LMFDB - Embedded newform 2160.4.a.w.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:	$2$
Coefficient field:	$\Q(\sqrt{401}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 100 $ x^2 - x - 100
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 270)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-9.51249$ 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)

$f(q)$	$=$	$q-5.00000 q^{5} +29.5375 q^{7} +O(q^{10})$	$q-5.00000 q^{5} +29.5375 q^{7} +46.5375 q^{11} +92.0750 q^{13} +4.53748 q^{17} -87.5375 q^{19} +160.687 q^{23} +25.0000 q^{25} +241.762 q^{29} +2.68738 q^{31} -147.687 q^{35} -20.6124 q^{37} +501.225 q^{41} -294.687 q^{43} +478.987 q^{47} +529.463 q^{49} +243.075 q^{53} -232.687 q^{55} -383.700 q^{59} +132.612 q^{61} -460.375 q^{65} -582.612 q^{67} -566.775 q^{71} -839.388 q^{73} +1374.60 q^{77} -451.775 q^{79} -301.049 q^{83} -22.6874 q^{85} -739.349 q^{89} +2719.66 q^{91} +437.687 q^{95} -1146.14 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 10 q^{5} - q^{7}+O(q^{10}) $ 2 * q - 10 * q^5 - q^7	$ 2 q - 10 q^{5} - q^{7} + 33 q^{11} + 64 q^{13} - 51 q^{17} - 115 q^{19} + 21 q^{23} + 50 q^{25} + 63 q^{29} - 295 q^{31} + 5 q^{35} + 139 q^{37} + 642 q^{41} - 289 q^{43} + 177 q^{47} + 1119 q^{49} + 366 q^{53} - 165 q^{55} - 1248 q^{59} + 85 q^{61} - 320 q^{65} - 985 q^{67} - 1494 q^{71} - 1859 q^{73} + 1788 q^{77} - 1264 q^{79} + 1080 q^{83} + 255 q^{85} + 684 q^{89} + 3577 q^{91} + 575 q^{95} - 1271 q^{97}+O(q^{100}) $ 2 * q - 10 * q^5 - q^7 + 33 * q^11 + 64 * q^13 - 51 * q^17 - 115 * q^19 + 21 * q^23 + 50 * q^25 + 63 * q^29 - 295 * q^31 + 5 * q^35 + 139 * q^37 + 642 * q^41 - 289 * q^43 + 177 * q^47 + 1119 * q^49 + 366 * q^53 - 165 * q^55 - 1248 * q^59 + 85 * q^61 - 320 * q^65 - 985 * q^67 - 1494 * q^71 - 1859 * q^73 + 1788 * q^77 - 1264 * q^79 + 1080 * q^83 + 255 * q^85 + 684 * q^89 + 3577 * q^91 + 575 * q^95 - 1271 * 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
$7$	29.5375	1.59487	0.797437	−	0.603402i	$-0.206189\pi$
$7$	29.5375	1.59487	0.797437	+	0.603402i	$0.206189\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	46.5375	1.27560	0.637799	−	0.770203i	$-0.279845\pi$
$11$	46.5375	1.27560	0.637799	+	0.770203i	$0.279845\pi$
$12$	0	0
$13$	92.0750	1.96438	0.982192	−	0.187879i	$-0.0601612\pi$
$13$	92.0750	1.96438	0.982192	+	0.187879i	$0.0601612\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.53748	0.0647353	0.0323676	−	0.999476i	$-0.489695\pi$
$17$	4.53748	0.0647353	0.0323676	+	0.999476i	$0.489695\pi$
$18$	0	0
$19$	−87.5375	−1.05697	−0.528486	−	0.848942i	$-0.677240\pi$
$19$	−87.5375	−1.05697	−0.528486	+	0.848942i	$0.677240\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	160.687	1.45677	0.728383	−	0.685170i	$-0.240272\pi$
$23$	160.687	1.45677	0.728383	+	0.685170i	$0.240272\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	241.762	1.54807	0.774037	−	0.633141i	$-0.218234\pi$
$29$	241.762	1.54807	0.774037	+	0.633141i	$0.218234\pi$
$30$	0	0
$31$	2.68738	0.0155699	0.00778497	−	0.999970i	$-0.497522\pi$
$31$	2.68738	0.0155699	0.00778497	+	0.999970i	$0.497522\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−147.687	−0.713249
$36$	0	0
$37$	−20.6124	−0.0915855	−0.0457927	−	0.998951i	$-0.514581\pi$
$37$	−20.6124	−0.0915855	−0.0457927	+	0.998951i	$0.514581\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	501.225	1.90922	0.954612	−	0.297853i	$-0.0962704\pi$
$41$	501.225	1.90922	0.954612	+	0.297853i	$0.0962704\pi$
$42$	0	0
$43$	−294.687	−1.04510	−0.522551	−	0.852608i	$-0.675020\pi$
$43$	−294.687	−1.04510	−0.522551	+	0.852608i	$0.675020\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	478.987	1.48654	0.743271	−	0.668991i	$-0.233273\pi$
$47$	478.987	1.48654	0.743271	+	0.668991i	$0.233273\pi$
$48$	0	0
$49$	529.463	1.54362
$50$	0	0
$51$	0	0
$52$	0	0
$53$	243.075	0.629979	0.314990	−	0.949095i	$-0.397999\pi$
$53$	243.075	0.629979	0.314990	+	0.949095i	$0.397999\pi$
$54$	0	0
$55$	−232.687	−0.570465
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−383.700	−0.846670	−0.423335	−	0.905973i	$-0.639141\pi$
$59$	−383.700	−0.846670	−0.423335	+	0.905973i	$0.639141\pi$
$60$	0	0
$61$	132.612	0.278349	0.139174	−	0.990268i	$-0.455555\pi$
$61$	132.612	0.278349	0.139174	+	0.990268i	$0.455555\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−460.375	−0.878499
$66$	0	0
$67$	−582.612	−1.06235	−0.531175	−	0.847262i	$-0.678249\pi$
$67$	−582.612	−1.06235	−0.531175	+	0.847262i	$0.678249\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−566.775	−0.947378	−0.473689	−	0.880692i	$-0.657078\pi$
$71$	−566.775	−0.947378	−0.473689	+	0.880692i	$0.657078\pi$
$72$	0	0
$73$	−839.388	−1.34579	−0.672896	−	0.739737i	$-0.734950\pi$
$73$	−839.388	−1.34579	−0.672896	+	0.739737i	$0.734950\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1374.60	2.03442
$78$	0	0
$79$	−451.775	−0.643401	−0.321700	−	0.946841i	$-0.604254\pi$
$79$	−451.775	−0.643401	−0.321700	+	0.946841i	$0.604254\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−301.049	−0.398126	−0.199063	−	0.979987i	$-0.563790\pi$
$83$	−301.049	−0.398126	−0.199063	+	0.979987i	$0.563790\pi$
$84$	0	0
$85$	−22.6874	−0.0289505
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−739.349	−0.880571	−0.440286	−	0.897858i	$-0.645123\pi$
$89$	−739.349	−0.880571	−0.440286	+	0.897858i	$0.645123\pi$
$90$	0	0
$91$	2719.66	3.13295
$92$	0	0
$93$	0	0
$94$	0	0
$95$	437.687	0.472692
$96$	0	0
$97$	−1146.14	−1.19972	−0.599859	−	0.800106i	$-0.704777\pi$
$97$	−1146.14	−1.19972	−0.599859	+	0.800106i	$0.704777\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	511.463	0.503885	0.251943	−	0.967742i	$-0.418931\pi$
$101$	511.463	0.503885	0.251943	+	0.967742i	$0.418931\pi$
$102$	0	0
$103$	−421.512	−0.403231	−0.201616	−	0.979465i	$-0.564619\pi$
$103$	−421.512	−0.403231	−0.201616	+	0.979465i	$0.564619\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1639.50	−1.48127	−0.740637	−	0.671905i	$-0.765476\pi$
$107$	−1639.50	−1.48127	−0.740637	+	0.671905i	$0.765476\pi$
$108$	0	0
$109$	−690.475	−0.606748	−0.303374	−	0.952872i	$-0.598113\pi$
$109$	−690.475	−0.606748	−0.303374	+	0.952872i	$0.598113\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−970.238	−0.807719	−0.403860	−	0.914821i	$-0.632332\pi$
$113$	−970.238	−0.807719	−0.403860	+	0.914821i	$0.632332\pi$
$114$	0	0
$115$	−803.437	−0.651486
$116$	0	0
$117$	0	0
$118$	0	0
$119$	134.026	0.103245
$120$	0	0
$121$	834.737	0.627150
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−805.024	−0.562475	−0.281237	−	0.959638i	$-0.590745\pi$
$127$	−805.024	−0.562475	−0.281237	+	0.959638i	$0.590745\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	214.687	0.143186	0.0715928	−	0.997434i	$-0.477192\pi$
$131$	214.687	0.143186	0.0715928	+	0.997434i	$0.477192\pi$
$132$	0	0
$133$	−2585.64	−1.68574
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−367.799	−0.229366	−0.114683	−	0.993402i	$-0.536585\pi$
$137$	−367.799	−0.229366	−0.114683	+	0.993402i	$0.536585\pi$
$138$	0	0
$139$	1427.31	0.870956	0.435478	−	0.900199i	$-0.356579\pi$
$139$	1427.31	0.870956	0.435478	+	0.900199i	$0.356579\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4284.94	2.50576
$144$	0	0
$145$	−1208.81	−0.692320
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1757.74	0.966439	0.483219	−	0.875499i	$-0.339467\pi$
$149$	1757.74	0.966439	0.483219	+	0.875499i	$0.339467\pi$
$150$	0	0
$151$	537.775	0.289825	0.144912	−	0.989445i	$-0.453710\pi$
$151$	537.775	0.289825	0.144912	+	0.989445i	$0.453710\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−13.4369	−0.00696309
$156$	0	0
$157$	671.788	0.341494	0.170747	−	0.985315i	$-0.445382\pi$
$157$	671.788	0.341494	0.170747	+	0.985315i	$0.445382\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4746.30	2.32336
$162$	0	0
$163$	2501.42	1.20200	0.601002	−	0.799248i	$-0.294768\pi$
$163$	2501.42	1.20200	0.601002	+	0.799248i	$0.294768\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	751.649	0.348289	0.174145	−	0.984720i	$-0.444284\pi$
$167$	751.649	0.348289	0.174145	+	0.984720i	$0.444284\pi$
$168$	0	0
$169$	6280.80	2.85881
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2374.27	−1.04343	−0.521713	−	0.853121i	$-0.674707\pi$
$173$	−2374.27	−1.04343	−0.521713	+	0.853121i	$0.674707\pi$
$174$	0	0
$175$	738.437	0.318975
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2424.52	−1.01239	−0.506194	−	0.862420i	$-0.668948\pi$
$179$	−2424.52	−1.01239	−0.506194	+	0.862420i	$0.668948\pi$
$180$	0	0
$181$	1179.46	0.484357	0.242179	−	0.970232i	$-0.422138\pi$
$181$	1179.46	0.484357	0.242179	+	0.970232i	$0.422138\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	103.062	0.0409583
$186$	0	0
$187$	211.163	0.0825762
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−344.475	−0.130499	−0.0652496	−	0.997869i	$-0.520784\pi$
$191$	−344.475	−0.130499	−0.0652496	+	0.997869i	$0.520784\pi$
$192$	0	0
$193$	1941.84	0.724231	0.362115	−	0.932133i	$-0.382055\pi$
$193$	1941.84	0.724231	0.362115	+	0.932133i	$0.382055\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1780.95	0.644099	0.322049	−	0.946723i	$-0.395628\pi$
$197$	1780.95	0.644099	0.322049	+	0.946723i	$0.395628\pi$
$198$	0	0
$199$	808.375	0.287961	0.143980	−	0.989581i	$-0.454010\pi$
$199$	808.375	0.287961	0.143980	+	0.989581i	$0.454010\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	7141.05	2.46898
$204$	0	0
$205$	−2506.12	−0.853831
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4073.77	−1.34827
$210$	0	0
$211$	−1223.04	−0.399039	−0.199520	−	0.979894i	$-0.563938\pi$
$211$	−1223.04	−0.399039	−0.199520	+	0.979894i	$0.563938\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1473.44	0.467384
$216$	0	0
$217$	79.3785	0.0248321
$218$	0	0
$219$	0	0
$220$	0	0
$221$	417.788	0.127165
$222$	0	0
$223$	−1061.00	−0.318610	−0.159305	−	0.987229i	$-0.550925\pi$
$223$	−1061.00	−0.318610	−0.159305	+	0.987229i	$0.550925\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3573.07	−1.04473	−0.522364	−	0.852723i	$-0.674950\pi$
$227$	−3573.07	−1.04473	−0.522364	+	0.852723i	$0.674950\pi$
$228$	0	0
$229$	1001.82	0.289094	0.144547	−	0.989498i	$-0.453828\pi$
$229$	1001.82	0.289094	0.144547	+	0.989498i	$0.453828\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6082.42	1.71018	0.855091	−	0.518477i	$-0.173501\pi$
$233$	6082.42	1.71018	0.855091	+	0.518477i	$0.173501\pi$
$234$	0	0
$235$	−2394.94	−0.664802
$236$	0	0
$237$	0	0
$238$	0	0
$239$	212.399	0.0574850	0.0287425	−	0.999587i	$-0.490850\pi$
$239$	212.399	0.0574850	0.0287425	+	0.999587i	$0.490850\pi$
$240$	0	0
$241$	−2944.45	−0.787006	−0.393503	−	0.919323i	$-0.628737\pi$
$241$	−2944.45	−0.787006	−0.393503	+	0.919323i	$0.628737\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2647.31	−0.690329
$246$	0	0
$247$	−8060.01	−2.07630
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1130.36	−0.284254	−0.142127	−	0.989848i	$-0.545394\pi$
$251$	−1130.36	−0.284254	−0.142127	+	0.989848i	$0.545394\pi$
$252$	0	0
$253$	7477.99	1.85825
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1218.34	0.295711	0.147856	−	0.989009i	$-0.452763\pi$
$257$	1218.34	0.295711	0.147856	+	0.989009i	$0.452763\pi$
$258$	0	0
$259$	−608.839	−0.146067
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6937.50	1.62656	0.813279	−	0.581875i	$-0.197681\pi$
$263$	6937.50	1.62656	0.813279	+	0.581875i	$0.197681\pi$
$264$	0	0
$265$	−1215.37	−0.281735
$266$	0	0
$267$	0	0
$268$	0	0
$269$	515.214	0.116778	0.0583888	−	0.998294i	$-0.481404\pi$
$269$	515.214	0.116778	0.0583888	+	0.998294i	$0.481404\pi$
$270$	0	0
$271$	652.411	0.146240	0.0731202	−	0.997323i	$-0.476704\pi$
$271$	652.411	0.146240	0.0731202	+	0.997323i	$0.476704\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1163.44	0.255120
$276$	0	0
$277$	2938.85	0.637467	0.318733	−	0.947844i	$-0.396743\pi$
$277$	2938.85	0.637467	0.318733	+	0.947844i	$0.396743\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5351.77	1.13616	0.568078	−	0.822974i	$-0.307687\pi$
$281$	5351.77	1.13616	0.568078	+	0.822974i	$0.307687\pi$
$282$	0	0
$283$	−1479.95	−0.310862	−0.155431	−	0.987847i	$-0.549677\pi$
$283$	−1479.95	−0.310862	−0.155431	+	0.987847i	$0.549677\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	14804.9	3.04497
$288$	0	0
$289$	−4892.41	−0.995809
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−5178.90	−1.03261	−0.516305	−	0.856405i	$-0.672693\pi$
$293$	−5178.90	−1.03261	−0.516305	+	0.856405i	$0.672693\pi$
$294$	0	0
$295$	1918.50	0.378642
$296$	0	0
$297$	0	0
$298$	0	0
$299$	14795.3	2.86165
$300$	0	0
$301$	−8704.32	−1.66681
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−663.062	−0.124481
$306$	0	0
$307$	−10306.0	−1.91594	−0.957970	−	0.286868i	$-0.907386\pi$
$307$	−10306.0	−1.91594	−0.957970	+	0.286868i	$0.907386\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2062.65	−0.376084	−0.188042	−	0.982161i	$-0.560214\pi$
$311$	−2062.65	−0.376084	−0.188042	+	0.982161i	$0.560214\pi$
$312$	0	0
$313$	9247.96	1.67005	0.835025	−	0.550212i	$-0.185453\pi$
$313$	9247.96	1.67005	0.835025	+	0.550212i	$0.185453\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	610.347	0.108140	0.0540702	−	0.998537i	$-0.482781\pi$
$317$	610.347	0.108140	0.0540702	+	0.998537i	$0.482781\pi$
$318$	0	0
$319$	11251.0	1.97472
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−397.199	−0.0684234
$324$	0	0
$325$	2301.87	0.392877
$326$	0	0
$327$	0	0
$328$	0	0
$329$	14148.1	2.37085
$330$	0	0
$331$	2089.26	0.346937	0.173469	−	0.984839i	$-0.444502\pi$
$331$	2089.26	0.346937	0.173469	+	0.984839i	$0.444502\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2913.06	0.475097
$336$	0	0
$337$	−9426.36	−1.52370	−0.761849	−	0.647754i	$-0.775708\pi$
$337$	−9426.36	−1.52370	−0.761849	+	0.647754i	$0.775708\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	125.064	0.0198610
$342$	0	0
$343$	5507.63	0.867009
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1221.22	−0.188930	−0.0944651	−	0.995528i	$-0.530114\pi$
$347$	−1221.22	−0.188930	−0.0944651	+	0.995528i	$0.530114\pi$
$348$	0	0
$349$	−4188.29	−0.642389	−0.321195	−	0.947013i	$-0.604084\pi$
$349$	−4188.29	−0.642389	−0.321195	+	0.947013i	$0.604084\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4469.36	0.673882	0.336941	−	0.941526i	$-0.390608\pi$
$353$	4469.36	0.673882	0.336941	+	0.941526i	$0.390608\pi$
$354$	0	0
$355$	2833.88	0.423680
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−643.572	−0.0946140	−0.0473070	−	0.998880i	$-0.515064\pi$
$359$	−643.572	−0.0946140	−0.0473070	+	0.998880i	$0.515064\pi$
$360$	0	0
$361$	803.810	0.117191
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4196.94	0.601857
$366$	0	0
$367$	5934.74	0.844116	0.422058	−	0.906569i	$-0.361308\pi$
$367$	5934.74	0.844116	0.422058	+	0.906569i	$0.361308\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	7179.82	1.00474
$372$	0	0
$373$	−6187.83	−0.858964	−0.429482	−	0.903075i	$-0.641304\pi$
$373$	−6187.83	−0.858964	−0.429482	+	0.903075i	$0.641304\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	22260.3	3.04101
$378$	0	0
$379$	−10428.5	−1.41340	−0.706699	−	0.707515i	$-0.749816\pi$
$379$	−10428.5	−1.41340	−0.706699	+	0.707515i	$0.749816\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−10246.7	−1.36705	−0.683526	−	0.729926i	$-0.739554\pi$
$383$	−10246.7	−1.36705	−0.683526	+	0.729926i	$0.739554\pi$
$384$	0	0
$385$	−6873.00	−0.909819
$386$	0	0
$387$	0	0
$388$	0	0
$389$	8508.48	1.10899	0.554495	−	0.832187i	$-0.312912\pi$
$389$	8508.48	1.10899	0.554495	+	0.832187i	$0.312912\pi$
$390$	0	0
$391$	729.115	0.0943042
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2258.88	0.287738
$396$	0	0
$397$	3775.96	0.477356	0.238678	−	0.971099i	$-0.423286\pi$
$397$	3775.96	0.477356	0.238678	+	0.971099i	$0.423286\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−7556.02	−0.940971	−0.470486	−	0.882408i	$-0.655921\pi$
$401$	−7556.02	−0.940971	−0.470486	+	0.882408i	$0.655921\pi$
$402$	0	0
$403$	247.441	0.0305854
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−959.250	−0.116826
$408$	0	0
$409$	8963.22	1.08362	0.541812	−	0.840499i	$-0.317738\pi$
$409$	8963.22	1.08362	0.541812	+	0.840499i	$0.317738\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−11333.5	−1.35033
$414$	0	0
$415$	1505.25	0.178047
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−14153.8	−1.65026	−0.825130	−	0.564943i	$-0.808898\pi$
$419$	−14153.8	−1.65026	−0.825130	+	0.564943i	$0.808898\pi$
$420$	0	0
$421$	−14727.3	−1.70490	−0.852451	−	0.522807i	$-0.824885\pi$
$421$	−14727.3	−1.70490	−0.852451	+	0.522807i	$0.824885\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	113.437	0.0129471
$426$	0	0
$427$	3917.04	0.443931
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3048.98	0.340752	0.170376	−	0.985379i	$-0.445502\pi$
$431$	3048.98	0.340752	0.170376	+	0.985379i	$0.445502\pi$
$432$	0	0
$433$	17218.4	1.91100	0.955500	−	0.294990i	$-0.0953164\pi$
$433$	17218.4	1.91100	0.955500	+	0.294990i	$0.0953164\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−14066.2	−1.53976
$438$	0	0
$439$	−1667.91	−0.181332	−0.0906660	−	0.995881i	$-0.528900\pi$
$439$	−1667.91	−0.181332	−0.0906660	+	0.995881i	$0.528900\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9223.20	−0.989182	−0.494591	−	0.869126i	$-0.664682\pi$
$443$	−9223.20	−0.989182	−0.494591	+	0.869126i	$0.664682\pi$
$444$	0	0
$445$	3696.75	0.393803
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2636.18	0.277080	0.138540	−	0.990357i	$-0.455759\pi$
$449$	2636.18	0.277080	0.138540	+	0.990357i	$0.455759\pi$
$450$	0	0
$451$	23325.7	2.43540
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−13598.3	−1.40110
$456$	0	0
$457$	−3859.67	−0.395072	−0.197536	−	0.980296i	$-0.563294\pi$
$457$	−3859.67	−0.395072	−0.197536	+	0.980296i	$0.563294\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6615.07	0.668318	0.334159	−	0.942517i	$-0.391548\pi$
$461$	6615.07	0.668318	0.334159	+	0.942517i	$0.391548\pi$
$462$	0	0
$463$	17571.7	1.76377	0.881887	−	0.471460i	$-0.156273\pi$
$463$	17571.7	1.76377	0.881887	+	0.471460i	$0.156273\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10076.8	−0.998502	−0.499251	−	0.866457i	$-0.666391\pi$
$467$	−10076.8	−0.998502	−0.499251	+	0.866457i	$0.666391\pi$
$468$	0	0
$469$	−17208.9	−1.69431
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−13714.0	−1.33313
$474$	0	0
$475$	−2188.44	−0.211394
$476$	0	0
$477$	0	0
$478$	0	0
$479$	17945.8	1.71182	0.855912	−	0.517122i	$-0.172997\pi$
$479$	17945.8	1.71182	0.855912	+	0.517122i	$0.172997\pi$
$480$	0	0
$481$	−1897.89	−0.179909
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5730.69	0.536530
$486$	0	0
$487$	−17356.9	−1.61502	−0.807511	−	0.589853i	$-0.799186\pi$
$487$	−17356.9	−1.61502	−0.807511	+	0.589853i	$0.799186\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	8393.17	0.771443	0.385722	−	0.922615i	$-0.373953\pi$
$491$	8393.17	0.771443	0.385722	+	0.922615i	$0.373953\pi$
$492$	0	0
$493$	1096.99	0.100215
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−16741.1	−1.51095
$498$	0	0
$499$	−16421.1	−1.47317	−0.736585	−	0.676345i	$-0.763563\pi$
$499$	−16421.1	−1.47317	−0.736585	+	0.676345i	$0.763563\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−16014.9	−1.41962	−0.709808	−	0.704395i	$-0.751218\pi$
$503$	−16014.9	−1.41962	−0.709808	+	0.704395i	$0.751218\pi$
$504$	0	0
$505$	−2557.31	−0.225344
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−13695.9	−1.19265	−0.596327	−	0.802741i	$-0.703374\pi$
$509$	−13695.9	−1.19265	−0.596327	+	0.802741i	$0.703374\pi$
$510$	0	0
$511$	−24793.4	−2.14637
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2107.56	0.180330
$516$	0	0
$517$	22290.9	1.89623
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5887.12	0.495047	0.247523	−	0.968882i	$-0.420383\pi$
$521$	5887.12	0.495047	0.247523	+	0.968882i	$0.420383\pi$
$522$	0	0
$523$	−8544.57	−0.714394	−0.357197	−	0.934029i	$-0.616267\pi$
$523$	−8544.57	−0.714394	−0.357197	+	0.934029i	$0.616267\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12.1939	0.00100792
$528$	0	0
$529$	13653.4	1.12217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	46150.3	3.75045
$534$	0	0
$535$	8197.50	0.662446
$536$	0	0
$537$	0	0
$538$	0	0
$539$	24639.8	1.96904
$540$	0	0
$541$	10236.5	0.813499	0.406749	−	0.913540i	$-0.366662\pi$
$541$	10236.5	0.813499	0.406749	+	0.913540i	$0.366662\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3452.38	0.271346
$546$	0	0
$547$	4785.31	0.374050	0.187025	−	0.982355i	$-0.440115\pi$
$547$	4785.31	0.374050	0.187025	+	0.982355i	$0.440115\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−21163.3	−1.63627
$552$	0	0
$553$	−13344.3	−1.02614
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2031.75	−0.154557	−0.0772783	−	0.997010i	$-0.524623\pi$
$557$	−2031.75	−0.154557	−0.0772783	+	0.997010i	$0.524623\pi$
$558$	0	0
$559$	−27133.3	−2.05298
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12162.4	0.910449	0.455224	−	0.890377i	$-0.349559\pi$
$563$	12162.4	0.910449	0.455224	+	0.890377i	$0.349559\pi$
$564$	0	0
$565$	4851.19	0.361223
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−144.753	−0.0106650	−0.00533249	−	0.999986i	$-0.501697\pi$
$569$	−144.753	−0.0106650	−0.00533249	+	0.999986i	$0.501697\pi$
$570$	0	0
$571$	7479.28	0.548158	0.274079	−	0.961707i	$-0.411627\pi$
$571$	7479.28	0.548158	0.274079	+	0.961707i	$0.411627\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4017.18	0.291353
$576$	0	0
$577$	−6562.94	−0.473516	−0.236758	−	0.971569i	$-0.576085\pi$
$577$	−6562.94	−0.473516	−0.236758	+	0.971569i	$0.576085\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−8892.24	−0.634961
$582$	0	0
$583$	11312.1	0.803601
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4227.89	0.297281	0.148640	−	0.988891i	$-0.452510\pi$
$587$	4227.89	0.297281	0.148640	+	0.988891i	$0.452510\pi$
$588$	0	0
$589$	−235.247	−0.0164570
$590$	0	0
$591$	0	0
$592$	0	0
$593$	28746.9	1.99071	0.995356	−	0.0962641i	$-0.0306893\pi$
$593$	28746.9	1.99071	0.995356	+	0.0962641i	$0.0306893\pi$
$594$	0	0
$595$	−670.128	−0.0461724
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−21714.9	−1.48121	−0.740607	−	0.671939i	$-0.765462\pi$
$599$	−21714.9	−1.48121	−0.740607	+	0.671939i	$0.765462\pi$
$600$	0	0
$601$	6055.82	0.411018	0.205509	−	0.978655i	$-0.434115\pi$
$601$	6055.82	0.411018	0.205509	+	0.978655i	$0.434115\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−4173.68	−0.280470
$606$	0	0
$607$	−18260.0	−1.22100	−0.610502	−	0.792015i	$-0.709032\pi$
$607$	−18260.0	−1.22100	−0.610502	+	0.792015i	$0.709032\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	44102.7	2.92014
$612$	0	0
$613$	−16207.4	−1.06788	−0.533941	−	0.845521i	$-0.679290\pi$
$613$	−16207.4	−1.06788	−0.533941	+	0.845521i	$0.679290\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	14003.1	0.913686	0.456843	−	0.889547i	$-0.348980\pi$
$617$	14003.1	0.913686	0.456843	+	0.889547i	$0.348980\pi$
$618$	0	0
$619$	269.684	0.0175113	0.00875565	−	0.999962i	$-0.497213\pi$
$619$	269.684	0.0175113	0.00875565	+	0.999962i	$0.497213\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−21838.5	−1.40440
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−93.5284	−0.00592881
$630$	0	0
$631$	21240.2	1.34003	0.670016	−	0.742347i	$-0.266287\pi$
$631$	21240.2	1.34003	0.670016	+	0.742347i	$0.266287\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4025.12	0.251546
$636$	0	0
$637$	48750.2	3.03227
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−19210.9	−1.18375	−0.591875	−	0.806030i	$-0.701612\pi$
$641$	−19210.9	−1.18375	−0.591875	+	0.806030i	$0.701612\pi$
$642$	0	0
$643$	−7731.07	−0.474158	−0.237079	−	0.971490i	$-0.576190\pi$
$643$	−7731.07	−0.474158	−0.237079	+	0.971490i	$0.576190\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	23347.4	1.41867	0.709337	−	0.704870i	$-0.248995\pi$
$647$	23347.4	1.41867	0.709337	+	0.704870i	$0.248995\pi$
$648$	0	0
$649$	−17856.4	−1.08001
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−7667.71	−0.459511	−0.229755	−	0.973248i	$-0.573793\pi$
$653$	−7667.71	−0.459511	−0.229755	+	0.973248i	$0.573793\pi$
$654$	0	0
$655$	−1073.44	−0.0640346
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−14673.6	−0.867378	−0.433689	−	0.901063i	$-0.642788\pi$
$659$	−14673.6	−0.867378	−0.433689	+	0.901063i	$0.642788\pi$
$660$	0	0
$661$	25446.9	1.49738	0.748691	−	0.662919i	$-0.230683\pi$
$661$	25446.9	1.49738	0.748691	+	0.662919i	$0.230683\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	12928.2	0.753885
$666$	0	0
$667$	38848.2	2.25518
$668$	0	0
$669$	0	0
$670$	0	0
$671$	6171.45	0.355061
$672$	0	0
$673$	−12239.5	−0.701035	−0.350518	−	0.936556i	$-0.613994\pi$
$673$	−12239.5	−0.701035	−0.350518	+	0.936556i	$0.613994\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10457.5	0.593672	0.296836	−	0.954929i	$-0.404069\pi$
$677$	10457.5	0.593672	0.296836	+	0.954929i	$0.404069\pi$
$678$	0	0
$679$	−33854.0	−1.91340
$680$	0	0
$681$	0	0
$682$	0	0
$683$	304.720	0.0170714	0.00853572	−	0.999964i	$-0.497283\pi$
$683$	304.720	0.0170714	0.00853572	+	0.999964i	$0.497283\pi$
$684$	0	0
$685$	1838.99	0.102576
$686$	0	0
$687$	0	0
$688$	0	0
$689$	22381.1	1.23752
$690$	0	0
$691$	14937.8	0.822373	0.411186	−	0.911551i	$-0.365115\pi$
$691$	14937.8	0.822373	0.411186	+	0.911551i	$0.365115\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7136.55	−0.389503
$696$	0	0
$697$	2274.30	0.123594
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−13752.9	−0.741000	−0.370500	−	0.928832i	$-0.620814\pi$
$701$	−13752.9	−0.741000	−0.370500	+	0.928832i	$0.620814\pi$
$702$	0	0
$703$	1804.36	0.0968033
$704$	0	0
$705$	0	0
$706$	0	0
$707$	15107.3	0.803634
$708$	0	0
$709$	−31225.5	−1.65402	−0.827008	−	0.562190i	$-0.809959\pi$
$709$	−31225.5	−1.65402	−0.827008	+	0.562190i	$0.809959\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	431.829	0.0226818
$714$	0	0
$715$	−21424.7	−1.12061
$716$	0	0
$717$	0	0
$718$	0	0
$719$	13945.5	0.723337	0.361669	−	0.932307i	$-0.382207\pi$
$719$	13945.5	0.723337	0.361669	+	0.932307i	$0.382207\pi$
$720$	0	0
$721$	−12450.4	−0.643103
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6044.06	0.309615
$726$	0	0
$727$	37585.1	1.91741	0.958704	−	0.284407i	$-0.0917966\pi$
$727$	37585.1	1.91741	0.958704	+	0.284407i	$0.0917966\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1337.14	−0.0676550
$732$	0	0
$733$	2848.70	0.143546	0.0717729	−	0.997421i	$-0.477134\pi$
$733$	2848.70	0.143546	0.0717729	+	0.997421i	$0.477134\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−27113.3	−1.35513
$738$	0	0
$739$	−1828.99	−0.0910428	−0.0455214	−	0.998963i	$-0.514495\pi$
$739$	−1828.99	−0.0910428	−0.0455214	+	0.998963i	$0.514495\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	32798.9	1.61948	0.809740	−	0.586789i	$-0.199608\pi$
$743$	32798.9	1.61948	0.809740	+	0.586789i	$0.199608\pi$
$744$	0	0
$745$	−8788.68	−0.432205
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−48426.7	−2.36245
$750$	0	0
$751$	16225.2	0.788371	0.394186	−	0.919031i	$-0.371027\pi$
$751$	16225.2	0.788371	0.394186	+	0.919031i	$0.371027\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−2688.88	−0.129614
$756$	0	0
$757$	−14461.0	−0.694312	−0.347156	−	0.937807i	$-0.612853\pi$
$757$	−14461.0	−0.694312	−0.347156	+	0.937807i	$0.612853\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−22060.5	−1.05084	−0.525422	−	0.850842i	$-0.676093\pi$
$761$	−22060.5	−1.05084	−0.525422	+	0.850842i	$0.676093\pi$
$762$	0	0
$763$	−20394.9	−0.967687
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−35329.2	−1.66318
$768$	0	0
$769$	−1675.74	−0.0785810	−0.0392905	−	0.999228i	$-0.512510\pi$
$769$	−1675.74	−0.0785810	−0.0392905	+	0.999228i	$0.512510\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−10878.6	−0.506179	−0.253089	−	0.967443i	$-0.581447\pi$
$773$	−10878.6	−0.506179	−0.253089	+	0.967443i	$0.581447\pi$
$774$	0	0
$775$	67.1846	0.00311399
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−43876.0	−2.01800
$780$	0	0
$781$	−26376.3	−1.20847
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−3358.94	−0.152721
$786$	0	0
$787$	27293.2	1.23621	0.618105	−	0.786096i	$-0.287901\pi$
$787$	27293.2	1.23621	0.618105	+	0.786096i	$0.287901\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−28658.4	−1.28821
$792$	0	0
$793$	12210.3	0.546784
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−38925.4	−1.73000	−0.865000	−	0.501772i	$-0.832682\pi$
$797$	−38925.4	−1.73000	−0.865000	+	0.501772i	$0.832682\pi$
$798$	0	0
$799$	2173.39	0.0962317
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−39063.0	−1.71669
$804$	0	0
$805$	−23731.5	−1.03904
$806$	0	0
$807$	0	0
$808$	0	0
$809$	22806.9	0.991159	0.495579	−	0.868563i	$-0.334956\pi$
$809$	22806.9	0.991159	0.495579	+	0.868563i	$0.334956\pi$
$810$	0	0
$811$	42523.0	1.84117	0.920583	−	0.390547i	$-0.127714\pi$
$811$	42523.0	1.84117	0.920583	+	0.390547i	$0.127714\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−12507.1	−0.537553
$816$	0	0
$817$	25796.2	1.10464
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−32644.4	−1.38770	−0.693848	−	0.720122i	$-0.744086\pi$
$821$	−32644.4	−1.38770	−0.693848	+	0.720122i	$0.744086\pi$
$822$	0	0
$823$	−4471.81	−0.189402	−0.0947008	−	0.995506i	$-0.530189\pi$
$823$	−4471.81	−0.189402	−0.0947008	+	0.995506i	$0.530189\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	29667.5	1.24745	0.623725	−	0.781644i	$-0.285619\pi$
$827$	29667.5	1.24745	0.623725	+	0.781644i	$0.285619\pi$
$828$	0	0
$829$	5130.74	0.214955	0.107478	−	0.994207i	$-0.465723\pi$
$829$	5130.74	0.214955	0.107478	+	0.994207i	$0.465723\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2402.42	0.0999268
$834$	0	0
$835$	−3758.24	−0.155760
$836$	0	0
$837$	0	0
$838$	0	0
$839$	24205.9	0.996042	0.498021	−	0.867165i	$-0.334060\pi$
$839$	24205.9	0.996042	0.498021	+	0.867165i	$0.334060\pi$
$840$	0	0
$841$	34060.0	1.39653
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−31404.0	−1.27850
$846$	0	0
$847$	24656.0	1.00023
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−3312.16	−0.133419
$852$	0	0
$853$	21843.2	0.876784	0.438392	−	0.898784i	$-0.355548\pi$
$853$	21843.2	0.876784	0.438392	+	0.898784i	$0.355548\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2770.50	−0.110430	−0.0552151	−	0.998474i	$-0.517584\pi$
$857$	−2770.50	−0.110430	−0.0552151	+	0.998474i	$0.517584\pi$
$858$	0	0
$859$	18439.6	0.732421	0.366211	−	0.930532i	$-0.380655\pi$
$859$	18439.6	0.732421	0.366211	+	0.930532i	$0.380655\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	22073.1	0.870659	0.435329	−	0.900271i	$-0.356632\pi$
$863$	22073.1	0.870659	0.435329	+	0.900271i	$0.356632\pi$
$864$	0	0
$865$	11871.4	0.466634
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−21024.5	−0.820721
$870$	0	0
$871$	−53644.0	−2.08686
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3692.18	−0.142650
$876$	0	0
$877$	−8446.67	−0.325227	−0.162613	−	0.986690i	$-0.551992\pi$
$877$	−8446.67	−0.325227	−0.162613	+	0.986690i	$0.551992\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	16665.8	0.637325	0.318663	−	0.947868i	$-0.396766\pi$
$881$	16665.8	0.637325	0.318663	+	0.947868i	$0.396766\pi$
$882$	0	0
$883$	−37742.6	−1.43844	−0.719219	−	0.694784i	$-0.755500\pi$
$883$	−37742.6	−1.43844	−0.719219	+	0.694784i	$0.755500\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−19447.2	−0.736158	−0.368079	−	0.929795i	$-0.619984\pi$
$887$	−19447.2	−0.736158	−0.368079	+	0.929795i	$0.619984\pi$
$888$	0	0
$889$	−23778.4	−0.897076
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−41929.3	−1.57123
$894$	0	0
$895$	12122.6	0.452753
$896$	0	0
$897$	0	0
$898$	0	0
$899$	649.708	0.0241034
$900$	0	0
$901$	1102.95	0.0407819
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5897.30	−0.216611
$906$	0	0
$907$	−15428.9	−0.564838	−0.282419	−	0.959291i	$-0.591137\pi$
$907$	−15428.9	−0.564838	−0.282419	+	0.959291i	$0.591137\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	688.190	0.0250283	0.0125141	−	0.999922i	$-0.496017\pi$
$911$	688.190	0.0250283	0.0125141	+	0.999922i	$0.496017\pi$
$912$	0	0
$913$	−14010.1	−0.507849
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6341.32	0.228363
$918$	0	0
$919$	−6809.05	−0.244407	−0.122203	−	0.992505i	$-0.538996\pi$
$919$	−6809.05	−0.244407	−0.122203	+	0.992505i	$0.538996\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−52185.8	−1.86101
$924$	0	0
$925$	−515.311	−0.0183171
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−47682.2	−1.68396	−0.841982	−	0.539506i	$-0.818611\pi$
$929$	−47682.2	−1.68396	−0.841982	+	0.539506i	$0.818611\pi$
$930$	0	0
$931$	−46347.8	−1.63157
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1055.81	−0.0369292
$936$	0	0
$937$	−30516.3	−1.06395	−0.531977	−	0.846759i	$-0.678551\pi$
$937$	−30516.3	−1.06395	−0.531977	+	0.846759i	$0.678551\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−31348.8	−1.08602	−0.543009	−	0.839727i	$-0.682715\pi$
$941$	−31348.8	−1.08602	−0.543009	+	0.839727i	$0.682715\pi$
$942$	0	0
$943$	80540.5	2.78129
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−34598.6	−1.18723	−0.593614	−	0.804750i	$-0.702299\pi$
$947$	−34598.6	−1.18723	−0.593614	+	0.804750i	$0.702299\pi$
$948$	0	0
$949$	−77286.6	−2.64365
$950$	0	0
$951$	0	0
$952$	0	0
$953$	26766.7	0.909820	0.454910	−	0.890537i	$-0.349671\pi$
$953$	26766.7	0.909820	0.454910	+	0.890537i	$0.349671\pi$
$954$	0	0
$955$	1722.38	0.0583610
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10863.9	−0.365810
$960$	0	0
$961$	−29783.8	−0.999758
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−9709.19	−0.323886
$966$	0	0
$967$	−8764.26	−0.291458	−0.145729	−	0.989325i	$-0.546553\pi$
$967$	−8764.26	−0.291458	−0.145729	+	0.989325i	$0.546553\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−35417.3	−1.17054	−0.585270	−	0.810838i	$-0.699012\pi$
$971$	−35417.3	−1.17054	−0.585270	+	0.810838i	$0.699012\pi$
$972$	0	0
$973$	42159.2	1.38907
$974$	0	0
$975$	0	0
$976$	0	0
$977$	26629.0	0.871993	0.435997	−	0.899948i	$-0.356396\pi$
$977$	26629.0	0.871993	0.435997	+	0.899948i	$0.356396\pi$
$978$	0	0
$979$	−34407.4	−1.12326
$980$	0	0
$981$	0	0
$982$	0	0
$983$	16809.2	0.545402	0.272701	−	0.962099i	$-0.412083\pi$
$983$	16809.2	0.545402	0.272701	+	0.962099i	$0.412083\pi$
$984$	0	0
$985$	−8904.75	−0.288050
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−47352.5	−1.52247
$990$	0	0
$991$	−53758.7	−1.72321	−0.861606	−	0.507578i	$-0.830541\pi$
$991$	−53758.7	−1.72321	−0.861606	+	0.507578i	$0.830541\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−4041.87	−0.128780
$996$	0	0
$997$	41575.1	1.32066	0.660330	−	0.750976i	$-0.270417\pi$
$997$	41575.1	1.32066	0.660330	+	0.750976i	$0.270417\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.a.w.1.2		2
3.2	odd	2		2160.4.a.bb.1.2		2
4.3	odd	2		270.4.a.m.1.1	✓	2
12.11	even	2		270.4.a.n.1.1	yes	2
20.3	even	4		1350.4.c.u.649.4		4
20.7	even	4		1350.4.c.u.649.1		4
20.19	odd	2		1350.4.a.bm.1.2		2
36.7	odd	6		810.4.e.bd.271.2		4
36.11	even	6		810.4.e.z.271.2		4
36.23	even	6		810.4.e.z.541.2		4
36.31	odd	6		810.4.e.bd.541.2		4
60.23	odd	4		1350.4.c.bb.649.2		4
60.47	odd	4		1350.4.c.bb.649.3		4
60.59	even	2		1350.4.a.bf.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.4.a.m.1.1	✓	2	4.3	odd	2
270.4.a.n.1.1	yes	2	12.11	even	2
810.4.e.z.271.2		4	36.11	even	6
810.4.e.z.541.2		4	36.23	even	6
810.4.e.bd.271.2		4	36.7	odd	6
810.4.e.bd.541.2		4	36.31	odd	6
1350.4.a.bf.1.2		2	60.59	even	2
1350.4.a.bm.1.2		2	20.19	odd	2
1350.4.c.u.649.1		4	20.7	even	4
1350.4.c.u.649.4		4	20.3	even	4
1350.4.c.bb.649.2		4	60.23	odd	4
1350.4.c.bb.649.3		4	60.47	odd	4
2160.4.a.w.1.2		2	1.1	even	1	trivial
2160.4.a.bb.1.2		2	3.2	odd	2

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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)

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} +29.5375 q^{7} +O(q^{10})\)	\(q-5.00000 q^{5} +29.5375 q^{7} +46.5375 q^{11} +92.0750 q^{13} +4.53748 q^{17} -87.5375 q^{19} +160.687 q^{23} +25.0000 q^{25} +241.762 q^{29} +2.68738 q^{31} -147.687 q^{35} -20.6124 q^{37} +501.225 q^{41} -294.687 q^{43} +478.987 q^{47} +529.463 q^{49} +243.075 q^{53} -232.687 q^{55} -383.700 q^{59} +132.612 q^{61} -460.375 q^{65} -582.612 q^{67} -566.775 q^{71} -839.388 q^{73} +1374.60 q^{77} -451.775 q^{79} -301.049 q^{83} -22.6874 q^{85} -739.349 q^{89} +2719.66 q^{91} +437.687 q^{95} -1146.14 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 10 q^{5} - q^{7}+O(q^{10}) \) 2 * q - 10 * q^5 - q^7	\( 2 q - 10 q^{5} - q^{7} + 33 q^{11} + 64 q^{13} - 51 q^{17} - 115 q^{19} + 21 q^{23} + 50 q^{25} + 63 q^{29} - 295 q^{31} + 5 q^{35} + 139 q^{37} + 642 q^{41} - 289 q^{43} + 177 q^{47} + 1119 q^{49} + 366 q^{53} - 165 q^{55} - 1248 q^{59} + 85 q^{61} - 320 q^{65} - 985 q^{67} - 1494 q^{71} - 1859 q^{73} + 1788 q^{77} - 1264 q^{79} + 1080 q^{83} + 255 q^{85} + 684 q^{89} + 3577 q^{91} + 575 q^{95} - 1271 q^{97}+O(q^{100}) \) 2 * q - 10 * q^5 - q^7 + 33 * q^11 + 64 * q^13 - 51 * q^17 - 115 * q^19 + 21 * q^23 + 50 * q^25 + 63 * q^29 - 295 * q^31 + 5 * q^35 + 139 * q^37 + 642 * q^41 - 289 * q^43 + 177 * q^47 + 1119 * q^49 + 366 * q^53 - 165 * q^55 - 1248 * q^59 + 85 * q^61 - 320 * q^65 - 985 * q^67 - 1494 * q^71 - 1859 * q^73 + 1788 * q^77 - 1264 * q^79 + 1080 * q^83 + 255 * q^85 + 684 * q^89 + 3577 * q^91 + 575 * q^95 - 1271 * q^97

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