	0.839093i	$0.683086\pi$
$522$	0	0
$523$	−9800.41	−0.819392	−0.409696	−	0.912222i	$-0.634365\pi$
$523$	−9800.41	−0.819392	−0.409696	+	0.912222i	$0.634365\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3619.11	0.299148
$528$	0	0
$529$	−9631.66	−0.791621
$530$	0	0
$531$	0	0
$532$	0	0
$533$	15381.6	1.25000
$534$	0	0
$535$	9432.11	0.762217
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1858.95	−0.148554
$540$	0	0
$541$	19852.4	1.57767	0.788836	−	0.614604i	$-0.210684\pi$
$541$	19852.4	1.57767	0.788836	+	0.614604i	$0.210684\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3023.37	0.237627
$546$	0	0
$547$	23140.5	1.80880	0.904402	−	0.426681i	$-0.140317\pi$
$547$	23140.5	1.80880	0.904402	+	0.426681i	$0.140317\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−4322.57	−0.334207
$552$	0	0
$553$	−8658.78	−0.665838
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2127.55	−0.161844	−0.0809222	−	0.996720i	$-0.525787\pi$
$557$	−2127.55	−0.161844	−0.0809222	+	0.996720i	$0.525787\pi$
$558$	0	0
$559$	−13634.1	−1.03160
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−17206.7	−1.28806	−0.644028	−	0.765002i	$-0.722738\pi$
$563$	−17206.7	−1.28806	−0.644028	+	0.765002i	$0.722738\pi$
$564$	0	0
$565$	−7925.47	−0.590136
$566$	0	0
$567$	0	0
$568$	0	0
$569$	13928.6	1.02622	0.513108	−	0.858324i	$-0.328494\pi$
$569$	13928.6	1.02622	0.513108	+	0.858324i	$0.328494\pi$
$570$	0	0
$571$	201.457	0.0147648	0.00738239	−	0.999973i	$-0.497650\pi$
$571$	201.457	0.0147648	0.00738239	+	0.999973i	$0.497650\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1258.81	0.0912970
$576$	0	0
$577$	164.727	0.0118851	0.00594253	−	0.999982i	$-0.498108\pi$
$577$	164.727	0.0118851	0.00594253	+	0.999982i	$0.498108\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1099.71	0.0785261
$582$	0	0
$583$	−2398.18	−0.170364
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14327.9	−1.00746	−0.503728	−	0.863862i	$-0.668039\pi$
$587$	−14327.9	−1.00746	−0.503728	+	0.863862i	$0.668039\pi$
$588$	0	0
$589$	−1739.73	−0.121705
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−13213.8	−0.915049	−0.457525	−	0.889197i	$-0.651264\pi$
$593$	−13213.8	−0.915049	−0.457525	+	0.889197i	$0.651264\pi$
$594$	0	0
$595$	9127.33	0.628881
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−13780.3	−0.939979	−0.469990	−	0.882672i	$-0.655742\pi$
$599$	−13780.3	−0.939979	−0.469990	+	0.882672i	$0.655742\pi$
$600$	0	0
$601$	1473.27	0.0999933	0.0499966	−	0.998749i	$-0.484079\pi$
$601$	1473.27	0.0999933	0.0499966	+	0.998749i	$0.484079\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−5682.79	−0.381881
$606$	0	0
$607$	9450.54	0.631937	0.315968	−	0.948770i	$-0.397671\pi$
$607$	9450.54	0.631937	0.315968	+	0.948770i	$0.397671\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	161.837	0.0107156
$612$	0	0
$613$	17768.1	1.17071	0.585356	−	0.810777i	$-0.300955\pi$
$613$	17768.1	1.17071	0.585356	+	0.810777i	$0.300955\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−5429.29	−0.354255	−0.177127	−	0.984188i	$-0.556680\pi$
$617$	−5429.29	−0.354255	−0.177127	+	0.984188i	$0.556680\pi$
$618$	0	0
$619$	−6441.62	−0.418273	−0.209136	−	0.977886i	$-0.567065\pi$
$619$	−6441.62	−0.418273	−0.209136	+	0.977886i	$0.567065\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−11580.7	−0.744740
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	9658.51	0.612257
$630$	0	0
$631$	−21418.5	−1.35128	−0.675639	−	0.737232i	$-0.736132\pi$
$631$	−21418.5	−1.35128	−0.675639	+	0.737232i	$0.736132\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	7321.81	0.457570
$636$	0	0
$637$	5176.24	0.321962
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1413.11	−0.0870740	−0.0435370	−	0.999052i	$-0.513863\pi$
$641$	−1413.11	−0.0870740	−0.0435370	+	0.999052i	$0.513863\pi$
$642$	0	0
$643$	−23835.3	−1.46185	−0.730926	−	0.682456i	$-0.760912\pi$
$643$	−23835.3	−1.46185	−0.730926	+	0.682456i	$0.760912\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−13237.4	−0.804350	−0.402175	−	0.915563i	$-0.631746\pi$
$647$	−13237.4	−0.804350	−0.402175	+	0.915563i	$0.631746\pi$
$648$	0	0
$649$	1626.37	0.0983677
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−23830.7	−1.42813	−0.714064	−	0.700081i	$-0.753147\pi$
$653$	−23830.7	−1.42813	−0.714064	+	0.700081i	$0.753147\pi$
$654$	0	0
$655$	−2103.33	−0.125472
$656$	0	0
$657$	0	0
$658$	0	0
$659$	8249.82	0.487659	0.243830	−	0.969818i	$-0.421596\pi$
$659$	8249.82	0.487659	0.243830	+	0.969818i	$0.421596\pi$
$660$	0	0
$661$	13951.8	0.820970	0.410485	−	0.911867i	$-0.365359\pi$
$661$	13951.8	0.820970	0.410485	+	0.911867i	$0.365359\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−4387.57	−0.255854
$666$	0	0
$667$	3591.63	0.208499
$668$	0	0
$669$	0	0
$670$	0	0
$671$	7306.17	0.420345
$672$	0	0
$673$	−11395.9	−0.652717	−0.326358	−	0.945246i	$-0.605822\pi$
$673$	−11395.9	−0.652717	−0.326358	+	0.945246i	$0.605822\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−3944.88	−0.223950	−0.111975	−	0.993711i	$-0.535718\pi$
$677$	−3944.88	−0.223950	−0.111975	+	0.993711i	$0.535718\pi$
$678$	0	0
$679$	6445.63	0.364301
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−30972.3	−1.73517	−0.867584	−	0.497290i	$-0.834328\pi$
$683$	−30972.3	−1.73517	−0.867584	+	0.497290i	$0.834328\pi$
$684$	0	0
$685$	773.153	0.0431250
$686$	0	0
$687$	0	0
$688$	0	0
$689$	6677.70	0.369231
$690$	0	0
$691$	−21561.2	−1.18701	−0.593506	−	0.804830i	$-0.702257\pi$
$691$	−21561.2	−1.18701	−0.593506	+	0.804830i	$0.702257\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	9194.13	0.501803
$696$	0	0
$697$	−49940.0	−2.71394
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−10381.1	−0.559326	−0.279663	−	0.960098i	$-0.590223\pi$
$701$	−10381.1	−0.559326	−0.279663	+	0.960098i	$0.590223\pi$
$702$	0	0
$703$	−4642.91	−0.249091
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2926.33	0.155666
$708$	0	0
$709$	15336.2	0.812357	0.406179	−	0.913794i	$-0.366861\pi$
$709$	15336.2	0.812357	0.406179	+	0.913794i	$0.366861\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1445.55	0.0759272
$714$	0	0
$715$	−2707.11	−0.141595
$716$	0	0
$717$	0	0
$718$	0	0
$719$	26655.6	1.38260	0.691298	−	0.722570i	$-0.257039\pi$
$719$	26655.6	1.38260	0.691298	+	0.722570i	$0.257039\pi$
$720$	0	0
$721$	−11333.6	−0.585416
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1783.25	0.0913495
$726$	0	0
$727$	31166.6	1.58997	0.794983	−	0.606631i	$-0.207480\pi$
$727$	31166.6	1.58997	0.794983	+	0.606631i	$0.207480\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	44266.5	2.23975
$732$	0	0
$733$	5735.03	0.288988	0.144494	−	0.989506i	$-0.453845\pi$
$733$	5735.03	0.288988	0.144494	+	0.989506i	$0.453845\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12165.3	0.608024
$738$	0	0
$739$	−28634.6	−1.42536	−0.712680	−	0.701489i	$-0.752519\pi$
$739$	−28634.6	−1.42536	−0.712680	+	0.701489i	$0.752519\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	14988.8	0.740086	0.370043	−	0.929015i	$-0.379343\pi$
$743$	14988.8	0.740086	0.370043	+	0.929015i	$0.379343\pi$
$744$	0	0
$745$	−8238.03	−0.405125
$746$	0	0
$747$	0	0
$748$	0	0
$749$	27316.5	1.33261
$750$	0	0
$751$	19998.2	0.971696	0.485848	−	0.874043i	$-0.338511\pi$
$751$	19998.2	0.971696	0.485848	+	0.874043i	$0.338511\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−5116.68	−0.246642
$756$	0	0
$757$	−31663.4	−1.52025	−0.760123	−	0.649779i	$-0.774861\pi$
$757$	−31663.4	−1.52025	−0.760123	+	0.649779i	$0.774861\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	21227.4	1.01116	0.505581	−	0.862779i	$-0.331278\pi$
$761$	21227.4	1.01116	0.505581	+	0.862779i	$0.331278\pi$
$762$	0	0
$763$	8756.01	0.415451
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4528.61	−0.213193
$768$	0	0
$769$	21304.5	0.999037	0.499518	−	0.866303i	$-0.333510\pi$
$769$	21304.5	0.999037	0.499518	+	0.866303i	$0.333510\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−16993.1	−0.790686	−0.395343	−	0.918534i	$-0.629374\pi$
$773$	−16993.1	−0.790686	−0.395343	+	0.918534i	$0.629374\pi$
$774$	0	0
$775$	717.717	0.0332660
$776$	0	0
$777$	0	0
$778$	0	0
$779$	24006.5	1.10414
$780$	0	0
$781$	11188.5	0.512618
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−4401.05	−0.200102
$786$	0	0
$787$	4191.03	0.189827	0.0949137	−	0.995486i	$-0.469742\pi$
$787$	4191.03	0.189827	0.0949137	+	0.995486i	$0.469742\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−22953.0	−1.03175
$792$	0	0
$793$	−20343.9	−0.911015
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−34926.1	−1.55225	−0.776126	−	0.630578i	$-0.782818\pi$
$797$	−34926.1	−1.55225	−0.776126	+	0.630578i	$0.782818\pi$
$798$	0	0
$799$	−525.443	−0.0232651
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−188.615	−0.00828903
$804$	0	0
$805$	3645.64	0.159617
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1388.34	0.0603355	0.0301677	−	0.999545i	$-0.490396\pi$
$809$	1388.34	0.0603355	0.0301677	+	0.999545i	$0.490396\pi$
$810$	0	0
$811$	11611.7	0.502766	0.251383	−	0.967888i	$-0.419115\pi$
$811$	11611.7	0.502766	0.251383	+	0.967888i	$0.419115\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−6733.84	−0.289418
$816$	0	0
$817$	−21279.2	−0.911220
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−5754.09	−0.244603	−0.122302	−	0.992493i	$-0.539027\pi$
$821$	−5754.09	−0.244603	−0.122302	+	0.992493i	$0.539027\pi$
$822$	0	0
$823$	−31328.1	−1.32689	−0.663443	−	0.748227i	$-0.730906\pi$
$823$	−31328.1	−1.32689	−0.663443	+	0.748227i	$0.730906\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	22192.5	0.933143	0.466572	−	0.884483i	$-0.345489\pi$
$827$	22192.5	0.933143	0.466572	+	0.884483i	$0.345489\pi$
$828$	0	0
$829$	26744.5	1.12048	0.560238	−	0.828331i	$-0.310710\pi$
$829$	26744.5	1.12048	0.560238	+	0.828331i	$0.310710\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−16805.9	−0.699028
$834$	0	0
$835$	12899.4	0.534612
$836$	0	0
$837$	0	0
$838$	0	0
$839$	40896.3	1.68283	0.841416	−	0.540388i	$-0.181723\pi$
$839$	40896.3	1.68283	0.841416	+	0.540388i	$0.181723\pi$
$840$	0	0
$841$	−19301.0	−0.791381
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−3447.08	−0.140335
$846$	0	0
$847$	−16458.0	−0.667655
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3857.80	0.155398
$852$	0	0
$853$	2596.52	0.104224	0.0521120	−	0.998641i	$-0.483405\pi$
$853$	2596.52	0.104224	0.0521120	+	0.998641i	$0.483405\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−19284.5	−0.768665	−0.384332	−	0.923195i	$-0.625568\pi$
$857$	−19284.5	−0.768665	−0.384332	+	0.923195i	$0.625568\pi$
$858$	0	0
$859$	−12590.5	−0.500098	−0.250049	−	0.968233i	$-0.580447\pi$
$859$	−12590.5	−0.500098	−0.250049	+	0.968233i	$0.580447\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	28897.6	1.13984	0.569921	−	0.821699i	$-0.306974\pi$
$863$	28897.6	1.13984	0.569921	+	0.821699i	$0.306974\pi$
$864$	0	0
$865$	−14388.6	−0.565579
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−8338.09	−0.325489
$870$	0	0
$871$	−33874.1	−1.31777
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1810.07	0.0699332
$876$	0	0
$877$	−38957.0	−1.49998	−0.749991	−	0.661448i	$-0.769942\pi$
$877$	−38957.0	−1.49998	−0.749991	+	0.661448i	$0.769942\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−21416.2	−0.818989	−0.409495	−	0.912313i	$-0.634295\pi$
$881$	−21416.2	−0.818989	−0.409495	+	0.912313i	$0.634295\pi$
$882$	0	0
$883$	−12270.0	−0.467631	−0.233816	−	0.972281i	$-0.575121\pi$
$883$	−12270.0	−0.467631	−0.233816	+	0.972281i	$0.575121\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−23430.6	−0.886947	−0.443473	−	0.896288i	$-0.646254\pi$
$887$	−23430.6	−0.886947	−0.443473	+	0.896288i	$0.646254\pi$
$888$	0	0
$889$	21204.8	0.799984
$890$	0	0
$891$	0	0
$892$	0	0
$893$	252.584	0.00946519
$894$	0	0
$895$	−9533.97	−0.356073
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2047.80	0.0759709
$900$	0	0
$901$	−21680.7	−0.801654
$902$	0	0
$903$	0	0
$904$	0	0
$905$	24205.5	0.889081
$906$	0	0
$907$	−14548.3	−0.532599	−0.266300	−	0.963890i	$-0.585801\pi$
$907$	−14548.3	−0.532599	−0.266300	+	0.963890i	$0.585801\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	19448.2	0.707297	0.353649	−	0.935378i	$-0.384941\pi$
$911$	19448.2	0.707297	0.353649	+	0.935378i	$0.384941\pi$
$912$	0	0
$913$	1058.98	0.0383868
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6091.49	−0.219366
$918$	0	0
$919$	−23106.0	−0.829376	−0.414688	−	0.909964i	$-0.636109\pi$
$919$	−23106.0	−0.829376	−0.414688	+	0.909964i	$0.636109\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−31154.2	−1.11100
$924$	0	0
$925$	1915.41	0.0680846
$926$	0	0
$927$	0	0
$928$	0	0
$929$	687.069	0.0242648	0.0121324	−	0.999926i	$-0.496138\pi$
$929$	687.069	0.0242648	0.0121324	+	0.999926i	$0.496138\pi$
$930$	0	0
$931$	8078.72	0.284392
$932$	0	0
$933$	0	0
$934$	0	0
$935$	8789.29	0.307423
$936$	0	0
$937$	−17926.3	−0.625003	−0.312502	−	0.949917i	$-0.601167\pi$
$937$	−17926.3	−0.625003	−0.312502	+	0.949917i	$0.601167\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	11253.2	0.389843	0.194921	−	0.980819i	$-0.437555\pi$
$941$	11253.2	0.389843	0.194921	+	0.980819i	$0.437555\pi$
$942$	0	0
$943$	−19947.1	−0.688829
$944$	0	0
$945$	0	0
$946$	0	0
$947$	43243.3	1.48386	0.741931	−	0.670476i	$-0.233910\pi$
$947$	43243.3	1.48386	0.741931	+	0.670476i	$0.233910\pi$
$948$	0	0
$949$	525.198	0.0179648
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−44192.2	−1.50213	−0.751063	−	0.660230i	$-0.770459\pi$
$953$	−44192.2	−1.50213	−0.751063	+	0.660230i	$0.770459\pi$
$954$	0	0
$955$	520.319	0.0176305
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2239.14	0.0753968
$960$	0	0
$961$	−28966.8	−0.972334
$962$	0	0
$963$	0	0
$964$	0	0
$965$	19519.8	0.651155
$966$	0	0
$967$	−2689.33	−0.0894345	−0.0447172	−	0.999000i	$-0.514239\pi$
$967$	−2689.33	−0.0894345	−0.0447172	+	0.999000i	$0.514239\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−49178.9	−1.62536	−0.812680	−	0.582710i	$-0.801992\pi$
$971$	−49178.9	−1.62536	−0.812680	+	0.582710i	$0.801992\pi$
$972$	0	0
$973$	26627.2	0.877318
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−60933.9	−1.99534	−0.997670	−	0.0682270i	$-0.978266\pi$
$977$	−60933.9	−1.99534	−0.997670	+	0.0682270i	$0.978266\pi$
$978$	0	0
$979$	−11151.8	−0.364059
$980$	0	0
$981$	0	0
$982$	0	0
$983$	255.703	0.00829672	0.00414836	−	0.999991i	$-0.498680\pi$
$983$	255.703	0.00829672	0.00414836	+	0.999991i	$0.498680\pi$
$984$	0	0
$985$	−24224.6	−0.783615
$986$	0	0
$987$	0	0
$988$	0	0
$989$	17681.0	0.568475
$990$	0	0
$991$	−33864.4	−1.08551	−0.542754	−	0.839892i	$-0.682618\pi$
$991$	−33864.4	−1.08551	−0.542754	+	0.839892i	$0.682618\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4738.72	0.150982
$996$	0	0
$997$	−52881.3	−1.67981	−0.839904	−	0.542735i	$-0.817389\pi$
$997$	−52881.3	−1.67981	−0.839904	+	0.542735i	$0.817389\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.a.bt.1.2		3
3.2	odd	2		2160.4.a.bl.1.2		3
4.3	odd	2		1080.4.a.i.1.2	yes	3
12.11	even	2		1080.4.a.c.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.a.c.1.2	✓	3	12.11	even	2
1080.4.a.i.1.2	yes	3	4.3	odd	2
2160.4.a.bl.1.2		3	3.2	odd	2
2160.4.a.bt.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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+5.00000 q^{5} +14.4806 q^{7} +O(q^{10})\)	\(q+5.00000 q^{5} +14.4806 q^{7} +13.9443 q^{11} -38.8276 q^{13} +126.063 q^{17} -60.5995 q^{19} +50.3522 q^{23} +25.0000 q^{25} +71.3302 q^{29} +28.7087 q^{31} +72.4028 q^{35} +76.6164 q^{37} -396.151 q^{41} +351.146 q^{43} -4.16809 q^{47} -133.313 q^{49} -171.983 q^{53} +69.7213 q^{55} +116.634 q^{59} +523.955 q^{61} -194.138 q^{65} +872.421 q^{67} +802.371 q^{71} -13.5264 q^{73} +201.921 q^{77} -597.959 q^{79} +75.9439 q^{83} +630.316 q^{85} -799.744 q^{89} -562.246 q^{91} -302.998 q^{95} +445.123 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 15 q^{5} + 24 q^{7}+O(q^{10}) \) 3 * q + 15 * q^5 + 24 * q^7	\( 3 q + 15 q^{5} + 24 q^{7} - 6 q^{11} + 48 q^{13} - 27 q^{17} + 195 q^{19} + 27 q^{23} + 75 q^{25} + 60 q^{29} + 279 q^{31} + 120 q^{35} - 138 q^{37} + 66 q^{41} - 222 q^{43} + 264 q^{47} - 237 q^{49} - 507 q^{53} - 30 q^{55} + 960 q^{59} + 543 q^{61} + 240 q^{65} + 1086 q^{67} + 1818 q^{71} + 1362 q^{73} - 1776 q^{77} + 129 q^{79} + 1569 q^{83} - 135 q^{85} - 1770 q^{89} - 1488 q^{91} + 975 q^{95} - 336 q^{97}+O(q^{100}) \) 3 * q + 15 * q^5 + 24 * q^7 - 6 * q^11 + 48 * q^13 - 27 * q^17 + 195 * q^19 + 27 * q^23 + 75 * q^25 + 60 * q^29 + 279 * q^31 + 120 * q^35 - 138 * q^37 + 66 * q^41 - 222 * q^43 + 264 * q^47 - 237 * q^49 - 507 * q^53 - 30 * q^55 + 960 * q^59 + 543 * q^61 + 240 * q^65 + 1086 * q^67 + 1818 * q^71 + 1362 * q^73 - 1776 * q^77 + 129 * q^79 + 1569 * q^83 - 135 * q^85 - 1770 * q^89 - 1488 * q^91 + 975 * q^95 - 336 * q^97

Introduction

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