LMFDB - Embedded newform 2160.4.a.bg.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:	$1$
Dimension:	$3$
Coefficient field:	3.3.985.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 1 $ x^3 - x^2 - 6*x + 1
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			$2.93080$ 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)

$f(q)$	$=$	$q-5.00000 q^{5} +7.95278 q^{7} +O(q^{10})$	$q-5.00000 q^{5} +7.95278 q^{7} -37.1224 q^{11} -35.1224 q^{13} +99.1976 q^{17} +44.6505 q^{19} +102.603 q^{23} +25.0000 q^{25} -285.404 q^{29} -238.593 q^{31} -39.7639 q^{35} +339.168 q^{37} +423.799 q^{41} -144.330 q^{43} +418.414 q^{47} -279.753 q^{49} -186.933 q^{53} +185.612 q^{55} +293.783 q^{59} -701.657 q^{61} +175.612 q^{65} +292.697 q^{67} +738.742 q^{71} +453.214 q^{73} -295.226 q^{77} -892.759 q^{79} -66.4113 q^{83} -495.988 q^{85} -868.816 q^{89} -279.321 q^{91} -223.253 q^{95} +112.488 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 15 q^{5} - 6 q^{7}+O(q^{10}) $ 3 * q - 15 * q^5 - 6 * q^7	$ 3 q - 15 q^{5} - 6 q^{7} + 12 q^{11} + 18 q^{13} + 21 q^{17} - 57 q^{19} + 87 q^{23} + 75 q^{25} - 138 q^{29} - 117 q^{31} + 30 q^{35} + 150 q^{37} - 180 q^{43} + 684 q^{47} - 81 q^{49} - 87 q^{53} - 60 q^{55} + 714 q^{59} - 513 q^{61} - 90 q^{65} + 174 q^{67} + 768 q^{71} - 252 q^{73} - 888 q^{77} - 207 q^{79} + 1689 q^{83} - 105 q^{85} - 312 q^{89} - 900 q^{91} + 285 q^{95} - 1080 q^{97}+O(q^{100}) $ 3 * q - 15 * q^5 - 6 * q^7 + 12 * q^11 + 18 * q^13 + 21 * q^17 - 57 * q^19 + 87 * q^23 + 75 * q^25 - 138 * q^29 - 117 * q^31 + 30 * q^35 + 150 * q^37 - 180 * q^43 + 684 * q^47 - 81 * q^49 - 87 * q^53 - 60 * q^55 + 714 * q^59 - 513 * q^61 - 90 * q^65 + 174 * q^67 + 768 * q^71 - 252 * q^73 - 888 * q^77 - 207 * q^79 + 1689 * q^83 - 105 * q^85 - 312 * q^89 - 900 * q^91 + 285 * q^95 - 1080 * 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
$7$	7.95278	0.429410	0.214705	−	0.976679i	$-0.431121\pi$
$7$	7.95278	0.429410	0.214705	+	0.976679i	$0.431121\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−37.1224	−1.01753	−0.508765	−	0.860906i	$-0.669898\pi$
$11$	−37.1224	−1.01753	−0.508765	+	0.860906i	$0.669898\pi$
$12$	0	0
$13$	−35.1224	−0.749323	−0.374662	−	0.927162i	$-0.622241\pi$
$13$	−35.1224	−0.749323	−0.374662	+	0.927162i	$0.622241\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	99.1976	1.41523	0.707616	−	0.706597i	$-0.249771\pi$
$17$	99.1976	1.41523	0.707616	+	0.706597i	$0.249771\pi$
$18$	0	0
$19$	44.6505	0.539133	0.269567	−	0.962982i	$-0.413120\pi$
$19$	44.6505	0.539133	0.269567	+	0.962982i	$0.413120\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	102.603	0.930186	0.465093	−	0.885262i	$-0.346021\pi$
$23$	102.603	0.930186	0.465093	+	0.885262i	$0.346021\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−285.404	−1.82752	−0.913762	−	0.406249i	$-0.866837\pi$
$29$	−285.404	−1.82752	−0.913762	+	0.406249i	$0.866837\pi$
$30$	0	0
$31$	−238.593	−1.38234	−0.691171	−	0.722692i	$-0.742905\pi$
$31$	−238.593	−1.38234	−0.691171	+	0.722692i	$0.742905\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−39.7639	−0.192038
$36$	0	0
$37$	339.168	1.50700	0.753498	−	0.657450i	$-0.228365\pi$
$37$	339.168	1.50700	0.753498	+	0.657450i	$0.228365\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	423.799	1.61430	0.807150	−	0.590346i	$-0.201009\pi$
$41$	423.799	1.61430	0.807150	+	0.590346i	$0.201009\pi$
$42$	0	0
$43$	−144.330	−0.511863	−0.255931	−	0.966695i	$-0.582382\pi$
$43$	−144.330	−0.511863	−0.255931	+	0.966695i	$0.582382\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	418.414	1.29855	0.649276	−	0.760553i	$-0.275072\pi$
$47$	418.414	1.29855	0.649276	+	0.760553i	$0.275072\pi$
$48$	0	0
$49$	−279.753	−0.815607
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−186.933	−0.484476	−0.242238	−	0.970217i	$-0.577882\pi$
$53$	−186.933	−0.484476	−0.242238	+	0.970217i	$0.577882\pi$
$54$	0	0
$55$	185.612	0.455053
$56$	0	0
$57$	0	0
$58$	0	0
$59$	293.783	0.648259	0.324129	−	0.946013i	$-0.394929\pi$
$59$	293.783	0.648259	0.324129	+	0.946013i	$0.394929\pi$
$60$	0	0
$61$	−701.657	−1.47275	−0.736377	−	0.676571i	$-0.763465\pi$
$61$	−701.657	−1.47275	−0.736377	+	0.676571i	$0.763465\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	175.612	0.335107
$66$	0	0
$67$	292.697	0.533710	0.266855	−	0.963737i	$-0.414016\pi$
$67$	292.697	0.533710	0.266855	+	0.963737i	$0.414016\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	738.742	1.23482	0.617412	−	0.786640i	$-0.288181\pi$
$71$	738.742	1.23482	0.617412	+	0.786640i	$0.288181\pi$
$72$	0	0
$73$	453.214	0.726639	0.363320	−	0.931665i	$-0.381643\pi$
$73$	453.214	0.726639	0.363320	+	0.931665i	$0.381643\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−295.226	−0.436937
$78$	0	0
$79$	−892.759	−1.27143	−0.635717	−	0.771922i	$-0.719295\pi$
$79$	−892.759	−1.27143	−0.635717	+	0.771922i	$0.719295\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−66.4113	−0.0878264	−0.0439132	−	0.999035i	$-0.513982\pi$
$83$	−66.4113	−0.0878264	−0.0439132	+	0.999035i	$0.513982\pi$
$84$	0	0
$85$	−495.988	−0.632911
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−868.816	−1.03477	−0.517384	−	0.855753i	$-0.673094\pi$
$89$	−868.816	−1.03477	−0.517384	+	0.855753i	$0.673094\pi$
$90$	0	0
$91$	−279.321	−0.321767
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−223.253	−0.241108
$96$	0	0
$97$	112.488	0.117747	0.0588735	−	0.998265i	$-0.481249\pi$
$97$	112.488	0.117747	0.0588735	+	0.998265i	$0.481249\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1581.13	−1.55770	−0.778852	−	0.627207i	$-0.784198\pi$
$101$	−1581.13	−1.55770	−0.778852	+	0.627207i	$0.784198\pi$
$102$	0	0
$103$	905.167	0.865910	0.432955	−	0.901416i	$-0.357471\pi$
$103$	905.167	0.865910	0.432955	+	0.901416i	$0.357471\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1127.93	−1.01908	−0.509539	−	0.860447i	$-0.670184\pi$
$107$	−1127.93	−1.01908	−0.509539	+	0.860447i	$0.670184\pi$
$108$	0	0
$109$	−1947.65	−1.71148	−0.855739	−	0.517408i	$-0.826897\pi$
$109$	−1947.65	−1.71148	−0.855739	+	0.517408i	$0.826897\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1724.52	−1.43566	−0.717829	−	0.696220i	$-0.754864\pi$
$113$	−1724.52	−1.43566	−0.717829	+	0.696220i	$0.754864\pi$
$114$	0	0
$115$	−513.017	−0.415992
$116$	0	0
$117$	0	0
$118$	0	0
$119$	788.896	0.607714
$120$	0	0
$121$	47.0724	0.0353662
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−645.720	−0.451168	−0.225584	−	0.974224i	$-0.572429\pi$
$127$	−645.720	−0.451168	−0.225584	+	0.974224i	$0.572429\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2055.92	1.37119	0.685597	−	0.727982i	$-0.259541\pi$
$131$	2055.92	1.37119	0.685597	+	0.727982i	$0.259541\pi$
$132$	0	0
$133$	355.096	0.231509
$134$	0	0
$135$	0	0
$136$	0	0
$137$	244.549	0.152505	0.0762525	−	0.997089i	$-0.475704\pi$
$137$	244.549	0.152505	0.0762525	+	0.997089i	$0.475704\pi$
$138$	0	0
$139$	−1068.67	−0.652111	−0.326056	−	0.945351i	$-0.605720\pi$
$139$	−1068.67	−0.652111	−0.326056	+	0.945351i	$0.605720\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1303.83	0.762458
$144$	0	0
$145$	1427.02	0.817294
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1295.84	0.712481	0.356240	−	0.934394i	$-0.384058\pi$
$149$	1295.84	0.712481	0.356240	+	0.934394i	$0.384058\pi$
$150$	0	0
$151$	2727.32	1.46984	0.734921	−	0.678153i	$-0.237219\pi$
$151$	2727.32	1.46984	0.734921	+	0.678153i	$0.237219\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1192.97	0.618202
$156$	0	0
$157$	867.700	0.441083	0.220541	−	0.975378i	$-0.429218\pi$
$157$	867.700	0.441083	0.220541	+	0.975378i	$0.429218\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	815.981	0.399431
$162$	0	0
$163$	−1217.40	−0.584995	−0.292497	−	0.956266i	$-0.594486\pi$
$163$	−1217.40	−0.584995	−0.292497	+	0.956266i	$0.594486\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3266.76	1.51371	0.756854	−	0.653583i	$-0.226735\pi$
$167$	3266.76	1.51371	0.756854	+	0.653583i	$0.226735\pi$
$168$	0	0
$169$	−963.417	−0.438515
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1421.71	−0.624802	−0.312401	−	0.949950i	$-0.601133\pi$
$173$	−1421.71	−0.624802	−0.312401	+	0.949950i	$0.601133\pi$
$174$	0	0
$175$	198.819	0.0858819
$176$	0	0
$177$	0	0
$178$	0	0
$179$	949.492	0.396471	0.198236	−	0.980154i	$-0.436479\pi$
$179$	949.492	0.396471	0.198236	+	0.980154i	$0.436479\pi$
$180$	0	0
$181$	1748.25	0.717938	0.358969	−	0.933350i	$-0.383128\pi$
$181$	1748.25	0.717938	0.358969	+	0.933350i	$0.383128\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1695.84	−0.673950
$186$	0	0
$187$	−3682.45	−1.44004
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1418.37	−0.537327	−0.268664	−	0.963234i	$-0.586582\pi$
$191$	−1418.37	−0.537327	−0.268664	+	0.963234i	$0.586582\pi$
$192$	0	0
$193$	−3733.49	−1.39245	−0.696223	−	0.717825i	$-0.745138\pi$
$193$	−3733.49	−1.39245	−0.696223	+	0.717825i	$0.745138\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2294.67	−0.829889	−0.414945	−	0.909847i	$-0.636199\pi$
$197$	−2294.67	−0.829889	−0.414945	+	0.909847i	$0.636199\pi$
$198$	0	0
$199$	−3713.93	−1.32298	−0.661492	−	0.749952i	$-0.730076\pi$
$199$	−3713.93	−1.32298	−0.661492	+	0.749952i	$0.730076\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2269.76	−0.784757
$204$	0	0
$205$	−2119.00	−0.721937
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1657.53	−0.548584
$210$	0	0
$211$	199.877	0.0652139	0.0326069	−	0.999468i	$-0.489619\pi$
$211$	199.877	0.0652139	0.0326069	+	0.999468i	$0.489619\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	721.649	0.228912
$216$	0	0
$217$	−1897.48	−0.593591
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3484.06	−1.06047
$222$	0	0
$223$	−879.739	−0.264178	−0.132089	−	0.991238i	$-0.542168\pi$
$223$	−879.739	−0.264178	−0.132089	+	0.991238i	$0.542168\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4331.67	1.26653	0.633267	−	0.773934i	$-0.281714\pi$
$227$	4331.67	1.26653	0.633267	+	0.773934i	$0.281714\pi$
$228$	0	0
$229$	−2995.80	−0.864489	−0.432244	−	0.901757i	$-0.642278\pi$
$229$	−2995.80	−0.864489	−0.432244	+	0.901757i	$0.642278\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2068.75	−0.581666	−0.290833	−	0.956774i	$-0.593932\pi$
$233$	−2068.75	−0.581666	−0.290833	+	0.956774i	$0.593932\pi$
$234$	0	0
$235$	−2092.07	−0.580730
$236$	0	0
$237$	0	0
$238$	0	0
$239$	311.612	0.0843368	0.0421684	−	0.999111i	$-0.486573\pi$
$239$	311.612	0.0843368	0.0421684	+	0.999111i	$0.486573\pi$
$240$	0	0
$241$	−4636.53	−1.23928	−0.619638	−	0.784888i	$-0.712720\pi$
$241$	−4636.53	−1.23928	−0.619638	+	0.784888i	$0.712720\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1398.77	0.364751
$246$	0	0
$247$	−1568.23	−0.403985
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1243.61	0.312732	0.156366	−	0.987699i	$-0.450022\pi$
$251$	1243.61	0.312732	0.156366	+	0.987699i	$0.450022\pi$
$252$	0	0
$253$	−3808.88	−0.946491
$254$	0	0
$255$	0	0
$256$	0	0
$257$	687.895	0.166964	0.0834819	−	0.996509i	$-0.473396\pi$
$257$	687.895	0.166964	0.0834819	+	0.996509i	$0.473396\pi$
$258$	0	0
$259$	2697.33	0.647119
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2819.20	−0.660986	−0.330493	−	0.943808i	$-0.607215\pi$
$263$	−2819.20	−0.660986	−0.330493	+	0.943808i	$0.607215\pi$
$264$	0	0
$265$	934.666	0.216664
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−4208.33	−0.953854	−0.476927	−	0.878943i	$-0.658249\pi$
$269$	−4208.33	−0.953854	−0.476927	+	0.878943i	$0.658249\pi$
$270$	0	0
$271$	−8395.09	−1.88179	−0.940896	−	0.338696i	$-0.890014\pi$
$271$	−8395.09	−1.88179	−0.940896	+	0.338696i	$0.890014\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−928.060	−0.203506
$276$	0	0
$277$	−1197.24	−0.259694	−0.129847	−	0.991534i	$-0.541449\pi$
$277$	−1197.24	−0.259694	−0.129847	+	0.991534i	$0.541449\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2335.97	−0.495916	−0.247958	−	0.968771i	$-0.579759\pi$
$281$	−2335.97	−0.495916	−0.247958	+	0.968771i	$0.579759\pi$
$282$	0	0
$283$	1028.83	0.216104	0.108052	−	0.994145i	$-0.465539\pi$
$283$	1028.83	0.216104	0.108052	+	0.994145i	$0.465539\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3370.38	0.693196
$288$	0	0
$289$	4927.16	1.00288
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3375.36	0.673006	0.336503	−	0.941682i	$-0.390756\pi$
$293$	3375.36	0.673006	0.336503	+	0.941682i	$0.390756\pi$
$294$	0	0
$295$	−1468.91	−0.289910
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3603.67	−0.697010
$300$	0	0
$301$	−1147.82	−0.219799
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3508.29	0.658636
$306$	0	0
$307$	−6518.27	−1.21178	−0.605892	−	0.795547i	$-0.707184\pi$
$307$	−6518.27	−1.21178	−0.605892	+	0.795547i	$0.707184\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−7327.27	−1.33598	−0.667992	−	0.744168i	$-0.732846\pi$
$311$	−7327.27	−1.33598	−0.667992	+	0.744168i	$0.732846\pi$
$312$	0	0
$313$	1834.06	0.331206	0.165603	−	0.986193i	$-0.447043\pi$
$313$	1834.06	0.331206	0.165603	+	0.986193i	$0.447043\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3431.64	−0.608013	−0.304006	−	0.952670i	$-0.598324\pi$
$317$	−3431.64	−0.608013	−0.304006	+	0.952670i	$0.598324\pi$
$318$	0	0
$319$	10594.9	1.85956
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4429.22	0.762999
$324$	0	0
$325$	−878.060	−0.149865
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3327.55	0.557611
$330$	0	0
$331$	−7488.66	−1.24355	−0.621773	−	0.783197i	$-0.713588\pi$
$331$	−7488.66	−1.24355	−0.621773	+	0.783197i	$0.713588\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1463.48	−0.238682
$336$	0	0
$337$	−6037.63	−0.975936	−0.487968	−	0.872862i	$-0.662262\pi$
$337$	−6037.63	−0.975936	−0.487968	+	0.872862i	$0.662262\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	8857.15	1.40657
$342$	0	0
$343$	−4952.62	−0.779639
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4374.17	−0.676708	−0.338354	−	0.941019i	$-0.609870\pi$
$347$	−4374.17	−0.676708	−0.338354	+	0.941019i	$0.609870\pi$
$348$	0	0
$349$	−11777.8	−1.80645	−0.903223	−	0.429172i	$-0.858805\pi$
$349$	−11777.8	−1.80645	−0.903223	+	0.429172i	$0.858805\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1548.39	0.233463	0.116732	−	0.993163i	$-0.462758\pi$
$353$	1548.39	0.233463	0.116732	+	0.993163i	$0.462758\pi$
$354$	0	0
$355$	−3693.71	−0.552230
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4793.75	−0.704748	−0.352374	−	0.935859i	$-0.614626\pi$
$359$	−4793.75	−0.704748	−0.352374	+	0.935859i	$0.614626\pi$
$360$	0	0
$361$	−4865.33	−0.709335
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2266.07	−0.324963
$366$	0	0
$367$	−7183.40	−1.02172	−0.510859	−	0.859665i	$-0.670673\pi$
$367$	−7183.40	−1.02172	−0.510859	+	0.859665i	$0.670673\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1486.64	−0.208039
$372$	0	0
$373$	−6359.09	−0.882738	−0.441369	−	0.897326i	$-0.645507\pi$
$373$	−6359.09	−0.882738	−0.441369	+	0.897326i	$0.645507\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	10024.1	1.36941
$378$	0	0
$379$	7598.86	1.02989	0.514944	−	0.857224i	$-0.327813\pi$
$379$	7598.86	1.02989	0.514944	+	0.857224i	$0.327813\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	11674.6	1.55756	0.778778	−	0.627300i	$-0.215840\pi$
$383$	11674.6	1.55756	0.778778	+	0.627300i	$0.215840\pi$
$384$	0	0
$385$	1476.13	0.195404
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4324.51	0.563653	0.281827	−	0.959465i	$-0.409060\pi$
$389$	4324.51	0.563653	0.281827	+	0.959465i	$0.409060\pi$
$390$	0	0
$391$	10178.0	1.31643
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4463.80	0.568602
$396$	0	0
$397$	9834.45	1.24327	0.621633	−	0.783308i	$-0.286469\pi$
$397$	9834.45	1.24327	0.621633	+	0.783308i	$0.286469\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2593.68	−0.322998	−0.161499	−	0.986873i	$-0.551633\pi$
$401$	−2593.68	−0.322998	−0.161499	+	0.986873i	$0.551633\pi$
$402$	0	0
$403$	8379.96	1.03582
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−12590.7	−1.53341
$408$	0	0
$409$	−12202.2	−1.47520	−0.737602	−	0.675236i	$-0.764042\pi$
$409$	−12202.2	−1.47520	−0.737602	+	0.675236i	$0.764042\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2336.39	0.278369
$414$	0	0
$415$	332.057	0.0392771
$416$	0	0
$417$	0	0
$418$	0	0
$419$	13107.9	1.52831	0.764155	−	0.645033i	$-0.223156\pi$
$419$	13107.9	1.52831	0.764155	+	0.645033i	$0.223156\pi$
$420$	0	0
$421$	624.409	0.0722846	0.0361423	−	0.999347i	$-0.488493\pi$
$421$	624.409	0.0722846	0.0361423	+	0.999347i	$0.488493\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2479.94	0.283046
$426$	0	0
$427$	−5580.13	−0.632415
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6376.30	0.712612	0.356306	−	0.934369i	$-0.384036\pi$
$431$	6376.30	0.712612	0.356306	+	0.934369i	$0.384036\pi$
$432$	0	0
$433$	11244.7	1.24800	0.624000	−	0.781424i	$-0.285506\pi$
$433$	11244.7	1.24800	0.624000	+	0.781424i	$0.285506\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4581.29	0.501494
$438$	0	0
$439$	−11262.2	−1.22441	−0.612204	−	0.790700i	$-0.709717\pi$
$439$	−11262.2	−1.22441	−0.612204	+	0.790700i	$0.709717\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−14404.2	−1.54485	−0.772423	−	0.635109i	$-0.780955\pi$
$443$	−14404.2	−1.54485	−0.772423	+	0.635109i	$0.780955\pi$
$444$	0	0
$445$	4344.08	0.462762
$446$	0	0
$447$	0	0
$448$	0	0
$449$	11709.3	1.23073	0.615365	−	0.788242i	$-0.289008\pi$
$449$	11709.3	1.23073	0.615365	+	0.788242i	$0.289008\pi$
$450$	0	0
$451$	−15732.4	−1.64260
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1396.60	0.143898
$456$	0	0
$457$	15784.8	1.61572	0.807858	−	0.589377i	$-0.200627\pi$
$457$	15784.8	1.61572	0.807858	+	0.589377i	$0.200627\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−3956.19	−0.399692	−0.199846	−	0.979827i	$-0.564044\pi$
$461$	−3956.19	−0.399692	−0.199846	+	0.979827i	$0.564044\pi$
$462$	0	0
$463$	−9252.50	−0.928726	−0.464363	−	0.885645i	$-0.653717\pi$
$463$	−9252.50	−0.928726	−0.464363	+	0.885645i	$0.653717\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9116.74	−0.903367	−0.451684	−	0.892178i	$-0.649176\pi$
$467$	−9116.74	−0.903367	−0.451684	+	0.892178i	$0.649176\pi$
$468$	0	0
$469$	2327.75	0.229180
$470$	0	0
$471$	0	0
$472$	0	0
$473$	5357.87	0.520835
$474$	0	0
$475$	1116.26	0.107827
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2236.33	0.213321	0.106660	−	0.994296i	$-0.465984\pi$
$479$	2236.33	0.213321	0.106660	+	0.994296i	$0.465984\pi$
$480$	0	0
$481$	−11912.4	−1.12923
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−562.441	−0.0526580
$486$	0	0
$487$	−14371.9	−1.33728	−0.668638	−	0.743588i	$-0.733122\pi$
$487$	−14371.9	−1.33728	−0.668638	+	0.743588i	$0.733122\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−5423.17	−0.498461	−0.249230	−	0.968444i	$-0.580178\pi$
$491$	−5423.17	−0.498461	−0.249230	+	0.968444i	$0.580178\pi$
$492$	0	0
$493$	−28311.4	−2.58637
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5875.05	0.530246
$498$	0	0
$499$	4920.24	0.441403	0.220701	−	0.975341i	$-0.429165\pi$
$499$	4920.24	0.441403	0.220701	+	0.975341i	$0.429165\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−16004.1	−1.41866	−0.709330	−	0.704876i	$-0.751002\pi$
$503$	−16004.1	−1.41866	−0.709330	+	0.704876i	$0.751002\pi$
$504$	0	0
$505$	7905.64	0.696627
$506$	0	0
$507$	0	0
$508$	0	0
$509$	12973.0	1.12970	0.564852	−	0.825192i	$-0.308933\pi$
$509$	12973.0	1.12970	0.564852	+	0.825192i	$0.308933\pi$
$510$	0	0
$511$	3604.31	0.312026
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4525.83	−0.387247
$516$	0	0
$517$	−15532.5	−1.32132
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1933.35	−0.162575	−0.0812877	−	0.996691i	$-0.525903\pi$
$521$	−1933.35	−0.162575	−0.0812877	+	0.996691i	$0.525903\pi$
$522$	0	0
$523$	−12275.0	−1.02629	−0.513144	−	0.858303i	$-0.671519\pi$
$523$	−12275.0	−1.02629	−0.513144	+	0.858303i	$0.671519\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−23667.9	−1.95633
$528$	0	0
$529$	−1639.56	−0.134755
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−14884.8	−1.20963
$534$	0	0
$535$	5639.67	0.455746
$536$	0	0
$537$	0	0
$538$	0	0
$539$	10385.1	0.829904
$540$	0	0
$541$	6956.43	0.552829	0.276414	−	0.961039i	$-0.410854\pi$
$541$	6956.43	0.552829	0.276414	+	0.961039i	$0.410854\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	9738.25	0.765396
$546$	0	0
$547$	16829.2	1.31547	0.657737	−	0.753247i	$-0.271514\pi$
$547$	16829.2	1.31547	0.657737	+	0.753247i	$0.271514\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−12743.4	−0.985280
$552$	0	0
$553$	−7099.92	−0.545966
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1687.87	−0.128398	−0.0641988	−	0.997937i	$-0.520449\pi$
$557$	−1687.87	−0.128398	−0.0641988	+	0.997937i	$0.520449\pi$
$558$	0	0
$559$	5069.21	0.383551
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−19706.4	−1.47518	−0.737590	−	0.675249i	$-0.764036\pi$
$563$	−19706.4	−1.47518	−0.737590	+	0.675249i	$0.764036\pi$
$564$	0	0
$565$	8622.61	0.642046
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−19640.8	−1.44707	−0.723537	−	0.690286i	$-0.757485\pi$
$569$	−19640.8	−1.44707	−0.723537	+	0.690286i	$0.757485\pi$
$570$	0	0
$571$	5247.74	0.384608	0.192304	−	0.981335i	$-0.438404\pi$
$571$	5247.74	0.384608	0.192304	+	0.981335i	$0.438404\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2565.08	0.186037
$576$	0	0
$577$	25474.3	1.83797	0.918985	−	0.394293i	$-0.129011\pi$
$577$	25474.3	1.83797	0.918985	+	0.394293i	$0.129011\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−528.154	−0.0377135
$582$	0	0
$583$	6939.41	0.492969
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22502.7	1.58226	0.791129	−	0.611650i	$-0.209494\pi$
$587$	22502.7	1.58226	0.791129	+	0.611650i	$0.209494\pi$
$588$	0	0
$589$	−10653.3	−0.745266
$590$	0	0
$591$	0	0
$592$	0	0
$593$	4175.65	0.289163	0.144581	−	0.989493i	$-0.453816\pi$
$593$	4175.65	0.289163	0.144581	+	0.989493i	$0.453816\pi$
$594$	0	0
$595$	−3944.48	−0.271778
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−7096.57	−0.484070	−0.242035	−	0.970267i	$-0.577815\pi$
$599$	−7096.57	−0.484070	−0.242035	+	0.970267i	$0.577815\pi$
$600$	0	0
$601$	9677.83	0.656850	0.328425	−	0.944530i	$-0.393482\pi$
$601$	9677.83	0.656850	0.328425	+	0.944530i	$0.393482\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−235.362	−0.0158162
$606$	0	0
$607$	−24557.7	−1.64212	−0.821060	−	0.570842i	$-0.806617\pi$
$607$	−24557.7	−1.64212	−0.821060	+	0.570842i	$0.806617\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−14695.7	−0.973035
$612$	0	0
$613$	−22514.3	−1.48343	−0.741715	−	0.670716i	$-0.765987\pi$
$613$	−22514.3	−1.48343	−0.741715	+	0.670716i	$0.765987\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−14144.1	−0.922882	−0.461441	−	0.887171i	$-0.652667\pi$
$617$	−14144.1	−0.922882	−0.461441	+	0.887171i	$0.652667\pi$
$618$	0	0
$619$	24650.9	1.60065	0.800324	−	0.599567i	$-0.204661\pi$
$619$	24650.9	1.60065	0.800324	+	0.599567i	$0.204661\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−6909.50	−0.444339
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	33644.6	2.13275
$630$	0	0
$631$	21513.3	1.35726	0.678630	−	0.734481i	$-0.262574\pi$
$631$	21513.3	1.35726	0.678630	+	0.734481i	$0.262574\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	3228.60	0.201769
$636$	0	0
$637$	9825.61	0.611153
$638$	0	0
$639$	0	0
$640$	0	0
$641$	17483.0	1.07728	0.538640	−	0.842536i	$-0.318938\pi$
$641$	17483.0	1.07728	0.538640	+	0.842536i	$0.318938\pi$
$642$	0	0
$643$	25464.1	1.56175	0.780877	−	0.624685i	$-0.214773\pi$
$643$	25464.1	1.56175	0.780877	+	0.624685i	$0.214773\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	25974.4	1.57830	0.789148	−	0.614203i	$-0.210523\pi$
$647$	25974.4	1.57830	0.789148	+	0.614203i	$0.210523\pi$
$648$	0	0
$649$	−10905.9	−0.659622
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3532.46	0.211694	0.105847	−	0.994382i	$-0.466245\pi$
$653$	3532.46	0.211694	0.105847	+	0.994382i	$0.466245\pi$
$654$	0	0
$655$	−10279.6	−0.613216
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−29558.1	−1.74723	−0.873613	−	0.486622i	$-0.838229\pi$
$659$	−29558.1	−1.74723	−0.873613	+	0.486622i	$0.838229\pi$
$660$	0	0
$661$	−17632.5	−1.03755	−0.518777	−	0.854910i	$-0.673612\pi$
$661$	−17632.5	−1.03755	−0.518777	+	0.854910i	$0.673612\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1775.48	−0.103534
$666$	0	0
$667$	−29283.4	−1.69994
$668$	0	0
$669$	0	0
$670$	0	0
$671$	26047.2	1.49857
$672$	0	0
$673$	16119.9	0.923296	0.461648	−	0.887063i	$-0.347258\pi$
$673$	16119.9	0.923296	0.461648	+	0.887063i	$0.347258\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8987.07	−0.510194	−0.255097	−	0.966915i	$-0.582107\pi$
$677$	−8987.07	−0.510194	−0.255097	+	0.966915i	$0.582107\pi$
$678$	0	0
$679$	894.594	0.0505617
$680$	0	0
$681$	0	0
$682$	0	0
$683$	16085.7	0.901173	0.450586	−	0.892733i	$-0.351215\pi$
$683$	16085.7	0.901173	0.450586	+	0.892733i	$0.351215\pi$
$684$	0	0
$685$	−1222.74	−0.0682023
$686$	0	0
$687$	0	0
$688$	0	0
$689$	6565.54	0.363029
$690$	0	0
$691$	−2781.12	−0.153109	−0.0765547	−	0.997065i	$-0.524392\pi$
$691$	−2781.12	−0.153109	−0.0765547	+	0.997065i	$0.524392\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5343.35	0.291633
$696$	0	0
$697$	42039.9	2.28461
$698$	0	0
$699$	0	0
$700$	0	0
$701$	19690.3	1.06090	0.530450	−	0.847716i	$-0.322023\pi$
$701$	19690.3	1.06090	0.530450	+	0.847716i	$0.322023\pi$
$702$	0	0
$703$	15144.0	0.812472
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−12574.4	−0.668894
$708$	0	0
$709$	−14455.3	−0.765699	−0.382849	−	0.923811i	$-0.625057\pi$
$709$	−14455.3	−0.765699	−0.382849	+	0.923811i	$0.625057\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−24480.4	−1.28583
$714$	0	0
$715$	−6519.14	−0.340982
$716$	0	0
$717$	0	0
$718$	0	0
$719$	20466.1	1.06155	0.530776	−	0.847512i	$-0.321900\pi$
$719$	20466.1	1.06155	0.530776	+	0.847512i	$0.321900\pi$
$720$	0	0
$721$	7198.59	0.371830
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−7135.10	−0.365505
$726$	0	0
$727$	−14551.8	−0.742362	−0.371181	−	0.928561i	$-0.621047\pi$
$727$	−14551.8	−0.742362	−0.371181	+	0.928561i	$0.621047\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−14317.2	−0.724405
$732$	0	0
$733$	−7253.87	−0.365522	−0.182761	−	0.983157i	$-0.558503\pi$
$733$	−7253.87	−0.365522	−0.182761	+	0.983157i	$0.558503\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−10865.6	−0.543066
$738$	0	0
$739$	−19673.1	−0.979281	−0.489640	−	0.871925i	$-0.662872\pi$
$739$	−19673.1	−0.979281	−0.489640	+	0.871925i	$0.662872\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	16033.8	0.791686	0.395843	−	0.918318i	$-0.370453\pi$
$743$	16033.8	0.791686	0.395843	+	0.918318i	$0.370453\pi$
$744$	0	0
$745$	−6479.22	−0.318631
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−8970.20	−0.437602
$750$	0	0
$751$	4848.93	0.235606	0.117803	−	0.993037i	$-0.462415\pi$
$751$	4848.93	0.235606	0.117803	+	0.993037i	$0.462415\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−13636.6	−0.657333
$756$	0	0
$757$	31534.6	1.51406	0.757030	−	0.653380i	$-0.226650\pi$
$757$	31534.6	1.51406	0.757030	+	0.653380i	$0.226650\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−7873.75	−0.375063	−0.187532	−	0.982259i	$-0.560049\pi$
$761$	−7873.75	−0.375063	−0.187532	+	0.982259i	$0.560049\pi$
$762$	0	0
$763$	−15489.2	−0.734925
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−10318.4	−0.485755
$768$	0	0
$769$	−7134.10	−0.334541	−0.167271	−	0.985911i	$-0.553495\pi$
$769$	−7134.10	−0.334541	−0.167271	+	0.985911i	$0.553495\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−9224.32	−0.429205	−0.214603	−	0.976701i	$-0.568846\pi$
$773$	−9224.32	−0.429205	−0.214603	+	0.976701i	$0.568846\pi$
$774$	0	0
$775$	−5964.83	−0.276468
$776$	0	0
$777$	0	0
$778$	0	0
$779$	18922.9	0.870323
$780$	0	0
$781$	−27423.9	−1.25647
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−4338.50	−0.197258
$786$	0	0
$787$	30384.7	1.37623	0.688117	−	0.725599i	$-0.258437\pi$
$787$	30384.7	1.37623	0.688117	+	0.725599i	$0.258437\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−13714.7	−0.616485
$792$	0	0
$793$	24643.9	1.10357
$794$	0	0
$795$	0	0
$796$	0	0
$797$	9064.89	0.402879	0.201440	−	0.979501i	$-0.435438\pi$
$797$	9064.89	0.402879	0.201440	+	0.979501i	$0.435438\pi$
$798$	0	0
$799$	41505.7	1.83775
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−16824.4	−0.739377
$804$	0	0
$805$	−4079.91	−0.178631
$806$	0	0
$807$	0	0
$808$	0	0
$809$	11430.1	0.496737	0.248368	−	0.968666i	$-0.420106\pi$
$809$	11430.1	0.496737	0.248368	+	0.968666i	$0.420106\pi$
$810$	0	0
$811$	27003.1	1.16918	0.584591	−	0.811328i	$-0.301255\pi$
$811$	27003.1	1.16918	0.584591	+	0.811328i	$0.301255\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	6087.00	0.261618
$816$	0	0
$817$	−6444.40	−0.275962
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−32910.1	−1.39899	−0.699494	−	0.714638i	$-0.746591\pi$
$821$	−32910.1	−1.39899	−0.699494	+	0.714638i	$0.746591\pi$
$822$	0	0
$823$	29489.1	1.24900	0.624499	−	0.781026i	$-0.285303\pi$
$823$	29489.1	1.24900	0.624499	+	0.781026i	$0.285303\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	18645.6	0.784002	0.392001	−	0.919965i	$-0.371783\pi$
$827$	18645.6	0.784002	0.392001	+	0.919965i	$0.371783\pi$
$828$	0	0
$829$	−4919.89	−0.206122	−0.103061	−	0.994675i	$-0.532864\pi$
$829$	−4919.89	−0.206122	−0.103061	+	0.994675i	$0.532864\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−27750.9	−1.15427
$834$	0	0
$835$	−16333.8	−0.676951
$836$	0	0
$837$	0	0
$838$	0	0
$839$	45795.6	1.88444	0.942218	−	0.335001i	$-0.108737\pi$
$839$	45795.6	1.88444	0.942218	+	0.335001i	$0.108737\pi$
$840$	0	0
$841$	57066.5	2.33985
$842$	0	0
$843$	0	0
$844$	0	0
$845$	4817.09	0.196110
$846$	0	0
$847$	374.356	0.0151866
$848$	0	0
$849$	0	0
$850$	0	0
$851$	34799.8	1.40179
$852$	0	0
$853$	42136.7	1.69136	0.845681	−	0.533688i	$-0.179194\pi$
$853$	42136.7	1.69136	0.845681	+	0.533688i	$0.179194\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−27884.1	−1.11144	−0.555720	−	0.831370i	$-0.687557\pi$
$857$	−27884.1	−1.11144	−0.555720	+	0.831370i	$0.687557\pi$
$858$	0	0
$859$	28863.7	1.14647	0.573234	−	0.819392i	$-0.305689\pi$
$859$	28863.7	1.14647	0.573234	+	0.819392i	$0.305689\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−13061.5	−0.515203	−0.257601	−	0.966251i	$-0.582932\pi$
$863$	−13061.5	−0.515203	−0.257601	+	0.966251i	$0.582932\pi$
$864$	0	0
$865$	7108.56	0.279420
$866$	0	0
$867$	0	0
$868$	0	0
$869$	33141.4	1.29372
$870$	0	0
$871$	−10280.2	−0.399921
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−994.097	−0.0384076
$876$	0	0
$877$	6501.20	0.250319	0.125160	−	0.992137i	$-0.460056\pi$
$877$	6501.20	0.250319	0.125160	+	0.992137i	$0.460056\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−45486.0	−1.73946	−0.869729	−	0.493529i	$-0.835707\pi$
$881$	−45486.0	−1.73946	−0.869729	+	0.493529i	$0.835707\pi$
$882$	0	0
$883$	5963.30	0.227272	0.113636	−	0.993522i	$-0.463750\pi$
$883$	5963.30	0.227272	0.113636	+	0.993522i	$0.463750\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	14524.3	0.549807	0.274903	−	0.961472i	$-0.411354\pi$
$887$	14524.3	0.549807	0.274903	+	0.961472i	$0.411354\pi$
$888$	0	0
$889$	−5135.27	−0.193736
$890$	0	0
$891$	0	0
$892$	0	0
$893$	18682.4	0.700093
$894$	0	0
$895$	−4747.46	−0.177307
$896$	0	0
$897$	0	0
$898$	0	0
$899$	68095.5	2.52626
$900$	0	0
$901$	−18543.3	−0.685646
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−8741.27	−0.321072
$906$	0	0
$907$	−25024.4	−0.916119	−0.458060	−	0.888921i	$-0.651455\pi$
$907$	−25024.4	−0.916119	−0.458060	+	0.888921i	$0.651455\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−27898.0	−1.01460	−0.507300	−	0.861770i	$-0.669356\pi$
$911$	−27898.0	−1.01460	−0.507300	+	0.861770i	$0.669356\pi$
$912$	0	0
$913$	2465.35	0.0893659
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16350.3	0.588804
$918$	0	0
$919$	19703.7	0.707253	0.353627	−	0.935387i	$-0.384948\pi$
$919$	19703.7	0.707253	0.353627	+	0.935387i	$0.384948\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−25946.4	−0.925283
$924$	0	0
$925$	8479.20	0.301399
$926$	0	0
$927$	0	0
$928$	0	0
$929$	17982.7	0.635083	0.317542	−	0.948244i	$-0.397143\pi$
$929$	17982.7	0.635083	0.317542	+	0.948244i	$0.397143\pi$
$930$	0	0
$931$	−12491.1	−0.439721
$932$	0	0
$933$	0	0
$934$	0	0
$935$	18412.3	0.644006
$936$	0	0
$937$	25616.7	0.893128	0.446564	−	0.894752i	$-0.352647\pi$
$937$	25616.7	0.893128	0.446564	+	0.894752i	$0.352647\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−18854.2	−0.653166	−0.326583	−	0.945169i	$-0.605897\pi$
$941$	−18854.2	−0.653166	−0.326583	+	0.945169i	$0.605897\pi$
$942$	0	0
$943$	43483.2	1.50160
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−24298.8	−0.833798	−0.416899	−	0.908953i	$-0.636883\pi$
$947$	−24298.8	−0.833798	−0.416899	+	0.908953i	$0.636883\pi$
$948$	0	0
$949$	−15918.0	−0.544487
$950$	0	0
$951$	0	0
$952$	0	0
$953$	10472.4	0.355963	0.177982	−	0.984034i	$-0.443043\pi$
$953$	10472.4	0.355963	0.177982	+	0.984034i	$0.443043\pi$
$954$	0	0
$955$	7091.84	0.240300
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1944.84	0.0654871
$960$	0	0
$961$	27135.6	0.910867
$962$	0	0
$963$	0	0
$964$	0	0
$965$	18667.4	0.622721
$966$	0	0
$967$	−56840.3	−1.89024	−0.945120	−	0.326723i	$-0.894056\pi$
$967$	−56840.3	−1.89024	−0.945120	+	0.326723i	$0.894056\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	52095.6	1.72176	0.860879	−	0.508810i	$-0.169914\pi$
$971$	52095.6	1.72176	0.860879	+	0.508810i	$0.169914\pi$
$972$	0	0
$973$	−8498.90	−0.280023
$974$	0	0
$975$	0	0
$976$	0	0
$977$	24247.8	0.794017	0.397008	−	0.917815i	$-0.370048\pi$
$977$	24247.8	0.794017	0.397008	+	0.917815i	$0.370048\pi$
$978$	0	0
$979$	32252.5	1.05291
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−3529.10	−0.114508	−0.0572538	−	0.998360i	$-0.518234\pi$
$983$	−3529.10	−0.114508	−0.0572538	+	0.998360i	$0.518234\pi$
$984$	0	0
$985$	11473.3	0.371138
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−14808.7	−0.476127
$990$	0	0
$991$	−2131.22	−0.0683151	−0.0341576	−	0.999416i	$-0.510875\pi$
$991$	−2131.22	−0.0683151	−0.0341576	+	0.999416i	$0.510875\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	18569.7	0.591656
$996$	0	0
$997$	20296.8	0.644739	0.322370	−	0.946614i	$-0.395521\pi$
$997$	20296.8	0.644739	0.322370	+	0.946614i	$0.395521\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.a.bg.1.2		3
3.2	odd	2		2160.4.a.bo.1.2		3
4.3	odd	2		1080.4.a.g.1.2	✓	3
12.11	even	2		1080.4.a.m.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.a.g.1.2	✓	3	4.3	odd	2
1080.4.a.m.1.2	yes	3	12.11	even	2
2160.4.a.bg.1.2		3	1.1	even	1	trivial
2160.4.a.bo.1.2		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} +7.95278 q^{7} +O(q^{10})\)	\(q-5.00000 q^{5} +7.95278 q^{7} -37.1224 q^{11} -35.1224 q^{13} +99.1976 q^{17} +44.6505 q^{19} +102.603 q^{23} +25.0000 q^{25} -285.404 q^{29} -238.593 q^{31} -39.7639 q^{35} +339.168 q^{37} +423.799 q^{41} -144.330 q^{43} +418.414 q^{47} -279.753 q^{49} -186.933 q^{53} +185.612 q^{55} +293.783 q^{59} -701.657 q^{61} +175.612 q^{65} +292.697 q^{67} +738.742 q^{71} +453.214 q^{73} -295.226 q^{77} -892.759 q^{79} -66.4113 q^{83} -495.988 q^{85} -868.816 q^{89} -279.321 q^{91} -223.253 q^{95} +112.488 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 15 q^{5} - 6 q^{7}+O(q^{10}) \) 3 * q - 15 * q^5 - 6 * q^7	\( 3 q - 15 q^{5} - 6 q^{7} + 12 q^{11} + 18 q^{13} + 21 q^{17} - 57 q^{19} + 87 q^{23} + 75 q^{25} - 138 q^{29} - 117 q^{31} + 30 q^{35} + 150 q^{37} - 180 q^{43} + 684 q^{47} - 81 q^{49} - 87 q^{53} - 60 q^{55} + 714 q^{59} - 513 q^{61} - 90 q^{65} + 174 q^{67} + 768 q^{71} - 252 q^{73} - 888 q^{77} - 207 q^{79} + 1689 q^{83} - 105 q^{85} - 312 q^{89} - 900 q^{91} + 285 q^{95} - 1080 q^{97}+O(q^{100}) \) 3 * q - 15 * q^5 - 6 * q^7 + 12 * q^11 + 18 * q^13 + 21 * q^17 - 57 * q^19 + 87 * q^23 + 75 * q^25 - 138 * q^29 - 117 * q^31 + 30 * q^35 + 150 * q^37 - 180 * q^43 + 684 * q^47 - 81 * q^49 - 87 * q^53 - 60 * q^55 + 714 * q^59 - 513 * q^61 - 90 * q^65 + 174 * q^67 + 768 * q^71 - 252 * q^73 - 888 * q^77 - 207 * q^79 + 1689 * q^83 - 105 * q^85 - 312 * q^89 - 900 * q^91 + 285 * q^95 - 1080 * q^97

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