LMFDB - Embedded newform 2160.4.a.bu.1.4

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:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 141x^{2} + 200x + 3500 $ x^4 - x^3 - 141x^2 + 200x + 3500
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
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			1.4
Root			$7.09129$ 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)

$f(q)$	$=$	$q-5.00000 q^{5} +30.4893 q^{7} +O(q^{10})$	$q-5.00000 q^{5} +30.4893 q^{7} -37.9574 q^{11} -43.3934 q^{13} +29.8404 q^{17} +28.6063 q^{19} +51.2553 q^{23} +25.0000 q^{25} -178.457 q^{29} +237.595 q^{31} -152.446 q^{35} -12.7129 q^{37} -222.776 q^{41} -464.063 q^{43} +400.138 q^{47} +586.595 q^{49} -249.541 q^{53} +189.787 q^{55} -779.434 q^{59} +609.977 q^{61} +216.967 q^{65} +172.872 q^{67} -1044.33 q^{71} -38.6474 q^{73} -1157.29 q^{77} -1194.75 q^{79} +150.860 q^{83} -149.202 q^{85} -119.500 q^{89} -1323.03 q^{91} -143.031 q^{95} +1266.01 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 20 q^{5} - 14 q^{7}+O(q^{10}) $ 4 * q - 20 * q^5 - 14 * q^7	$ 4 q - 20 q^{5} - 14 q^{7} - 4 q^{11} + 30 q^{13} + 28 q^{17} - 78 q^{19} + 182 q^{23} + 100 q^{25} - 202 q^{29} + 76 q^{31} + 70 q^{35} + 302 q^{37} - 380 q^{41} - 178 q^{43} + 114 q^{47} + 958 q^{49} + 256 q^{53} + 20 q^{55} - 204 q^{59} + 766 q^{61} - 150 q^{65} - 330 q^{67} - 1060 q^{71} + 1442 q^{73} - 216 q^{77} - 742 q^{79} - 768 q^{83} - 140 q^{85} + 400 q^{89} - 3066 q^{91} + 390 q^{95} + 3338 q^{97}+O(q^{100}) $ 4 * q - 20 * q^5 - 14 * q^7 - 4 * q^11 + 30 * q^13 + 28 * q^17 - 78 * q^19 + 182 * q^23 + 100 * q^25 - 202 * q^29 + 76 * q^31 + 70 * q^35 + 302 * q^37 - 380 * q^41 - 178 * q^43 + 114 * q^47 + 958 * q^49 + 256 * q^53 + 20 * q^55 - 204 * q^59 + 766 * q^61 - 150 * q^65 - 330 * q^67 - 1060 * q^71 + 1442 * q^73 - 216 * q^77 - 742 * q^79 - 768 * q^83 - 140 * q^85 + 400 * q^89 - 3066 * q^91 + 390 * q^95 + 3338 * 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
$7$	30.4893	1.64627	0.823133	−	0.567849i	$-0.192224\pi$
$7$	30.4893	1.64627	0.823133	+	0.567849i	$0.192224\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−37.9574	−1.04042	−0.520209	−	0.854039i	$-0.674146\pi$
$11$	−37.9574	−1.04042	−0.520209	+	0.854039i	$0.674146\pi$
$12$	0	0
$13$	−43.3934	−0.925781	−0.462891	−	0.886415i	$-0.653188\pi$
$13$	−43.3934	−0.925781	−0.462891	+	0.886415i	$0.653188\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	29.8404	0.425727	0.212864	−	0.977082i	$-0.431721\pi$
$17$	29.8404	0.425727	0.212864	+	0.977082i	$0.431721\pi$
$18$	0	0
$19$	28.6063	0.345407	0.172703	−	0.984974i	$-0.444750\pi$
$19$	28.6063	0.345407	0.172703	+	0.984974i	$0.444750\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	51.2553	0.464672	0.232336	−	0.972636i	$-0.425363\pi$
$23$	51.2553	0.464672	0.232336	+	0.972636i	$0.425363\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−178.457	−1.14271	−0.571357	−	0.820702i	$-0.693583\pi$
$29$	−178.457	−1.14271	−0.571357	+	0.820702i	$0.693583\pi$
$30$	0	0
$31$	237.595	1.37656	0.688280	−	0.725445i	$-0.258366\pi$
$31$	237.595	1.37656	0.688280	+	0.725445i	$0.258366\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−152.446	−0.736232
$36$	0	0
$37$	−12.7129	−0.0564860	−0.0282430	−	0.999601i	$-0.508991\pi$
$37$	−12.7129	−0.0564860	−0.0282430	+	0.999601i	$0.508991\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−222.776	−0.848581	−0.424291	−	0.905526i	$-0.639476\pi$
$41$	−222.776	−0.848581	−0.424291	+	0.905526i	$0.639476\pi$
$42$	0	0
$43$	−464.063	−1.64579	−0.822894	−	0.568194i	$-0.807642\pi$
$43$	−464.063	−1.64579	−0.822894	+	0.568194i	$0.807642\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	400.138	1.24183	0.620916	−	0.783877i	$-0.286761\pi$
$47$	400.138	1.24183	0.620916	+	0.783877i	$0.286761\pi$
$48$	0	0
$49$	586.595	1.71019
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−249.541	−0.646738	−0.323369	−	0.946273i	$-0.604815\pi$
$53$	−249.541	−0.646738	−0.323369	+	0.946273i	$0.604815\pi$
$54$	0	0
$55$	189.787	0.465289
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−779.434	−1.71989	−0.859946	−	0.510385i	$-0.829503\pi$
$59$	−779.434	−1.71989	−0.859946	+	0.510385i	$0.829503\pi$
$60$	0	0
$61$	609.977	1.28032	0.640161	−	0.768241i	$-0.278868\pi$
$61$	609.977	1.28032	0.640161	+	0.768241i	$0.278868\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	216.967	0.414022
$66$	0	0
$67$	172.872	0.315218	0.157609	−	0.987502i	$-0.449621\pi$
$67$	172.872	0.315218	0.157609	+	0.987502i	$0.449621\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1044.33	−1.74562	−0.872809	−	0.488062i	$-0.837704\pi$
$71$	−1044.33	−1.74562	−0.872809	+	0.488062i	$0.837704\pi$
$72$	0	0
$73$	−38.6474	−0.0619635	−0.0309818	−	0.999520i	$-0.509863\pi$
$73$	−38.6474	−0.0619635	−0.0309818	+	0.999520i	$0.509863\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1157.29	−1.71280
$78$	0	0
$79$	−1194.75	−1.70152	−0.850761	−	0.525553i	$-0.823859\pi$
$79$	−1194.75	−1.70152	−0.850761	+	0.525553i	$0.823859\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	150.860	0.199507	0.0997535	−	0.995012i	$-0.468195\pi$
$83$	150.860	0.199507	0.0997535	+	0.995012i	$0.468195\pi$
$84$	0	0
$85$	−149.202	−0.190391
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−119.500	−0.142326	−0.0711628	−	0.997465i	$-0.522671\pi$
$89$	−119.500	−0.142326	−0.0711628	+	0.997465i	$0.522671\pi$
$90$	0	0
$91$	−1323.03	−1.52408
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−143.031	−0.154471
$96$	0	0
$97$	1266.01	1.32519	0.662596	−	0.748977i	$-0.269455\pi$
$97$	1266.01	1.32519	0.662596	+	0.748977i	$0.269455\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1762.34	1.73623	0.868114	−	0.496365i	$-0.165332\pi$
$101$	1762.34	1.73623	0.868114	+	0.496365i	$0.165332\pi$
$102$	0	0
$103$	−746.636	−0.714255	−0.357127	−	0.934056i	$-0.616244\pi$
$103$	−746.636	−0.714255	−0.357127	+	0.934056i	$0.616244\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	351.360	0.317451	0.158726	−	0.987323i	$-0.449261\pi$
$107$	351.360	0.317451	0.158726	+	0.987323i	$0.449261\pi$
$108$	0	0
$109$	−755.508	−0.663895	−0.331947	−	0.943298i	$-0.607706\pi$
$109$	−755.508	−0.663895	−0.331947	+	0.943298i	$0.607706\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	787.742	0.655792	0.327896	−	0.944714i	$-0.393660\pi$
$113$	787.742	0.655792	0.327896	+	0.944714i	$0.393660\pi$
$114$	0	0
$115$	−256.276	−0.207808
$116$	0	0
$117$	0	0
$118$	0	0
$119$	909.812	0.700860
$120$	0	0
$121$	109.765	0.0824684
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1415.65	0.989125	0.494563	−	0.869142i	$-0.335328\pi$
$127$	1415.65	0.989125	0.494563	+	0.869142i	$0.335328\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−882.082	−0.588304	−0.294152	−	0.955759i	$-0.595037\pi$
$131$	−882.082	−0.588304	−0.294152	+	0.955759i	$0.595037\pi$
$132$	0	0
$133$	872.184	0.568631
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−503.449	−0.313960	−0.156980	−	0.987602i	$-0.550176\pi$
$137$	−503.449	−0.313960	−0.156980	+	0.987602i	$0.550176\pi$
$138$	0	0
$139$	−587.011	−0.358199	−0.179099	−	0.983831i	$-0.557318\pi$
$139$	−587.011	−0.358199	−0.179099	+	0.983831i	$0.557318\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1647.10	0.963199
$144$	0	0
$145$	892.287	0.511037
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2927.36	−1.60952	−0.804760	−	0.593600i	$-0.797706\pi$
$149$	−2927.36	−1.60952	−0.804760	+	0.593600i	$0.797706\pi$
$150$	0	0
$151$	841.091	0.453291	0.226646	−	0.973977i	$-0.427224\pi$
$151$	841.091	0.453291	0.226646	+	0.973977i	$0.427224\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1187.98	−0.615617
$156$	0	0
$157$	1763.63	0.896518	0.448259	−	0.893904i	$-0.352044\pi$
$157$	1763.63	0.896518	0.448259	+	0.893904i	$0.352044\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1562.74	0.764974
$162$	0	0
$163$	−2522.29	−1.21203	−0.606016	−	0.795452i	$-0.707233\pi$
$163$	−2522.29	−1.21203	−0.606016	+	0.795452i	$0.707233\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2527.79	1.17129	0.585647	−	0.810566i	$-0.300841\pi$
$167$	2527.79	1.17129	0.585647	+	0.810566i	$0.300841\pi$
$168$	0	0
$169$	−314.014	−0.142929
$170$	0	0
$171$	0	0
$172$	0	0
$173$	389.805	0.171308	0.0856541	−	0.996325i	$-0.472702\pi$
$173$	389.805	0.171308	0.0856541	+	0.996325i	$0.472702\pi$
$174$	0	0
$175$	762.232	0.329253
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2305.67	−0.962760	−0.481380	−	0.876512i	$-0.659864\pi$
$179$	−2305.67	−0.962760	−0.481380	+	0.876512i	$0.659864\pi$
$180$	0	0
$181$	2644.87	1.08614	0.543071	−	0.839687i	$-0.317262\pi$
$181$	2644.87	1.08614	0.543071	+	0.839687i	$0.317262\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	63.5643	0.0252613
$186$	0	0
$187$	−1132.67	−0.442934
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1152.77	0.436711	0.218356	−	0.975869i	$-0.429931\pi$
$191$	1152.77	0.436711	0.218356	+	0.975869i	$0.429931\pi$
$192$	0	0
$193$	−2033.47	−0.758406	−0.379203	−	0.925313i	$-0.623802\pi$
$193$	−2033.47	−0.758406	−0.379203	+	0.925313i	$0.623802\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1701.78	−0.615466	−0.307733	−	0.951473i	$-0.599570\pi$
$197$	−1701.78	−0.615466	−0.307733	+	0.951473i	$0.599570\pi$
$198$	0	0
$199$	−5087.64	−1.81233	−0.906163	−	0.422928i	$-0.861002\pi$
$199$	−5087.64	−1.81233	−0.906163	+	0.422928i	$0.861002\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5441.04	−1.88121
$204$	0	0
$205$	1113.88	0.379497
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1085.82	−0.359367
$210$	0	0
$211$	−128.178	−0.0418206	−0.0209103	−	0.999781i	$-0.506656\pi$
$211$	−128.178	−0.0418206	−0.0209103	+	0.999781i	$0.506656\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2320.31	0.736019
$216$	0	0
$217$	7244.11	2.26618
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1294.88	−0.394130
$222$	0	0
$223$	−2803.55	−0.841882	−0.420941	−	0.907088i	$-0.638300\pi$
$223$	−2803.55	−0.841882	−0.420941	+	0.907088i	$0.638300\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2485.68	−0.726786	−0.363393	−	0.931636i	$-0.618382\pi$
$227$	−2485.68	−0.726786	−0.363393	+	0.931636i	$0.618382\pi$
$228$	0	0
$229$	−1733.54	−0.500241	−0.250121	−	0.968215i	$-0.580470\pi$
$229$	−1733.54	−0.500241	−0.250121	+	0.968215i	$0.580470\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4163.48	−1.17064	−0.585319	−	0.810803i	$-0.699031\pi$
$233$	−4163.48	−1.17064	−0.585319	+	0.810803i	$0.699031\pi$
$234$	0	0
$235$	−2000.69	−0.555364
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5658.52	−1.53146	−0.765730	−	0.643162i	$-0.777622\pi$
$239$	−5658.52	−1.53146	−0.765730	+	0.643162i	$0.777622\pi$
$240$	0	0
$241$	3016.40	0.806239	0.403119	−	0.915147i	$-0.367926\pi$
$241$	3016.40	0.806239	0.403119	+	0.915147i	$0.367926\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2932.98	−0.764821
$246$	0	0
$247$	−1241.32	−0.319771
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−6422.21	−1.61501	−0.807503	−	0.589863i	$-0.799182\pi$
$251$	−6422.21	−1.61501	−0.807503	+	0.589863i	$0.799182\pi$
$252$	0	0
$253$	−1945.52	−0.483453
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6124.07	−1.48642	−0.743208	−	0.669060i	$-0.766697\pi$
$257$	−6124.07	−1.48642	−0.743208	+	0.669060i	$0.766697\pi$
$258$	0	0
$259$	−387.606	−0.0929910
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3487.57	−0.817692	−0.408846	−	0.912603i	$-0.634069\pi$
$263$	−3487.57	−0.817692	−0.408846	+	0.912603i	$0.634069\pi$
$264$	0	0
$265$	1247.71	0.289230
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2681.05	−0.607683	−0.303841	−	0.952723i	$-0.598269\pi$
$269$	−2681.05	−0.607683	−0.303841	+	0.952723i	$0.598269\pi$
$270$	0	0
$271$	−6504.70	−1.45805	−0.729027	−	0.684485i	$-0.760027\pi$
$271$	−6504.70	−1.45805	−0.729027	+	0.684485i	$0.760027\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−948.935	−0.208083
$276$	0	0
$277$	2919.77	0.633329	0.316664	−	0.948538i	$-0.397437\pi$
$277$	2919.77	0.633329	0.316664	+	0.948538i	$0.397437\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−8232.92	−1.74781	−0.873905	−	0.486097i	$-0.838420\pi$
$281$	−8232.92	−1.74781	−0.873905	+	0.486097i	$0.838420\pi$
$282$	0	0
$283$	−579.928	−0.121813	−0.0609066	−	0.998143i	$-0.519399\pi$
$283$	−579.928	−0.121813	−0.0609066	+	0.998143i	$0.519399\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6792.29	−1.39699
$288$	0	0
$289$	−4022.55	−0.818756
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4788.47	0.954763	0.477382	−	0.878696i	$-0.341586\pi$
$293$	4788.47	0.954763	0.477382	+	0.878696i	$0.341586\pi$
$294$	0	0
$295$	3897.17	0.769159
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2224.14	−0.430185
$300$	0	0
$301$	−14148.9	−2.70941
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3049.89	−0.572577
$306$	0	0
$307$	8910.36	1.65649	0.828243	−	0.560369i	$-0.189341\pi$
$307$	8910.36	1.65649	0.828243	+	0.560369i	$0.189341\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9512.00	1.73433	0.867164	−	0.498023i	$-0.165940\pi$
$311$	9512.00	1.73433	0.867164	+	0.498023i	$0.165940\pi$
$312$	0	0
$313$	−667.152	−0.120478	−0.0602391	−	0.998184i	$-0.519186\pi$
$313$	−667.152	−0.120478	−0.0602391	+	0.998184i	$0.519186\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4543.67	−0.805041	−0.402520	−	0.915411i	$-0.631866\pi$
$317$	−4543.67	−0.805041	−0.402520	+	0.915411i	$0.631866\pi$
$318$	0	0
$319$	6773.78	1.18890
$320$	0	0
$321$	0	0
$322$	0	0
$323$	853.623	0.147049
$324$	0	0
$325$	−1084.83	−0.185156
$326$	0	0
$327$	0	0
$328$	0	0
$329$	12199.9	2.04438
$330$	0	0
$331$	−473.632	−0.0786500	−0.0393250	−	0.999226i	$-0.512521\pi$
$331$	−473.632	−0.0786500	−0.0393250	+	0.999226i	$0.512521\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−864.358	−0.140970
$336$	0	0
$337$	−968.065	−0.156480	−0.0782401	−	0.996935i	$-0.524930\pi$
$337$	−968.065	−0.156480	−0.0782401	+	0.996935i	$0.524930\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−9018.50	−1.43220
$342$	0	0
$343$	7427.05	1.16916
$344$	0	0
$345$	0	0
$346$	0	0
$347$	646.162	0.0999648	0.0499824	−	0.998750i	$-0.484083\pi$
$347$	646.162	0.0999648	0.0499824	+	0.998750i	$0.484083\pi$
$348$	0	0
$349$	8806.56	1.35073	0.675365	−	0.737484i	$-0.263986\pi$
$349$	8806.56	1.35073	0.675365	+	0.737484i	$0.263986\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−237.161	−0.0357586	−0.0178793	−	0.999840i	$-0.505691\pi$
$353$	−237.161	−0.0357586	−0.0178793	+	0.999840i	$0.505691\pi$
$354$	0	0
$355$	5221.64	0.780664
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11440.6	−1.68192	−0.840962	−	0.541094i	$-0.818010\pi$
$359$	−11440.6	−1.68192	−0.840962	+	0.541094i	$0.818010\pi$
$360$	0	0
$361$	−6040.68	−0.880694
$362$	0	0
$363$	0	0
$364$	0	0
$365$	193.237	0.0277109
$366$	0	0
$367$	−4164.30	−0.592301	−0.296151	−	0.955141i	$-0.595703\pi$
$367$	−4164.30	−0.592301	−0.296151	+	0.955141i	$0.595703\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−7608.32	−1.06470
$372$	0	0
$373$	−2502.13	−0.347333	−0.173667	−	0.984804i	$-0.555562\pi$
$373$	−2502.13	−0.347333	−0.173667	+	0.984804i	$0.555562\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7743.87	1.05790
$378$	0	0
$379$	3861.08	0.523300	0.261650	−	0.965163i	$-0.415733\pi$
$379$	3861.08	0.523300	0.261650	+	0.965163i	$0.415733\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5093.32	−0.679521	−0.339760	−	0.940512i	$-0.610346\pi$
$383$	−5093.32	−0.679521	−0.339760	+	0.940512i	$0.610346\pi$
$384$	0	0
$385$	5786.47	0.765989
$386$	0	0
$387$	0	0
$388$	0	0
$389$	13388.1	1.74500	0.872500	−	0.488614i	$-0.162497\pi$
$389$	13388.1	1.74500	0.872500	+	0.488614i	$0.162497\pi$
$390$	0	0
$391$	1529.48	0.197824
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5973.77	0.760944
$396$	0	0
$397$	−10809.2	−1.36650	−0.683249	−	0.730185i	$-0.739434\pi$
$397$	−10809.2	−1.36650	−0.683249	+	0.730185i	$0.739434\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−7927.12	−0.987185	−0.493593	−	0.869693i	$-0.664317\pi$
$401$	−7927.12	−0.987185	−0.493593	+	0.869693i	$0.664317\pi$
$402$	0	0
$403$	−10310.1	−1.27439
$404$	0	0
$405$	0	0
$406$	0	0
$407$	482.548	0.0587690
$408$	0	0
$409$	−5186.81	−0.627069	−0.313535	−	0.949577i	$-0.601513\pi$
$409$	−5186.81	−0.627069	−0.313535	+	0.949577i	$0.601513\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−23764.4	−2.83140
$414$	0	0
$415$	−754.302	−0.0892223
$416$	0	0
$417$	0	0
$418$	0	0
$419$	8835.54	1.03018	0.515089	−	0.857137i	$-0.327759\pi$
$419$	8835.54	1.03018	0.515089	+	0.857137i	$0.327759\pi$
$420$	0	0
$421$	12713.5	1.47177	0.735886	−	0.677105i	$-0.236766\pi$
$421$	12713.5	1.47177	0.735886	+	0.677105i	$0.236766\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	746.010	0.0851455
$426$	0	0
$427$	18597.8	2.10775
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4171.84	−0.466242	−0.233121	−	0.972448i	$-0.574894\pi$
$431$	−4171.84	−0.466242	−0.233121	+	0.972448i	$0.574894\pi$
$432$	0	0
$433$	13112.5	1.45531	0.727653	−	0.685946i	$-0.240611\pi$
$433$	13112.5	1.45531	0.727653	+	0.685946i	$0.240611\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1466.22	0.160501
$438$	0	0
$439$	−9951.32	−1.08189	−0.540946	−	0.841057i	$-0.681934\pi$
$439$	−9951.32	−1.08189	−0.540946	+	0.841057i	$0.681934\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−724.845	−0.0777391	−0.0388696	−	0.999244i	$-0.512376\pi$
$443$	−724.845	−0.0777391	−0.0388696	+	0.999244i	$0.512376\pi$
$444$	0	0
$445$	597.500	0.0636500
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1392.90	0.146403	0.0732015	−	0.997317i	$-0.476678\pi$
$449$	1392.90	0.146403	0.0732015	+	0.997317i	$0.476678\pi$
$450$	0	0
$451$	8456.02	0.882879
$452$	0	0
$453$	0	0
$454$	0	0
$455$	6615.16	0.681590
$456$	0	0
$457$	−8721.35	−0.892708	−0.446354	−	0.894856i	$-0.647278\pi$
$457$	−8721.35	−0.892708	−0.446354	+	0.894856i	$0.647278\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−927.718	−0.0937269	−0.0468635	−	0.998901i	$-0.514923\pi$
$461$	−927.718	−0.0937269	−0.0468635	+	0.998901i	$0.514923\pi$
$462$	0	0
$463$	17034.5	1.70985	0.854926	−	0.518749i	$-0.173602\pi$
$463$	17034.5	1.70985	0.854926	+	0.518749i	$0.173602\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13675.8	1.35512	0.677561	−	0.735467i	$-0.263037\pi$
$467$	13675.8	1.35512	0.677561	+	0.735467i	$0.263037\pi$
$468$	0	0
$469$	5270.73	0.518933
$470$	0	0
$471$	0	0
$472$	0	0
$473$	17614.6	1.71231
$474$	0	0
$475$	715.157	0.0690813
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−6785.36	−0.647247	−0.323623	−	0.946186i	$-0.604901\pi$
$479$	−6785.36	−0.647247	−0.323623	+	0.946186i	$0.604901\pi$
$480$	0	0
$481$	551.654	0.0522937
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6330.04	−0.592644
$486$	0	0
$487$	−6642.50	−0.618070	−0.309035	−	0.951051i	$-0.600006\pi$
$487$	−6642.50	−0.618070	−0.309035	+	0.951051i	$0.600006\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	11103.2	1.02053	0.510265	−	0.860017i	$-0.329547\pi$
$491$	11103.2	1.02053	0.510265	+	0.860017i	$0.329547\pi$
$492$	0	0
$493$	−5325.24	−0.486485
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−31840.8	−2.87375
$498$	0	0
$499$	8157.07	0.731785	0.365892	−	0.930657i	$-0.380764\pi$
$499$	8157.07	0.731785	0.365892	+	0.930657i	$0.380764\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−21298.6	−1.88799	−0.943994	−	0.329962i	$-0.892964\pi$
$503$	−21298.6	−1.88799	−0.943994	+	0.329962i	$0.892964\pi$
$504$	0	0
$505$	−8811.68	−0.776465
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−4303.19	−0.374726	−0.187363	−	0.982291i	$-0.559994\pi$
$509$	−4303.19	−0.374726	−0.187363	+	0.982291i	$0.559994\pi$
$510$	0	0
$511$	−1178.33	−0.102008
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3733.18	0.319424
$516$	0	0
$517$	−15188.2	−1.29202
$518$	0	0
$519$	0	0
$520$	0	0
$521$	17772.1	1.49445	0.747225	−	0.664572i	$-0.231386\pi$
$521$	17772.1	1.49445	0.747225	+	0.664572i	$0.231386\pi$
$522$	0	0
$523$	−1629.44	−0.136234	−0.0681170	−	0.997677i	$-0.521699\pi$
$523$	−1629.44	−0.136234	−0.0681170	+	0.997677i	$0.521699\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7089.94	0.586039
$528$	0	0
$529$	−9539.90	−0.784080
$530$	0	0
$531$	0	0
$532$	0	0
$533$	9667.02	0.785601
$534$	0	0
$535$	−1756.80	−0.141968
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−22265.6	−1.77931
$540$	0	0
$541$	−15199.2	−1.20788	−0.603940	−	0.797030i	$-0.706403\pi$
$541$	−15199.2	−1.20788	−0.603940	+	0.797030i	$0.706403\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3777.54	0.296903
$546$	0	0
$547$	−22585.0	−1.76539	−0.882693	−	0.469950i	$-0.844272\pi$
$547$	−22585.0	−1.76539	−0.882693	+	0.469950i	$0.844272\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5105.00	−0.394701
$552$	0	0
$553$	−36427.2	−2.80116
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−24374.9	−1.85421	−0.927107	−	0.374798i	$-0.877712\pi$
$557$	−24374.9	−1.85421	−0.927107	+	0.374798i	$0.877712\pi$
$558$	0	0
$559$	20137.3	1.52364
$560$	0	0
$561$	0	0
$562$	0	0
$563$	2009.16	0.150401	0.0752006	−	0.997168i	$-0.476040\pi$
$563$	2009.16	0.150401	0.0752006	+	0.997168i	$0.476040\pi$
$564$	0	0
$565$	−3938.71	−0.293279
$566$	0	0
$567$	0	0
$568$	0	0
$569$	2085.35	0.153642	0.0768210	−	0.997045i	$-0.475523\pi$
$569$	2085.35	0.153642	0.0768210	+	0.997045i	$0.475523\pi$
$570$	0	0
$571$	15348.8	1.12492	0.562459	−	0.826825i	$-0.309855\pi$
$571$	15348.8	1.12492	0.562459	+	0.826825i	$0.309855\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1281.38	0.0929344
$576$	0	0
$577$	10894.2	0.786020	0.393010	−	0.919534i	$-0.371434\pi$
$577$	10894.2	0.786020	0.393010	+	0.919534i	$0.371434\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	4599.62	0.328442
$582$	0	0
$583$	9471.93	0.672877
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16008.1	1.12560	0.562800	−	0.826593i	$-0.309724\pi$
$587$	16008.1	1.12560	0.562800	+	0.826593i	$0.309724\pi$
$588$	0	0
$589$	6796.72	0.475473
$590$	0	0
$591$	0	0
$592$	0	0
$593$	9103.71	0.630429	0.315215	−	0.949020i	$-0.397923\pi$
$593$	9103.71	0.630429	0.315215	+	0.949020i	$0.397923\pi$
$594$	0	0
$595$	−4549.06	−0.313434
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−4319.26	−0.294624	−0.147312	−	0.989090i	$-0.547062\pi$
$599$	−4319.26	−0.294624	−0.147312	+	0.989090i	$0.547062\pi$
$600$	0	0
$601$	−376.731	−0.0255693	−0.0127847	−	0.999918i	$-0.504070\pi$
$601$	−376.731	−0.0255693	−0.0127847	+	0.999918i	$0.504070\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−548.827	−0.0368810
$606$	0	0
$607$	15923.7	1.06478	0.532390	−	0.846499i	$-0.321294\pi$
$607$	15923.7	1.06478	0.532390	+	0.846499i	$0.321294\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−17363.3	−1.14966
$612$	0	0
$613$	−2458.04	−0.161957	−0.0809784	−	0.996716i	$-0.525804\pi$
$613$	−2458.04	−0.161957	−0.0809784	+	0.996716i	$0.525804\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9810.85	0.640146	0.320073	−	0.947393i	$-0.396293\pi$
$617$	9810.85	0.640146	0.320073	+	0.947393i	$0.396293\pi$
$618$	0	0
$619$	10185.4	0.661368	0.330684	−	0.943742i	$-0.392721\pi$
$619$	10185.4	0.661368	0.330684	+	0.943742i	$0.392721\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3643.47	−0.234306
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−379.357	−0.0240476
$630$	0	0
$631$	−24370.2	−1.53750	−0.768750	−	0.639549i	$-0.779121\pi$
$631$	−24370.2	−1.53750	−0.768750	+	0.639549i	$0.779121\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−7078.27	−0.442350
$636$	0	0
$637$	−25454.4	−1.58326
$638$	0	0
$639$	0	0
$640$	0	0
$641$	26828.4	1.65313	0.826566	−	0.562840i	$-0.190291\pi$
$641$	26828.4	1.65313	0.826566	+	0.562840i	$0.190291\pi$
$642$	0	0
$643$	−28928.0	−1.77420	−0.887100	−	0.461577i	$-0.847284\pi$
$643$	−28928.0	−1.77420	−0.887100	+	0.461577i	$0.847284\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−5640.22	−0.342720	−0.171360	−	0.985208i	$-0.554816\pi$
$647$	−5640.22	−0.342720	−0.171360	+	0.985208i	$0.554816\pi$
$648$	0	0
$649$	29585.3	1.78941
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−18973.7	−1.13705	−0.568527	−	0.822664i	$-0.692487\pi$
$653$	−18973.7	−1.13705	−0.568527	+	0.822664i	$0.692487\pi$
$654$	0	0
$655$	4410.41	0.263098
$656$	0	0
$657$	0	0
$658$	0	0
$659$	12979.7	0.767249	0.383625	−	0.923489i	$-0.374676\pi$
$659$	12979.7	0.767249	0.383625	+	0.923489i	$0.374676\pi$
$660$	0	0
$661$	−5073.66	−0.298551	−0.149276	−	0.988796i	$-0.547694\pi$
$661$	−5073.66	−0.298551	−0.149276	+	0.988796i	$0.547694\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−4360.92	−0.254300
$666$	0	0
$667$	−9146.88	−0.530987
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−23153.2	−1.33207
$672$	0	0
$673$	6222.04	0.356377	0.178189	−	0.983996i	$-0.442976\pi$
$673$	6222.04	0.356377	0.178189	+	0.983996i	$0.442976\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−24566.3	−1.39462	−0.697311	−	0.716769i	$-0.745620\pi$
$677$	−24566.3	−1.39462	−0.697311	+	0.716769i	$0.745620\pi$
$678$	0	0
$679$	38599.7	2.18162
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−5341.08	−0.299225	−0.149613	−	0.988745i	$-0.547803\pi$
$683$	−5341.08	−0.299225	−0.149613	+	0.988745i	$0.547803\pi$
$684$	0	0
$685$	2517.25	0.140407
$686$	0	0
$687$	0	0
$688$	0	0
$689$	10828.4	0.598738
$690$	0	0
$691$	31206.7	1.71803	0.859016	−	0.511948i	$-0.171076\pi$
$691$	31206.7	1.71803	0.859016	+	0.511948i	$0.171076\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2935.06	0.160191
$696$	0	0
$697$	−6647.74	−0.361264
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−26450.4	−1.42513	−0.712567	−	0.701604i	$-0.752467\pi$
$701$	−26450.4	−1.42513	−0.712567	+	0.701604i	$0.752467\pi$
$702$	0	0
$703$	−363.668	−0.0195107
$704$	0	0
$705$	0	0
$706$	0	0
$707$	53732.3	2.85829
$708$	0	0
$709$	9284.18	0.491784	0.245892	−	0.969297i	$-0.420919\pi$
$709$	9284.18	0.491784	0.245892	+	0.969297i	$0.420919\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	12178.0	0.639649
$714$	0	0
$715$	−8235.50	−0.430756
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−2340.29	−0.121388	−0.0606941	−	0.998156i	$-0.519331\pi$
$719$	−2340.29	−0.121388	−0.0606941	+	0.998156i	$0.519331\pi$
$720$	0	0
$721$	−22764.4	−1.17585
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4461.44	−0.228543
$726$	0	0
$727$	−38028.0	−1.94000	−0.970000	−	0.243104i	$-0.921834\pi$
$727$	−38028.0	−1.94000	−0.970000	+	0.243104i	$0.921834\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−13847.8	−0.700657
$732$	0	0
$733$	19036.1	0.959229	0.479614	−	0.877479i	$-0.340777\pi$
$733$	19036.1	0.959229	0.479614	+	0.877479i	$0.340777\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6561.76	−0.327959
$738$	0	0
$739$	14047.6	0.699253	0.349626	−	0.936889i	$-0.386309\pi$
$739$	14047.6	0.699253	0.349626	+	0.936889i	$0.386309\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1692.66	−0.0835768	−0.0417884	−	0.999126i	$-0.513306\pi$
$743$	−1692.66	−0.0835768	−0.0417884	+	0.999126i	$0.513306\pi$
$744$	0	0
$745$	14636.8	0.719800
$746$	0	0
$747$	0	0
$748$	0	0
$749$	10712.7	0.522609
$750$	0	0
$751$	−34968.0	−1.69907	−0.849535	−	0.527533i	$-0.823117\pi$
$751$	−34968.0	−1.69907	−0.849535	+	0.527533i	$0.823117\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−4205.45	−0.202718
$756$	0	0
$757$	37589.3	1.80477	0.902383	−	0.430936i	$-0.141816\pi$
$757$	37589.3	1.80477	0.902383	+	0.430936i	$0.141816\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	32525.0	1.54932	0.774659	−	0.632380i	$-0.217922\pi$
$761$	32525.0	1.54932	0.774659	+	0.632380i	$0.217922\pi$
$762$	0	0
$763$	−23034.9	−1.09295
$764$	0	0
$765$	0	0
$766$	0	0
$767$	33822.3	1.59224
$768$	0	0
$769$	10695.3	0.501537	0.250768	−	0.968047i	$-0.419317\pi$
$769$	10695.3	0.501537	0.250768	+	0.968047i	$0.419317\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	18685.3	0.869421	0.434711	−	0.900570i	$-0.356851\pi$
$773$	18685.3	0.869421	0.434711	+	0.900570i	$0.356851\pi$
$774$	0	0
$775$	5939.88	0.275312
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6372.80	−0.293106
$780$	0	0
$781$	39640.0	1.81617
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−8818.17	−0.400935
$786$	0	0
$787$	−33119.1	−1.50009	−0.750044	−	0.661388i	$-0.769968\pi$
$787$	−33119.1	−1.50009	−0.750044	+	0.661388i	$0.769968\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	24017.7	1.07961
$792$	0	0
$793$	−26469.0	−1.18530
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−34755.1	−1.54465	−0.772327	−	0.635225i	$-0.780907\pi$
$797$	−34755.1	−1.54465	−0.772327	+	0.635225i	$0.780907\pi$
$798$	0	0
$799$	11940.3	0.528681
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1466.96	0.0644679
$804$	0	0
$805$	−7813.68	−0.342107
$806$	0	0
$807$	0	0
$808$	0	0
$809$	16938.8	0.736137	0.368069	−	0.929799i	$-0.380019\pi$
$809$	16938.8	0.736137	0.368069	+	0.929799i	$0.380019\pi$
$810$	0	0
$811$	−5733.26	−0.248239	−0.124120	−	0.992267i	$-0.539611\pi$
$811$	−5733.26	−0.248239	−0.124120	+	0.992267i	$0.539611\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	12611.5	0.542037
$816$	0	0
$817$	−13275.1	−0.568466
$818$	0	0
$819$	0	0
$820$	0	0
$821$	30811.9	1.30980	0.654899	−	0.755717i	$-0.272711\pi$
$821$	30811.9	1.30980	0.654899	+	0.755717i	$0.272711\pi$
$822$	0	0
$823$	15591.8	0.660384	0.330192	−	0.943914i	$-0.392886\pi$
$823$	15591.8	0.660384	0.330192	+	0.943914i	$0.392886\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	22010.3	0.925480	0.462740	−	0.886494i	$-0.346866\pi$
$827$	22010.3	0.925480	0.462740	+	0.886494i	$0.346866\pi$
$828$	0	0
$829$	43144.8	1.80757	0.903787	−	0.427982i	$-0.140775\pi$
$829$	43144.8	1.80757	0.903787	+	0.427982i	$0.140775\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	17504.3	0.728075
$834$	0	0
$835$	−12638.9	−0.523819
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−20108.6	−0.827444	−0.413722	−	0.910403i	$-0.635772\pi$
$839$	−20108.6	−0.827444	−0.413722	+	0.910403i	$0.635772\pi$
$840$	0	0
$841$	7458.04	0.305795
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1570.07	0.0639196
$846$	0	0
$847$	3346.67	0.135765
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−651.601	−0.0262475
$852$	0	0
$853$	−1628.47	−0.0653666	−0.0326833	−	0.999466i	$-0.510405\pi$
$853$	−1628.47	−0.0653666	−0.0326833	+	0.999466i	$0.510405\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	16965.0	0.676212	0.338106	−	0.941108i	$-0.390214\pi$
$857$	16965.0	0.676212	0.338106	+	0.941108i	$0.390214\pi$
$858$	0	0
$859$	7815.73	0.310442	0.155221	−	0.987880i	$-0.450391\pi$
$859$	7815.73	0.310442	0.155221	+	0.987880i	$0.450391\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	23398.8	0.922948	0.461474	−	0.887154i	$-0.347321\pi$
$863$	23398.8	0.922948	0.461474	+	0.887154i	$0.347321\pi$
$864$	0	0
$865$	−1949.03	−0.0766114
$866$	0	0
$867$	0	0
$868$	0	0
$869$	45349.7	1.77029
$870$	0	0
$871$	−7501.49	−0.291823
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3811.16	−0.147246
$876$	0	0
$877$	−2187.35	−0.0842209	−0.0421104	−	0.999113i	$-0.513408\pi$
$877$	−2187.35	−0.0842209	−0.0421104	+	0.999113i	$0.513408\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−32719.9	−1.25126	−0.625631	−	0.780119i	$-0.715158\pi$
$881$	−32719.9	−1.25126	−0.625631	+	0.780119i	$0.715158\pi$
$882$	0	0
$883$	−30656.1	−1.16836	−0.584178	−	0.811625i	$-0.698583\pi$
$883$	−30656.1	−1.16836	−0.584178	+	0.811625i	$0.698583\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3922.07	0.148467	0.0742335	−	0.997241i	$-0.476349\pi$
$887$	3922.07	0.148467	0.0742335	+	0.997241i	$0.476349\pi$
$888$	0	0
$889$	43162.2	1.62836
$890$	0	0
$891$	0	0
$892$	0	0
$893$	11446.4	0.428937
$894$	0	0
$895$	11528.4	0.430559
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−42400.6	−1.57302
$900$	0	0
$901$	−7446.41	−0.275334
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−13224.3	−0.485737
$906$	0	0
$907$	9849.07	0.360566	0.180283	−	0.983615i	$-0.442299\pi$
$907$	9849.07	0.360566	0.180283	+	0.983615i	$0.442299\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	38738.8	1.40886	0.704431	−	0.709773i	$-0.251202\pi$
$911$	38738.8	1.40886	0.704431	+	0.709773i	$0.251202\pi$
$912$	0	0
$913$	−5726.27	−0.207571
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−26894.0	−0.968505
$918$	0	0
$919$	−13655.6	−0.490161	−0.245080	−	0.969503i	$-0.578814\pi$
$919$	−13655.6	−0.490161	−0.245080	+	0.969503i	$0.578814\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	45316.9	1.61606
$924$	0	0
$925$	−317.822	−0.0112972
$926$	0	0
$927$	0	0
$928$	0	0
$929$	39109.3	1.38120	0.690600	−	0.723237i	$-0.257347\pi$
$929$	39109.3	1.38120	0.690600	+	0.723237i	$0.257347\pi$
$930$	0	0
$931$	16780.3	0.590711
$932$	0	0
$933$	0	0
$934$	0	0
$935$	5663.33	0.198086
$936$	0	0
$937$	−20164.6	−0.703041	−0.351521	−	0.936180i	$-0.614335\pi$
$937$	−20164.6	−0.703041	−0.351521	+	0.936180i	$0.614335\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	24512.7	0.849195	0.424597	−	0.905382i	$-0.360416\pi$
$941$	24512.7	0.849195	0.424597	+	0.905382i	$0.360416\pi$
$942$	0	0
$943$	−11418.5	−0.394312
$944$	0	0
$945$	0	0
$946$	0	0
$947$	49278.9	1.69097	0.845485	−	0.533999i	$-0.179311\pi$
$947$	49278.9	1.69097	0.845485	+	0.533999i	$0.179311\pi$
$948$	0	0
$949$	1677.04	0.0573647
$950$	0	0
$951$	0	0
$952$	0	0
$953$	44093.4	1.49877	0.749384	−	0.662136i	$-0.230350\pi$
$953$	44093.4	1.49877	0.749384	+	0.662136i	$0.230350\pi$
$954$	0	0
$955$	−5763.87	−0.195303
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−15349.8	−0.516862
$960$	0	0
$961$	26660.5	0.894919
$962$	0	0
$963$	0	0
$964$	0	0
$965$	10167.4	0.339170
$966$	0	0
$967$	7183.12	0.238876	0.119438	−	0.992842i	$-0.461891\pi$
$967$	7183.12	0.238876	0.119438	+	0.992842i	$0.461891\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	37526.1	1.24024	0.620119	−	0.784508i	$-0.287084\pi$
$971$	37526.1	1.24024	0.620119	+	0.784508i	$0.287084\pi$
$972$	0	0
$973$	−17897.5	−0.589691
$974$	0	0
$975$	0	0
$976$	0	0
$977$	39517.3	1.29403	0.647017	−	0.762476i	$-0.276016\pi$
$977$	39517.3	1.29403	0.647017	+	0.762476i	$0.276016\pi$
$978$	0	0
$979$	4535.91	0.148078
$980$	0	0
$981$	0	0
$982$	0	0
$983$	55997.2	1.81692	0.908461	−	0.417971i	$-0.137258\pi$
$983$	55997.2	1.81692	0.908461	+	0.417971i	$0.137258\pi$
$984$	0	0
$985$	8508.90	0.275245
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−23785.7	−0.764752
$990$	0	0
$991$	−18626.7	−0.597069	−0.298535	−	0.954399i	$-0.596498\pi$
$991$	−18626.7	−0.597069	−0.298535	+	0.954399i	$0.596498\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	25438.2	0.810497
$996$	0	0
$997$	−1966.50	−0.0624671	−0.0312335	−	0.999512i	$-0.509944\pi$
$997$	−1966.50	−0.0624671	−0.0312335	+	0.999512i	$0.509944\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.a.bu.1.4		4
3.2	odd	2		2160.4.a.bv.1.4		4
4.3	odd	2		1080.4.a.o.1.1	✓	4
12.11	even	2		1080.4.a.p.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.a.o.1.1	✓	4	4.3	odd	2
1080.4.a.p.1.1	yes	4	12.11	even	2
2160.4.a.bu.1.4		4	1.1	even	1	trivial
2160.4.a.bv.1.4		4	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)

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} +30.4893 q^{7} +O(q^{10})\)	\(q-5.00000 q^{5} +30.4893 q^{7} -37.9574 q^{11} -43.3934 q^{13} +29.8404 q^{17} +28.6063 q^{19} +51.2553 q^{23} +25.0000 q^{25} -178.457 q^{29} +237.595 q^{31} -152.446 q^{35} -12.7129 q^{37} -222.776 q^{41} -464.063 q^{43} +400.138 q^{47} +586.595 q^{49} -249.541 q^{53} +189.787 q^{55} -779.434 q^{59} +609.977 q^{61} +216.967 q^{65} +172.872 q^{67} -1044.33 q^{71} -38.6474 q^{73} -1157.29 q^{77} -1194.75 q^{79} +150.860 q^{83} -149.202 q^{85} -119.500 q^{89} -1323.03 q^{91} -143.031 q^{95} +1266.01 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 20 q^{5} - 14 q^{7}+O(q^{10}) \) 4 * q - 20 * q^5 - 14 * q^7	\( 4 q - 20 q^{5} - 14 q^{7} - 4 q^{11} + 30 q^{13} + 28 q^{17} - 78 q^{19} + 182 q^{23} + 100 q^{25} - 202 q^{29} + 76 q^{31} + 70 q^{35} + 302 q^{37} - 380 q^{41} - 178 q^{43} + 114 q^{47} + 958 q^{49} + 256 q^{53} + 20 q^{55} - 204 q^{59} + 766 q^{61} - 150 q^{65} - 330 q^{67} - 1060 q^{71} + 1442 q^{73} - 216 q^{77} - 742 q^{79} - 768 q^{83} - 140 q^{85} + 400 q^{89} - 3066 q^{91} + 390 q^{95} + 3338 q^{97}+O(q^{100}) \) 4 * q - 20 * q^5 - 14 * q^7 - 4 * q^11 + 30 * q^13 + 28 * q^17 - 78 * q^19 + 182 * q^23 + 100 * q^25 - 202 * q^29 + 76 * q^31 + 70 * q^35 + 302 * q^37 - 380 * q^41 - 178 * q^43 + 114 * q^47 + 958 * q^49 + 256 * q^53 + 20 * q^55 - 204 * q^59 + 766 * q^61 - 150 * q^65 - 330 * q^67 - 1060 * q^71 + 1442 * q^73 - 216 * q^77 - 742 * q^79 - 768 * q^83 - 140 * q^85 + 400 * q^89 - 3066 * q^91 + 390 * q^95 + 3338 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more