LMFDB - Embedded newform 2160.4.a.bg.1.1

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			1.1
Root			$0.162962$ 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)

$f(q)$	$=$	$q-5.00000 q^{5} -26.8184 q^{7} +O(q^{10})$	$q-5.00000 q^{5} -26.8184 q^{7} +30.8629 q^{11} +32.8629 q^{13} -71.5442 q^{17} +49.3219 q^{19} +72.5035 q^{23} +25.0000 q^{25} -54.4628 q^{29} -146.736 q^{31} +134.092 q^{35} -65.6294 q^{37} -148.626 q^{41} +453.009 q^{43} -171.139 q^{47} +376.228 q^{49} +440.506 q^{53} -154.314 q^{55} -128.143 q^{59} +395.450 q^{61} -164.314 q^{65} +380.925 q^{67} +490.158 q^{71} -288.396 q^{73} -827.694 q^{77} +395.847 q^{79} +845.940 q^{83} +357.721 q^{85} +743.631 q^{89} -881.331 q^{91} -246.610 q^{95} -1840.93 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−26.8184	−1.44806	−0.724030	−	0.689769i	$-0.757712\pi$
$7$	−26.8184	−1.44806	−0.724030	+	0.689769i	$0.757712\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	30.8629	0.845956	0.422978	−	0.906140i	$-0.360985\pi$
$11$	30.8629	0.845956	0.422978	+	0.906140i	$0.360985\pi$
$12$	0	0
$13$	32.8629	0.701117	0.350559	−	0.936541i	$-0.385992\pi$
$13$	32.8629	0.701117	0.350559	+	0.936541i	$0.385992\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−71.5442	−1.02071	−0.510354	−	0.859965i	$-0.670485\pi$
$17$	−71.5442	−1.02071	−0.510354	+	0.859965i	$0.670485\pi$
$18$	0	0
$19$	49.3219	0.595538	0.297769	−	0.954638i	$-0.403757\pi$
$19$	49.3219	0.595538	0.297769	+	0.954638i	$0.403757\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	72.5035	0.657305	0.328653	−	0.944451i	$-0.393405\pi$
$23$	72.5035	0.657305	0.328653	+	0.944451i	$0.393405\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−54.4628	−0.348741	−0.174370	−	0.984680i	$-0.555789\pi$
$29$	−54.4628	−0.348741	−0.174370	+	0.984680i	$0.555789\pi$
$30$	0	0
$31$	−146.736	−0.850150	−0.425075	−	0.905158i	$-0.639752\pi$
$31$	−146.736	−0.850150	−0.425075	+	0.905158i	$0.639752\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	134.092	0.647592
$36$	0	0
$37$	−65.6294	−0.291606	−0.145803	−	0.989314i	$-0.546576\pi$
$37$	−65.6294	−0.291606	−0.145803	+	0.989314i	$0.546576\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−148.626	−0.566132	−0.283066	−	0.959100i	$-0.591352\pi$
$41$	−148.626	−0.566132	−0.283066	+	0.959100i	$0.591352\pi$
$42$	0	0
$43$	453.009	1.60659	0.803294	−	0.595583i	$-0.203079\pi$
$43$	453.009	1.60659	0.803294	+	0.595583i	$0.203079\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−171.139	−0.531133	−0.265566	−	0.964093i	$-0.585559\pi$
$47$	−171.139	−0.531133	−0.265566	+	0.964093i	$0.585559\pi$
$48$	0	0
$49$	376.228	1.09688
$50$	0	0
$51$	0	0
$52$	0	0
$53$	440.506	1.14166	0.570831	−	0.821067i	$-0.306621\pi$
$53$	440.506	1.14166	0.570831	+	0.821067i	$0.306621\pi$
$54$	0	0
$55$	−154.314	−0.378323
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−128.143	−0.282760	−0.141380	−	0.989955i	$-0.545154\pi$
$59$	−128.143	−0.282760	−0.141380	+	0.989955i	$0.545154\pi$
$60$	0	0
$61$	395.450	0.830036	0.415018	−	0.909813i	$-0.363775\pi$
$61$	395.450	0.830036	0.415018	+	0.909813i	$0.363775\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−164.314	−0.313549
$66$	0	0
$67$	380.925	0.694587	0.347294	−	0.937756i	$-0.387101\pi$
$67$	380.925	0.694587	0.347294	+	0.937756i	$0.387101\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	490.158	0.819311	0.409655	−	0.912240i	$-0.365649\pi$
$71$	490.158	0.819311	0.409655	+	0.912240i	$0.365649\pi$
$72$	0	0
$73$	−288.396	−0.462386	−0.231193	−	0.972908i	$-0.574263\pi$
$73$	−288.396	−0.462386	−0.231193	+	0.972908i	$0.574263\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−827.694	−1.22499
$78$	0	0
$79$	395.847	0.563750	0.281875	−	0.959451i	$-0.409044\pi$
$79$	395.847	0.563750	0.281875	+	0.959451i	$0.409044\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	845.940	1.11872	0.559361	−	0.828924i	$-0.311046\pi$
$83$	845.940	1.11872	0.559361	+	0.828924i	$0.311046\pi$
$84$	0	0
$85$	357.721	0.456474
$86$	0	0
$87$	0	0
$88$	0	0
$89$	743.631	0.885671	0.442835	−	0.896603i	$-0.353973\pi$
$89$	743.631	0.885671	0.442835	+	0.896603i	$0.353973\pi$
$90$	0	0
$91$	−881.331	−1.01526
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−246.610	−0.266333
$96$	0	0
$97$	−1840.93	−1.92699	−0.963494	−	0.267728i	$-0.913727\pi$
$97$	−1840.93	−1.92699	−0.963494	+	0.267728i	$0.913727\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	9.00402	0.00887062	0.00443531	−	0.999990i	$-0.498588\pi$
$101$	9.00402	0.00887062	0.00443531	+	0.999990i	$0.498588\pi$
$102$	0	0
$103$	−1181.11	−1.12988	−0.564941	−	0.825131i	$-0.691101\pi$
$103$	−1181.11	−1.12988	−0.564941	+	0.825131i	$0.691101\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1198.10	1.08247	0.541235	−	0.840872i	$-0.317957\pi$
$107$	1198.10	1.08247	0.541235	+	0.840872i	$0.317957\pi$
$108$	0	0
$109$	−674.102	−0.592360	−0.296180	−	0.955132i	$-0.595713\pi$
$109$	−674.102	−0.592360	−0.296180	+	0.955132i	$0.595713\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−779.998	−0.649346	−0.324673	−	0.945826i	$-0.605254\pi$
$113$	−779.998	−0.649346	−0.324673	+	0.945826i	$0.605254\pi$
$114$	0	0
$115$	−362.517	−0.293956
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1918.70	1.47804
$120$	0	0
$121$	−378.482	−0.284359
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1180.26	−0.824654	−0.412327	−	0.911036i	$-0.635284\pi$
$127$	−1180.26	−0.824654	−0.412327	+	0.911036i	$0.635284\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	721.126	0.480955	0.240477	−	0.970655i	$-0.422696\pi$
$131$	721.126	0.480955	0.240477	+	0.970655i	$0.422696\pi$
$132$	0	0
$133$	−1322.74	−0.862375
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2988.13	−1.86345	−0.931727	−	0.363159i	$-0.881698\pi$
$137$	−2988.13	−1.86345	−0.931727	+	0.363159i	$0.881698\pi$
$138$	0	0
$139$	2231.99	1.36198	0.680990	−	0.732293i	$-0.261550\pi$
$139$	2231.99	1.36198	0.680990	+	0.732293i	$0.261550\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1014.24	0.593114
$144$	0	0
$145$	272.314	0.155962
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1683.57	−0.925659	−0.462829	−	0.886447i	$-0.653166\pi$
$149$	−1683.57	−0.925659	−0.462829	+	0.886447i	$0.653166\pi$
$150$	0	0
$151$	−3430.80	−1.84897	−0.924485	−	0.381218i	$-0.875505\pi$
$151$	−3430.80	−1.84897	−0.924485	+	0.381218i	$0.875505\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	733.682	0.380199
$156$	0	0
$157$	−1729.76	−0.879296	−0.439648	−	0.898170i	$-0.644897\pi$
$157$	−1729.76	−0.879296	−0.439648	+	0.898170i	$0.644897\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1944.43	−0.951817
$162$	0	0
$163$	−1101.14	−0.529130	−0.264565	−	0.964368i	$-0.585228\pi$
$163$	−1101.14	−0.529130	−0.264565	+	0.964368i	$0.585228\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1871.07	−0.866991	−0.433495	−	0.901156i	$-0.642720\pi$
$167$	−1871.07	−0.866991	−0.433495	+	0.901156i	$0.642720\pi$
$168$	0	0
$169$	−1117.03	−0.508434
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2686.45	−1.18062	−0.590308	−	0.807178i	$-0.700994\pi$
$173$	−2686.45	−1.18062	−0.590308	+	0.807178i	$0.700994\pi$
$174$	0	0
$175$	−670.461	−0.289612
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3620.13	1.51163	0.755814	−	0.654786i	$-0.227241\pi$
$179$	3620.13	1.51163	0.755814	+	0.654786i	$0.227241\pi$
$180$	0	0
$181$	−444.917	−0.182710	−0.0913548	−	0.995818i	$-0.529120\pi$
$181$	−444.917	−0.182710	−0.0913548	+	0.995818i	$0.529120\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	328.147	0.130410
$186$	0	0
$187$	−2208.06	−0.863473
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2086.82	−0.790561	−0.395280	−	0.918561i	$-0.629353\pi$
$191$	−2086.82	−0.790561	−0.395280	+	0.918561i	$0.629353\pi$
$192$	0	0
$193$	−2750.05	−1.02566	−0.512831	−	0.858489i	$-0.671403\pi$
$193$	−2750.05	−1.02566	−0.512831	+	0.858489i	$0.671403\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2617.99	−0.946824	−0.473412	−	0.880841i	$-0.656978\pi$
$197$	−2617.99	−0.946824	−0.473412	+	0.880841i	$0.656978\pi$
$198$	0	0
$199$	−2697.80	−0.961014	−0.480507	−	0.876991i	$-0.659547\pi$
$199$	−2697.80	−0.961014	−0.480507	+	0.876991i	$0.659547\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1460.61	0.504997
$204$	0	0
$205$	743.128	0.253182
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1522.22	0.503799
$210$	0	0
$211$	1577.75	0.514773	0.257386	−	0.966309i	$-0.417139\pi$
$211$	1577.75	0.514773	0.257386	+	0.966309i	$0.417139\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2265.05	−0.718488
$216$	0	0
$217$	3935.24	1.23107
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2351.15	−0.715635
$222$	0	0
$223$	−3316.84	−0.996018	−0.498009	−	0.867172i	$-0.665935\pi$
$223$	−3316.84	−0.996018	−0.498009	+	0.867172i	$0.665935\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−6667.43	−1.94948	−0.974742	−	0.223332i	$-0.928306\pi$
$227$	−6667.43	−1.94948	−0.974742	+	0.223332i	$0.928306\pi$
$228$	0	0
$229$	−4170.75	−1.20354	−0.601770	−	0.798669i	$-0.705538\pi$
$229$	−4170.75	−1.20354	−0.601770	+	0.798669i	$0.705538\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5684.72	1.59836	0.799181	−	0.601090i	$-0.205267\pi$
$233$	5684.72	1.59836	0.799181	+	0.601090i	$0.205267\pi$
$234$	0	0
$235$	855.697	0.237530
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2980.17	0.806573	0.403286	−	0.915074i	$-0.367868\pi$
$239$	2980.17	0.806573	0.403286	+	0.915074i	$0.367868\pi$
$240$	0	0
$241$	2392.44	0.639462	0.319731	−	0.947508i	$-0.396407\pi$
$241$	2392.44	0.639462	0.319731	+	0.947508i	$0.396407\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1881.14	−0.490538
$246$	0	0
$247$	1620.86	0.417542
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2328.26	0.585491	0.292746	−	0.956190i	$-0.405431\pi$
$251$	2328.26	0.585491	0.292746	+	0.956190i	$0.405431\pi$
$252$	0	0
$253$	2237.67	0.556051
$254$	0	0
$255$	0	0
$256$	0	0
$257$	136.227	0.0330646	0.0165323	−	0.999863i	$-0.494737\pi$
$257$	136.227	0.0330646	0.0165323	+	0.999863i	$0.494737\pi$
$258$	0	0
$259$	1760.08	0.422262
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4436.05	1.04007	0.520036	−	0.854145i	$-0.325919\pi$
$263$	4436.05	1.04007	0.520036	+	0.854145i	$0.325919\pi$
$264$	0	0
$265$	−2202.53	−0.510567
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1679.91	0.380766	0.190383	−	0.981710i	$-0.439027\pi$
$269$	1679.91	0.380766	0.190383	+	0.981710i	$0.439027\pi$
$270$	0	0
$271$	−846.654	−0.189781	−0.0948904	−	0.995488i	$-0.530250\pi$
$271$	−846.654	−0.189781	−0.0948904	+	0.995488i	$0.530250\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	771.572	0.169191
$276$	0	0
$277$	554.305	0.120235	0.0601173	−	0.998191i	$-0.480853\pi$
$277$	554.305	0.120235	0.0601173	+	0.998191i	$0.480853\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1715.63	0.364221	0.182110	−	0.983278i	$-0.441707\pi$
$281$	1715.63	0.364221	0.182110	+	0.983278i	$0.441707\pi$
$282$	0	0
$283$	3327.36	0.698907	0.349454	−	0.936954i	$-0.386367\pi$
$283$	3327.36	0.698907	0.349454	+	0.936954i	$0.386367\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3985.91	0.819793
$288$	0	0
$289$	205.574	0.0418429
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1291.74	−0.257557	−0.128778	−	0.991673i	$-0.541106\pi$
$293$	−1291.74	−0.257557	−0.128778	+	0.991673i	$0.541106\pi$
$294$	0	0
$295$	640.716	0.126454
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2382.67	0.460848
$300$	0	0
$301$	−12149.0	−2.32643
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1977.25	−0.371203
$306$	0	0
$307$	−10514.4	−1.95468	−0.977339	−	0.211678i	$-0.932107\pi$
$307$	−10514.4	−1.95468	−0.977339	+	0.211678i	$0.932107\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8245.53	1.50341	0.751706	−	0.659498i	$-0.229231\pi$
$311$	8245.53	1.50341	0.751706	+	0.659498i	$0.229231\pi$
$312$	0	0
$313$	−9926.89	−1.79265	−0.896327	−	0.443393i	$-0.853775\pi$
$313$	−9926.89	−1.79265	−0.896327	+	0.443393i	$0.853775\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2204.92	0.390665	0.195333	−	0.980737i	$-0.437421\pi$
$317$	2204.92	0.390665	0.195333	+	0.980737i	$0.437421\pi$
$318$	0	0
$319$	−1680.88	−0.295019
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3528.70	−0.607870
$324$	0	0
$325$	821.572	0.140223
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4589.69	0.769112
$330$	0	0
$331$	8359.73	1.38819	0.694097	−	0.719882i	$-0.255804\pi$
$331$	8359.73	1.38819	0.694097	+	0.719882i	$0.255804\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1904.62	−0.310629
$336$	0	0
$337$	12063.6	1.95000	0.974998	−	0.222213i	$-0.0713282\pi$
$337$	12063.6	1.95000	0.974998	+	0.222213i	$0.0713282\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−4528.71	−0.719189
$342$	0	0
$343$	−891.130	−0.140281
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−151.823	−0.0234878	−0.0117439	−	0.999931i	$-0.503738\pi$
$347$	−151.823	−0.0234878	−0.0117439	+	0.999931i	$0.503738\pi$
$348$	0	0
$349$	−5990.20	−0.918763	−0.459381	−	0.888239i	$-0.651929\pi$
$349$	−5990.20	−0.918763	−0.459381	+	0.888239i	$0.651929\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−9917.82	−1.49539	−0.747695	−	0.664043i	$-0.768839\pi$
$353$	−9917.82	−1.49539	−0.747695	+	0.664043i	$0.768839\pi$
$354$	0	0
$355$	−2450.79	−0.366407
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−852.720	−0.125362	−0.0626809	−	0.998034i	$-0.519965\pi$
$359$	−852.720	−0.125362	−0.0626809	+	0.998034i	$0.519965\pi$
$360$	0	0
$361$	−4426.35	−0.645334
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1441.98	0.206785
$366$	0	0
$367$	−6306.86	−0.897045	−0.448522	−	0.893772i	$-0.648050\pi$
$367$	−6306.86	−0.897045	−0.448522	+	0.893772i	$0.648050\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−11813.7	−1.65320
$372$	0	0
$373$	−5201.75	−0.722082	−0.361041	−	0.932550i	$-0.617579\pi$
$373$	−5201.75	−0.722082	−0.361041	+	0.932550i	$0.617579\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1789.80	−0.244508
$378$	0	0
$379$	−13112.8	−1.77720	−0.888599	−	0.458684i	$-0.848321\pi$
$379$	−13112.8	−1.77720	−0.888599	+	0.458684i	$0.848321\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−12502.2	−1.66797	−0.833987	−	0.551785i	$-0.813947\pi$
$383$	−12502.2	−1.66797	−0.833987	+	0.551785i	$0.813947\pi$
$384$	0	0
$385$	4138.47	0.547834
$386$	0	0
$387$	0	0
$388$	0	0
$389$	7814.63	1.01855	0.509277	−	0.860603i	$-0.329913\pi$
$389$	7814.63	1.01855	0.509277	+	0.860603i	$0.329913\pi$
$390$	0	0
$391$	−5187.20	−0.670916
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1979.23	−0.252117
$396$	0	0
$397$	2748.91	0.347516	0.173758	−	0.984788i	$-0.444409\pi$
$397$	2748.91	0.347516	0.173758	+	0.984788i	$0.444409\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	12676.6	1.57865	0.789325	−	0.613975i	$-0.210430\pi$
$401$	12676.6	1.57865	0.789325	+	0.613975i	$0.210430\pi$
$402$	0	0
$403$	−4822.19	−0.596055
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2025.51	−0.246685
$408$	0	0
$409$	−9710.08	−1.17392	−0.586959	−	0.809616i	$-0.699675\pi$
$409$	−9710.08	−1.17392	−0.586959	+	0.809616i	$0.699675\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3436.60	0.409453
$414$	0	0
$415$	−4229.70	−0.500308
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−3292.09	−0.383841	−0.191920	−	0.981411i	$-0.561472\pi$
$419$	−3292.09	−0.383841	−0.191920	+	0.981411i	$0.561472\pi$
$420$	0	0
$421$	−14438.3	−1.67144	−0.835722	−	0.549152i	$-0.814951\pi$
$421$	−14438.3	−1.67144	−0.835722	+	0.549152i	$0.814951\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1788.61	−0.204141
$426$	0	0
$427$	−10605.4	−1.20194
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1304.43	0.145783	0.0728914	−	0.997340i	$-0.476777\pi$
$431$	1304.43	0.145783	0.0728914	+	0.997340i	$0.476777\pi$
$432$	0	0
$433$	9495.26	1.05384	0.526921	−	0.849914i	$-0.323346\pi$
$433$	9495.26	1.05384	0.526921	+	0.849914i	$0.323346\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3576.01	0.391450
$438$	0	0
$439$	2076.89	0.225796	0.112898	−	0.993607i	$-0.463987\pi$
$439$	2076.89	0.225796	0.112898	+	0.993607i	$0.463987\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−406.522	−0.0435992	−0.0217996	−	0.999762i	$-0.506940\pi$
$443$	−406.522	−0.0435992	−0.0217996	+	0.999762i	$0.506940\pi$
$444$	0	0
$445$	−3718.15	−0.396084
$446$	0	0
$447$	0	0
$448$	0	0
$449$	9231.25	0.970266	0.485133	−	0.874440i	$-0.338771\pi$
$449$	9231.25	0.970266	0.485133	+	0.874440i	$0.338771\pi$
$450$	0	0
$451$	−4587.02	−0.478923
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4406.66	0.454038
$456$	0	0
$457$	−5411.09	−0.553873	−0.276936	−	0.960888i	$-0.589319\pi$
$457$	−5411.09	−0.553873	−0.276936	+	0.960888i	$0.589319\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	16604.5	1.67755	0.838774	−	0.544480i	$-0.183273\pi$
$461$	16604.5	1.67755	0.838774	+	0.544480i	$0.183273\pi$
$462$	0	0
$463$	13732.8	1.37843	0.689217	−	0.724555i	$-0.257955\pi$
$463$	13732.8	1.37843	0.689217	+	0.724555i	$0.257955\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6504.30	−0.644503	−0.322252	−	0.946654i	$-0.604440\pi$
$467$	−6504.30	−0.644503	−0.322252	+	0.946654i	$0.604440\pi$
$468$	0	0
$469$	−10215.8	−1.00580
$470$	0	0
$471$	0	0
$472$	0	0
$473$	13981.2	1.35910
$474$	0	0
$475$	1233.05	0.119108
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−8405.21	−0.801762	−0.400881	−	0.916130i	$-0.631296\pi$
$479$	−8405.21	−0.801762	−0.400881	+	0.916130i	$0.631296\pi$
$480$	0	0
$481$	−2156.77	−0.204450
$482$	0	0
$483$	0	0
$484$	0	0
$485$	9204.64	0.861776
$486$	0	0
$487$	−17940.9	−1.66936	−0.834682	−	0.550732i	$-0.814349\pi$
$487$	−17940.9	−1.66936	−0.834682	+	0.550732i	$0.814349\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−11905.2	−1.09424	−0.547122	−	0.837053i	$-0.684277\pi$
$491$	−11905.2	−1.09424	−0.547122	+	0.837053i	$0.684277\pi$
$492$	0	0
$493$	3896.50	0.355962
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−13145.3	−1.18641
$498$	0	0
$499$	12372.2	1.10993	0.554964	−	0.831874i	$-0.312732\pi$
$499$	12372.2	1.10993	0.554964	+	0.831874i	$0.312732\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−19878.8	−1.76213	−0.881065	−	0.472994i	$-0.843173\pi$
$503$	−19878.8	−1.76213	−0.881065	+	0.472994i	$0.843173\pi$
$504$	0	0
$505$	−45.0201	−0.00396706
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1624.65	−0.141476	−0.0707379	−	0.997495i	$-0.522535\pi$
$509$	−1624.65	−0.141476	−0.0707379	+	0.997495i	$0.522535\pi$
$510$	0	0
$511$	7734.33	0.669562
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5905.53	0.505298
$516$	0	0
$517$	−5281.86	−0.449315
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1131.62	−0.0951574	−0.0475787	−	0.998867i	$-0.515150\pi$
$521$	−1131.62	−0.0951574	−0.0475787	+	0.998867i	$0.515150\pi$
$522$	0	0
$523$	−4750.46	−0.397176	−0.198588	−	0.980083i	$-0.563636\pi$
$523$	−4750.46	−0.397176	−0.198588	+	0.980083i	$0.563636\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10498.1	0.867754
$528$	0	0
$529$	−6910.24	−0.567950
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−4884.27	−0.396925
$534$	0	0
$535$	−5990.48	−0.484095
$536$	0	0
$537$	0	0
$538$	0	0
$539$	11611.5	0.927908
$540$	0	0
$541$	22335.8	1.77503	0.887514	−	0.460781i	$-0.152431\pi$
$541$	22335.8	1.77503	0.887514	+	0.460781i	$0.152431\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3370.51	0.264911
$546$	0	0
$547$	−20641.1	−1.61343	−0.806717	−	0.590937i	$-0.798758\pi$
$547$	−20641.1	−1.61343	−0.806717	+	0.590937i	$0.798758\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2686.21	−0.207688
$552$	0	0
$553$	−10616.0	−0.816344
$554$	0	0
$555$	0	0
$556$	0	0
$557$	19431.2	1.47814	0.739071	−	0.673628i	$-0.235265\pi$
$557$	19431.2	1.47814	0.739071	+	0.673628i	$0.235265\pi$
$558$	0	0
$559$	14887.2	1.12641
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8084.58	0.605194	0.302597	−	0.953119i	$-0.402146\pi$
$563$	8084.58	0.605194	0.302597	+	0.953119i	$0.402146\pi$
$564$	0	0
$565$	3899.99	0.290396
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5122.63	−0.377420	−0.188710	−	0.982033i	$-0.560431\pi$
$569$	−5122.63	−0.377420	−0.188710	+	0.982033i	$0.560431\pi$
$570$	0	0
$571$	−6658.07	−0.487971	−0.243986	−	0.969779i	$-0.578455\pi$
$571$	−6658.07	−0.487971	−0.243986	+	0.969779i	$0.578455\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1812.59	0.131461
$576$	0	0
$577$	18436.5	1.33019	0.665097	−	0.746757i	$-0.268390\pi$
$577$	18436.5	1.33019	0.665097	+	0.746757i	$0.268390\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−22686.8	−1.61998
$582$	0	0
$583$	13595.3	0.965796
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−7573.74	−0.532541	−0.266271	−	0.963898i	$-0.585792\pi$
$587$	−7573.74	−0.532541	−0.266271	+	0.963898i	$0.585792\pi$
$588$	0	0
$589$	−7237.33	−0.506297
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−7594.14	−0.525892	−0.262946	−	0.964811i	$-0.584694\pi$
$593$	−7594.14	−0.525892	−0.262946	+	0.964811i	$0.584694\pi$
$594$	0	0
$595$	−9593.52	−0.661001
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−12198.7	−0.832094	−0.416047	−	0.909343i	$-0.636585\pi$
$599$	−12198.7	−0.832094	−0.416047	+	0.909343i	$0.636585\pi$
$600$	0	0
$601$	26299.6	1.78500	0.892498	−	0.451051i	$-0.148951\pi$
$601$	26299.6	1.78500	0.892498	+	0.451051i	$0.148951\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1892.41	0.127169
$606$	0	0
$607$	27293.4	1.82505	0.912523	−	0.409025i	$-0.134131\pi$
$607$	27293.4	1.82505	0.912523	+	0.409025i	$0.134131\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5624.13	−0.372386
$612$	0	0
$613$	−15607.9	−1.02838	−0.514191	−	0.857676i	$-0.671908\pi$
$613$	−15607.9	−1.02838	−0.514191	+	0.857676i	$0.671908\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	22973.4	1.49898	0.749492	−	0.662013i	$-0.230298\pi$
$617$	22973.4	1.49898	0.749492	+	0.662013i	$0.230298\pi$
$618$	0	0
$619$	4542.72	0.294971	0.147486	−	0.989064i	$-0.452882\pi$
$619$	4542.72	0.294971	0.147486	+	0.989064i	$0.452882\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−19943.0	−1.28250
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4695.40	0.297644
$630$	0	0
$631$	−6102.70	−0.385015	−0.192507	−	0.981296i	$-0.561662\pi$
$631$	−6102.70	−0.385015	−0.192507	+	0.981296i	$0.561662\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5901.30	0.368797
$636$	0	0
$637$	12363.9	0.769038
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−19133.6	−1.17899	−0.589493	−	0.807773i	$-0.700672\pi$
$641$	−19133.6	−1.17899	−0.589493	+	0.807773i	$0.700672\pi$
$642$	0	0
$643$	4281.30	0.262579	0.131289	−	0.991344i	$-0.458088\pi$
$643$	4281.30	0.262579	0.131289	+	0.991344i	$0.458088\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3061.84	0.186048	0.0930242	−	0.995664i	$-0.470347\pi$
$647$	3061.84	0.186048	0.0930242	+	0.995664i	$0.470347\pi$
$648$	0	0
$649$	−3954.87	−0.239202
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−19647.5	−1.17744	−0.588718	−	0.808338i	$-0.700367\pi$
$653$	−19647.5	−1.17744	−0.588718	+	0.808338i	$0.700367\pi$
$654$	0	0
$655$	−3605.63	−0.215089
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−4733.23	−0.279788	−0.139894	−	0.990166i	$-0.544676\pi$
$659$	−4733.23	−0.279788	−0.139894	+	0.990166i	$0.544676\pi$
$660$	0	0
$661$	−26923.2	−1.58425	−0.792127	−	0.610356i	$-0.791026\pi$
$661$	−26923.2	−1.58425	−0.792127	+	0.610356i	$0.791026\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	6613.68	0.385666
$666$	0	0
$667$	−3948.74	−0.229229
$668$	0	0
$669$	0	0
$670$	0	0
$671$	12204.7	0.702174
$672$	0	0
$673$	5071.60	0.290484	0.145242	−	0.989396i	$-0.453604\pi$
$673$	5071.60	0.290484	0.145242	+	0.989396i	$0.453604\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−218.456	−0.0124017	−0.00620086	−	0.999981i	$-0.501974\pi$
$677$	−218.456	−0.0124017	−0.00620086	+	0.999981i	$0.501974\pi$
$678$	0	0
$679$	49370.8	2.79039
$680$	0	0
$681$	0	0
$682$	0	0
$683$	20868.6	1.16913	0.584565	−	0.811347i	$-0.301265\pi$
$683$	20868.6	1.16913	0.584565	+	0.811347i	$0.301265\pi$
$684$	0	0
$685$	14940.7	0.833362
$686$	0	0
$687$	0	0
$688$	0	0
$689$	14476.3	0.800439
$690$	0	0
$691$	14869.6	0.818618	0.409309	−	0.912396i	$-0.365770\pi$
$691$	14869.6	0.818618	0.409309	+	0.912396i	$0.365770\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11160.0	−0.609096
$696$	0	0
$697$	10633.3	0.577855
$698$	0	0
$699$	0	0
$700$	0	0
$701$	20898.6	1.12600	0.563002	−	0.826455i	$-0.309646\pi$
$701$	20898.6	1.12600	0.563002	+	0.826455i	$0.309646\pi$
$702$	0	0
$703$	−3236.97	−0.173662
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−241.474	−0.0128452
$708$	0	0
$709$	−27866.7	−1.47610	−0.738051	−	0.674745i	$-0.764253\pi$
$709$	−27866.7	−1.47610	−0.738051	+	0.674745i	$0.764253\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−10638.9	−0.558808
$714$	0	0
$715$	−5071.22	−0.265249
$716$	0	0
$717$	0	0
$718$	0	0
$719$	19210.2	0.996412	0.498206	−	0.867059i	$-0.333992\pi$
$719$	19210.2	0.996412	0.498206	+	0.867059i	$0.333992\pi$
$720$	0	0
$721$	31675.4	1.63614
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1361.57	−0.0697482
$726$	0	0
$727$	−15121.7	−0.771433	−0.385716	−	0.922617i	$-0.626046\pi$
$727$	−15121.7	−0.771433	−0.385716	+	0.922617i	$0.626046\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−32410.2	−1.63986
$732$	0	0
$733$	−6552.13	−0.330162	−0.165081	−	0.986280i	$-0.552788\pi$
$733$	−6552.13	−0.330162	−0.165081	+	0.986280i	$0.552788\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11756.4	0.587590
$738$	0	0
$739$	13514.5	0.672719	0.336359	−	0.941734i	$-0.390804\pi$
$739$	13514.5	0.672719	0.336359	+	0.941734i	$0.390804\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−25986.4	−1.28311	−0.641553	−	0.767079i	$-0.721710\pi$
$743$	−25986.4	−1.28311	−0.641553	+	0.767079i	$0.721710\pi$
$744$	0	0
$745$	8417.83	0.413967
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−32131.0	−1.56748
$750$	0	0
$751$	−21586.1	−1.04885	−0.524425	−	0.851457i	$-0.675720\pi$
$751$	−21586.1	−1.04885	−0.524425	+	0.851457i	$0.675720\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	17154.0	0.826885
$756$	0	0
$757$	8240.02	0.395626	0.197813	−	0.980240i	$-0.436616\pi$
$757$	8240.02	0.395626	0.197813	+	0.980240i	$0.436616\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−21903.1	−1.04335	−0.521674	−	0.853145i	$-0.674692\pi$
$761$	−21903.1	−1.04335	−0.521674	+	0.853145i	$0.674692\pi$
$762$	0	0
$763$	18078.4	0.857772
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4211.15	−0.198248
$768$	0	0
$769$	8344.70	0.391310	0.195655	−	0.980673i	$-0.437317\pi$
$769$	8344.70	0.391310	0.195655	+	0.980673i	$0.437317\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	10226.3	0.475826	0.237913	−	0.971286i	$-0.423537\pi$
$773$	10226.3	0.475826	0.237913	+	0.971286i	$0.423537\pi$
$774$	0	0
$775$	−3668.41	−0.170030
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7330.50	−0.337153
$780$	0	0
$781$	15127.7	0.693100
$782$	0	0
$783$	0	0
$784$	0	0
$785$	8648.78	0.393233
$786$	0	0
$787$	−5355.21	−0.242557	−0.121279	−	0.992619i	$-0.538699\pi$
$787$	−5355.21	−0.242557	−0.121279	+	0.992619i	$0.538699\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	20918.3	0.940291
$792$	0	0
$793$	12995.6	0.581953
$794$	0	0
$795$	0	0
$796$	0	0
$797$	24006.7	1.06695	0.533476	−	0.845815i	$-0.320885\pi$
$797$	24006.7	1.06695	0.533476	+	0.845815i	$0.320885\pi$
$798$	0	0
$799$	12244.0	0.542131
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−8900.73	−0.391158
$804$	0	0
$805$	9722.15	0.425666
$806$	0	0
$807$	0	0
$808$	0	0
$809$	845.632	0.0367501	0.0183751	−	0.999831i	$-0.494151\pi$
$809$	845.632	0.0367501	0.0183751	+	0.999831i	$0.494151\pi$
$810$	0	0
$811$	−3745.89	−0.162190	−0.0810950	−	0.996706i	$-0.525842\pi$
$811$	−3745.89	−0.162190	−0.0810950	+	0.996706i	$0.525842\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5505.72	0.236634
$816$	0	0
$817$	22343.3	0.956784
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−27575.2	−1.17221	−0.586104	−	0.810236i	$-0.699339\pi$
$821$	−27575.2	−1.17221	−0.586104	+	0.810236i	$0.699339\pi$
$822$	0	0
$823$	5541.15	0.234693	0.117347	−	0.993091i	$-0.462561\pi$
$823$	5541.15	0.234693	0.117347	+	0.993091i	$0.462561\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	38373.8	1.61353	0.806765	−	0.590873i	$-0.201216\pi$
$827$	38373.8	1.61353	0.806765	+	0.590873i	$0.201216\pi$
$828$	0	0
$829$	−1413.64	−0.0592252	−0.0296126	−	0.999561i	$-0.509427\pi$
$829$	−1413.64	−0.0592252	−0.0296126	+	0.999561i	$0.509427\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−26917.0	−1.11959
$834$	0	0
$835$	9355.33	0.387730
$836$	0	0
$837$	0	0
$838$	0	0
$839$	15354.0	0.631798	0.315899	−	0.948793i	$-0.397694\pi$
$839$	15354.0	0.631798	0.315899	+	0.948793i	$0.397694\pi$
$840$	0	0
$841$	−21422.8	−0.878380
$842$	0	0
$843$	0	0
$844$	0	0
$845$	5585.15	0.227379
$846$	0	0
$847$	10150.3	0.411769
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−4758.36	−0.191674
$852$	0	0
$853$	−29717.0	−1.19284	−0.596419	−	0.802674i	$-0.703410\pi$
$853$	−29717.0	−1.19284	−0.596419	+	0.802674i	$0.703410\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−31742.7	−1.26524	−0.632619	−	0.774463i	$-0.718020\pi$
$857$	−31742.7	−1.26524	−0.632619	+	0.774463i	$0.718020\pi$
$858$	0	0
$859$	5520.83	0.219288	0.109644	−	0.993971i	$-0.465029\pi$
$859$	5520.83	0.219288	0.109644	+	0.993971i	$0.465029\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	19947.1	0.786799	0.393399	−	0.919368i	$-0.371299\pi$
$863$	19947.1	0.786799	0.393399	+	0.919368i	$0.371299\pi$
$864$	0	0
$865$	13432.2	0.527988
$866$	0	0
$867$	0	0
$868$	0	0
$869$	12217.0	0.476908
$870$	0	0
$871$	12518.3	0.486987
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3352.30	0.129518
$876$	0	0
$877$	24074.1	0.926937	0.463469	−	0.886113i	$-0.346605\pi$
$877$	24074.1	0.926937	0.463469	+	0.886113i	$0.346605\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	29201.8	1.11672	0.558362	−	0.829598i	$-0.311430\pi$
$881$	29201.8	1.11672	0.558362	+	0.829598i	$0.311430\pi$
$882$	0	0
$883$	7705.46	0.293669	0.146834	−	0.989161i	$-0.453092\pi$
$883$	7705.46	0.293669	0.146834	+	0.989161i	$0.453092\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4225.85	0.159966	0.0799832	−	0.996796i	$-0.474513\pi$
$887$	4225.85	0.159966	0.0799832	+	0.996796i	$0.474513\pi$
$888$	0	0
$889$	31652.7	1.19415
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8440.92	−0.316310
$894$	0	0
$895$	−18100.7	−0.676021
$896$	0	0
$897$	0	0
$898$	0	0
$899$	7991.68	0.296482
$900$	0	0
$901$	−31515.6	−1.16530
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2224.59	0.0817102
$906$	0	0
$907$	−16052.5	−0.587667	−0.293833	−	0.955857i	$-0.594931\pi$
$907$	−16052.5	−0.587667	−0.293833	+	0.955857i	$0.594931\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	911.735	0.0331582	0.0165791	−	0.999863i	$-0.494722\pi$
$911$	911.735	0.0331582	0.0165791	+	0.999863i	$0.494722\pi$
$912$	0	0
$913$	26108.2	0.946390
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−19339.5	−0.696451
$918$	0	0
$919$	−2367.26	−0.0849713	−0.0424856	−	0.999097i	$-0.513528\pi$
$919$	−2367.26	−0.0849713	−0.0424856	+	0.999097i	$0.513528\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	16108.0	0.574433
$924$	0	0
$925$	−1640.73	−0.0583211
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−14903.0	−0.526319	−0.263159	−	0.964752i	$-0.584765\pi$
$929$	−14903.0	−0.526319	−0.263159	+	0.964752i	$0.584765\pi$
$930$	0	0
$931$	18556.3	0.653231
$932$	0	0
$933$	0	0
$934$	0	0
$935$	11040.3	0.386157
$936$	0	0
$937$	44983.7	1.56836	0.784180	−	0.620534i	$-0.213084\pi$
$937$	44983.7	1.56836	0.784180	+	0.620534i	$0.213084\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−35749.8	−1.23848	−0.619240	−	0.785202i	$-0.712559\pi$
$941$	−35749.8	−1.23848	−0.619240	+	0.785202i	$0.712559\pi$
$942$	0	0
$943$	−10775.9	−0.372122
$944$	0	0
$945$	0	0
$946$	0	0
$947$	46315.7	1.58929	0.794646	−	0.607074i	$-0.207657\pi$
$947$	46315.7	1.58929	0.794646	+	0.607074i	$0.207657\pi$
$948$	0	0
$949$	−9477.52	−0.324187
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−39448.6	−1.34089	−0.670444	−	0.741960i	$-0.733896\pi$
$953$	−39448.6	−1.34089	−0.670444	+	0.741960i	$0.733896\pi$
$954$	0	0
$955$	10434.1	0.353549
$956$	0	0
$957$	0	0
$958$	0	0
$959$	80137.0	2.69839
$960$	0	0
$961$	−8259.40	−0.277245
$962$	0	0
$963$	0	0
$964$	0	0
$965$	13750.3	0.458690
$966$	0	0
$967$	−43691.9	−1.45299	−0.726493	−	0.687174i	$-0.758851\pi$
$967$	−43691.9	−1.45299	−0.726493	+	0.687174i	$0.758851\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−5070.26	−0.167572	−0.0837860	−	0.996484i	$-0.526701\pi$
$971$	−5070.26	−0.167572	−0.0837860	+	0.996484i	$0.526701\pi$
$972$	0	0
$973$	−59858.6	−1.97223
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−34289.3	−1.12284	−0.561419	−	0.827532i	$-0.689744\pi$
$977$	−34289.3	−1.12284	−0.561419	+	0.827532i	$0.689744\pi$
$978$	0	0
$979$	22950.6	0.749238
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−57986.0	−1.88145	−0.940725	−	0.339170i	$-0.889854\pi$
$983$	−57986.0	−1.88145	−0.940725	+	0.339170i	$0.889854\pi$
$984$	0	0
$985$	13090.0	0.423432
$986$	0	0
$987$	0	0
$988$	0	0
$989$	32844.8	1.05602
$990$	0	0
$991$	24526.7	0.786193	0.393096	−	0.919497i	$-0.371404\pi$
$991$	24526.7	0.786193	0.393096	+	0.919497i	$0.371404\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	13489.0	0.429778
$996$	0	0
$997$	−16061.9	−0.510215	−0.255108	−	0.966913i	$-0.582111\pi$
$997$	−16061.9	−0.510215	−0.255108	+	0.966913i	$0.582111\pi$
$998$	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.1.1		3
3.2	odd	2		2160.4.a.bo.1.1		3
4.3	odd	2		1080.4.a.g.1.3	✓	3
12.11	even	2		1080.4.a.m.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.a.g.1.3	✓	3	4.3	odd	2
1080.4.a.m.1.3	yes	3	12.11	even	2
2160.4.a.bg.1.1		3	1.1	even	1	trivial
2160.4.a.bo.1.1		3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} -26.8184 q^{7} +O(q^{10})\)	\(q-5.00000 q^{5} -26.8184 q^{7} +30.8629 q^{11} +32.8629 q^{13} -71.5442 q^{17} +49.3219 q^{19} +72.5035 q^{23} +25.0000 q^{25} -54.4628 q^{29} -146.736 q^{31} +134.092 q^{35} -65.6294 q^{37} -148.626 q^{41} +453.009 q^{43} -171.139 q^{47} +376.228 q^{49} +440.506 q^{53} -154.314 q^{55} -128.143 q^{59} +395.450 q^{61} -164.314 q^{65} +380.925 q^{67} +490.158 q^{71} -288.396 q^{73} -827.694 q^{77} +395.847 q^{79} +845.940 q^{83} +357.721 q^{85} +743.631 q^{89} -881.331 q^{91} -246.610 q^{95} -1840.93 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 15 q^{5} - 6 q^{7}+O(q^{10}) \) 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

Introduction

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