LMFDB - Embedded newform 1440.4.a.bk.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1440,4,Mod(1,1440)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1440.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1440.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,15,0,14,0,0,0,22,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$84.9627504083$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.16773.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 17x + 18 $ x^3 - x^2 - 17*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-4.14660$ of defining polynomial
Character	$\chi$	$=$	1440.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+5.00000 q^{5} +22.5864 q^{7} -6.39560 q^{11} +78.5364 q^{13} -56.1548 q^{17} +82.9180 q^{19} +10.2548 q^{23} +25.0000 q^{25} +135.836 q^{29} +44.2188 q^{31} +112.932 q^{35} +128.628 q^{37} -174.218 q^{41} -270.455 q^{43} +596.964 q^{47} +167.146 q^{49} -0.289677 q^{53} -31.9780 q^{55} -655.760 q^{59} +48.7433 q^{61} +392.682 q^{65} +358.755 q^{67} +97.3007 q^{71} -617.219 q^{73} -144.454 q^{77} +448.728 q^{79} -723.684 q^{83} -280.774 q^{85} +112.835 q^{89} +1773.86 q^{91} +414.590 q^{95} -396.743 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 15 q^{5} + 14 q^{7} + 22 q^{11} + 8 q^{13} + 34 q^{17} - 4 q^{19} + 176 q^{23} + 75 q^{25} - 98 q^{29} + 88 q^{31} + 70 q^{35} + 284 q^{37} + 8 q^{41} + 504 q^{43} + 280 q^{47} - 409 q^{49} + 150 q^{53}+ \cdots - 1394 q^{97}+O(q^{100}) $ 3 * q + 15 * q^5 + 14 * q^7 + 22 * q^11 + 8 * q^13 + 34 * q^17 - 4 * q^19 + 176 * q^23 + 75 * q^25 - 98 * q^29 + 88 * q^31 + 70 * q^35 + 284 * q^37 + 8 * q^41 + 504 * q^43 + 280 * q^47 - 409 * q^49 + 150 * q^53 + 110 * q^55 + 350 * q^59 + 350 * q^61 + 40 * q^65 + 804 * q^67 + 500 * q^71 - 486 * q^73 - 788 * q^77 + 1592 * q^79 - 684 * q^83 + 170 * q^85 + 668 * q^89 + 1920 * q^91 - 20 * q^95 - 1394 * q^97

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
$7$	22.5864	1.21955	0.609776	−	0.792574i	$-0.291260\pi$
$7$	22.5864	1.21955	0.609776	+	0.792574i	$0.291260\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−6.39560	−0.175304	−0.0876521	−	0.996151i	$-0.527936\pi$
$11$	−6.39560	−0.175304	−0.0876521	+	0.996151i	$0.527936\pi$
$12$	0	0
$13$	78.5364	1.67555	0.837773	−	0.546019i	$-0.183858\pi$
$13$	78.5364	1.67555	0.837773	+	0.546019i	$0.183858\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−56.1548	−0.801150	−0.400575	−	0.916264i	$-0.631190\pi$
$17$	−56.1548	−0.801150	−0.400575	+	0.916264i	$0.631190\pi$
$18$	0	0
$19$	82.9180	1.00119	0.500597	−	0.865680i	$-0.333114\pi$
$19$	82.9180	1.00119	0.500597	+	0.865680i	$0.333114\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	10.2548	0.0929681	0.0464841	−	0.998919i	$-0.485198\pi$
$23$	10.2548	0.0929681	0.0464841	+	0.998919i	$0.485198\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	135.836	0.869797	0.434899	−	0.900479i	$-0.356784\pi$
$29$	135.836	0.869797	0.434899	+	0.900479i	$0.356784\pi$
$30$	0	0
$31$	44.2188	0.256191	0.128096	−	0.991762i	$-0.459114\pi$
$31$	44.2188	0.256191	0.128096	+	0.991762i	$0.459114\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	112.932	0.545400
$36$	0	0
$37$	128.628	0.571524	0.285762	−	0.958301i	$-0.407753\pi$
$37$	128.628	0.571524	0.285762	+	0.958301i	$0.407753\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−174.218	−0.663615	−0.331808	−	0.943347i	$-0.607659\pi$
$41$	−174.218	−0.663615	−0.331808	+	0.943347i	$0.607659\pi$
$42$	0	0
$43$	−270.455	−0.959164	−0.479582	−	0.877497i	$-0.659212\pi$
$43$	−270.455	−0.959164	−0.479582	+	0.877497i	$0.659212\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	596.964	1.85268	0.926342	−	0.376684i	$-0.122936\pi$
$47$	596.964	1.85268	0.926342	+	0.376684i	$0.122936\pi$
$48$	0	0
$49$	167.146	0.487305
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−0.289677	−0.000750759 0	−0.000375379	−	1.00000i	$-0.500119\pi$
$53$	−0.289677	−0.000750759 0	−0.000375379		1.00000i	$0.500119\pi$
$54$	0	0
$55$	−31.9780	−0.0783984
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−655.760	−1.44699	−0.723497	−	0.690327i	$-0.757467\pi$
$59$	−655.760	−1.44699	−0.723497	+	0.690327i	$0.757467\pi$
$60$	0	0
$61$	48.7433	0.102310	0.0511552	−	0.998691i	$-0.483710\pi$
$61$	48.7433	0.102310	0.0511552	+	0.998691i	$0.483710\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	392.682	0.749327
$66$	0	0
$67$	358.755	0.654163	0.327082	−	0.944996i	$-0.393935\pi$
$67$	358.755	0.654163	0.327082	+	0.944996i	$0.393935\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	97.3007	0.162640	0.0813202	−	0.996688i	$-0.474086\pi$
$71$	97.3007	0.162640	0.0813202	+	0.996688i	$0.474086\pi$
$72$	0	0
$73$	−617.219	−0.989589	−0.494794	−	0.869010i	$-0.664757\pi$
$73$	−617.219	−0.989589	−0.494794	+	0.869010i	$0.664757\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−144.454	−0.213792
$78$	0	0
$79$	448.728	0.639062	0.319531	−	0.947576i	$-0.396475\pi$
$79$	448.728	0.639062	0.319531	+	0.947576i	$0.396475\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−723.684	−0.957044	−0.478522	−	0.878076i	$-0.658827\pi$
$83$	−723.684	−0.957044	−0.478522	+	0.878076i	$0.658827\pi$
$84$	0	0
$85$	−280.774	−0.358285
$86$	0	0
$87$	0	0
$88$	0	0
$89$	112.835	0.134388	0.0671939	−	0.997740i	$-0.478595\pi$
$89$	112.835	0.134388	0.0671939	+	0.997740i	$0.478595\pi$
$90$	0	0
$91$	1773.86	2.04341
$92$	0	0
$93$	0	0
$94$	0	0
$95$	414.590	0.447748
$96$	0	0
$97$	−396.743	−0.415290	−0.207645	−	0.978204i	$-0.566580\pi$
$97$	−396.743	−0.415290	−0.207645	+	0.978204i	$0.566580\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1816.33	−1.78942	−0.894712	−	0.446643i	$-0.852619\pi$
$101$	−1816.33	−1.78942	−0.894712	+	0.446643i	$0.852619\pi$
$102$	0	0
$103$	1467.15	1.40352	0.701759	−	0.712414i	$-0.252398\pi$
$103$	1467.15	1.40352	0.701759	+	0.712414i	$0.252398\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1216.78	1.09935	0.549674	−	0.835379i	$-0.314752\pi$
$107$	1216.78	1.09935	0.549674	+	0.835379i	$0.314752\pi$
$108$	0	0
$109$	−1210.40	−1.06363	−0.531813	−	0.846862i	$-0.678489\pi$
$109$	−1210.40	−1.06363	−0.531813	+	0.846862i	$0.678489\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2382.43	1.98336	0.991682	−	0.128712i	$-0.0410842\pi$
$113$	2382.43	1.98336	0.991682	+	0.128712i	$0.0410842\pi$
$114$	0	0
$115$	51.2738	0.0415766
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1268.34	−0.977043
$120$	0	0
$121$	−1290.10	−0.969268
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	1855.59	1.29651	0.648255	−	0.761423i	$-0.275499\pi$
$127$	1855.59	1.29651	0.648255	+	0.761423i	$0.275499\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−289.842	−0.193310	−0.0966550	−	0.995318i	$-0.530814\pi$
$131$	−289.842	−0.193310	−0.0966550	+	0.995318i	$0.530814\pi$
$132$	0	0
$133$	1872.82	1.22101
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1193.52	−0.744303	−0.372152	−	0.928172i	$-0.621380\pi$
$137$	−1193.52	−0.744303	−0.372152	+	0.928172i	$0.621380\pi$
$138$	0	0
$139$	709.062	0.432675	0.216338	−	0.976319i	$-0.430589\pi$
$139$	709.062	0.432675	0.216338	+	0.976319i	$0.430589\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−502.288	−0.293730
$144$	0	0
$145$	679.180	0.388985
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1658.38	−0.911809	−0.455905	−	0.890029i	$-0.650684\pi$
$149$	−1658.38	−0.911809	−0.455905	+	0.890029i	$0.650684\pi$
$150$	0	0
$151$	−1180.56	−0.636242	−0.318121	−	0.948050i	$-0.603052\pi$
$151$	−1180.56	−0.636242	−0.318121	+	0.948050i	$0.603052\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	221.094	0.114572
$156$	0	0
$157$	683.954	0.347678	0.173839	−	0.984774i	$-0.444383\pi$
$157$	683.954	0.347678	0.173839	+	0.984774i	$0.444383\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	231.618	0.113379
$162$	0	0
$163$	2452.67	1.17858	0.589288	−	0.807923i	$-0.299408\pi$
$163$	2452.67	1.17858	0.589288	+	0.807923i	$0.299408\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2855.65	−1.32321	−0.661606	−	0.749851i	$-0.730125\pi$
$167$	−2855.65	−1.32321	−0.661606	+	0.749851i	$0.730125\pi$
$168$	0	0
$169$	3970.97	1.80745
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3596.35	1.58049	0.790247	−	0.612789i	$-0.209952\pi$
$173$	3596.35	1.58049	0.790247	+	0.612789i	$0.209952\pi$
$174$	0	0
$175$	564.660	0.243910
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1281.58	0.535139	0.267570	−	0.963539i	$-0.413779\pi$
$179$	1281.58	0.535139	0.267570	+	0.963539i	$0.413779\pi$
$180$	0	0
$181$	2385.39	0.979583	0.489791	−	0.871840i	$-0.337073\pi$
$181$	2385.39	0.979583	0.489791	+	0.871840i	$0.337073\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	643.142	0.255593
$186$	0	0
$187$	359.144	0.140445
$188$	0	0
$189$	0	0
$190$	0	0
$191$	929.171	0.352002	0.176001	−	0.984390i	$-0.443684\pi$
$191$	929.171	0.352002	0.176001	+	0.984390i	$0.443684\pi$
$192$	0	0
$193$	−2126.98	−0.793282	−0.396641	−	0.917974i	$-0.629824\pi$
$193$	−2126.98	−0.793282	−0.396641	+	0.917974i	$0.629824\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1655.73	−0.598811	−0.299406	−	0.954126i	$-0.596788\pi$
$197$	−1655.73	−0.598811	−0.299406	+	0.954126i	$0.596788\pi$
$198$	0	0
$199$	5009.24	1.78440	0.892201	−	0.451639i	$-0.149160\pi$
$199$	5009.24	1.78440	0.892201	+	0.451639i	$0.149160\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3068.05	1.06076
$204$	0	0
$205$	−871.088	−0.296778
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−530.311	−0.175514
$210$	0	0
$211$	4958.57	1.61783	0.808915	−	0.587925i	$-0.200055\pi$
$211$	4958.57	1.61783	0.808915	+	0.587925i	$0.200055\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1352.28	−0.428951
$216$	0	0
$217$	998.743	0.312438
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4410.20	−1.34236
$222$	0	0
$223$	4155.51	1.24786	0.623931	−	0.781479i	$-0.285535\pi$
$223$	4155.51	1.24786	0.623931	+	0.781479i	$0.285535\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5168.08	−1.51109	−0.755545	−	0.655096i	$-0.772628\pi$
$227$	−5168.08	−1.51109	−0.755545	+	0.655096i	$0.772628\pi$
$228$	0	0
$229$	−4391.41	−1.26722	−0.633608	−	0.773654i	$-0.718427\pi$
$229$	−4391.41	−1.26722	−0.633608	+	0.773654i	$0.718427\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2756.05	0.774913	0.387456	−	0.921888i	$-0.373354\pi$
$233$	2756.05	0.774913	0.387456	+	0.921888i	$0.373354\pi$
$234$	0	0
$235$	2984.82	0.828545
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4742.41	1.28352	0.641759	−	0.766906i	$-0.278205\pi$
$239$	4742.41	1.28352	0.641759	+	0.766906i	$0.278205\pi$
$240$	0	0
$241$	1389.76	0.371461	0.185730	−	0.982601i	$-0.440535\pi$
$241$	1389.76	0.371461	0.185730	+	0.982601i	$0.440535\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	835.729	0.217930
$246$	0	0
$247$	6512.09	1.67755
$248$	0	0
$249$	0	0
$250$	0	0
$251$	5966.89	1.50051	0.750253	−	0.661151i	$-0.229932\pi$
$251$	5966.89	1.50051	0.750253	+	0.661151i	$0.229932\pi$
$252$	0	0
$253$	−65.5854	−0.0162977
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4707.74	1.14265	0.571324	−	0.820725i	$-0.306430\pi$
$257$	4707.74	1.14265	0.571324	+	0.820725i	$0.306430\pi$
$258$	0	0
$259$	2905.25	0.697002
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1137.88	0.266787	0.133393	−	0.991063i	$-0.457413\pi$
$263$	1137.88	0.266787	0.133393	+	0.991063i	$0.457413\pi$
$264$	0	0
$265$	−1.44839	−0.000335749 0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2700.80	−0.612158	−0.306079	−	0.952006i	$-0.599017\pi$
$269$	−2700.80	−0.612158	−0.306079	+	0.952006i	$0.599017\pi$
$270$	0	0
$271$	521.725	0.116947	0.0584734	−	0.998289i	$-0.481377\pi$
$271$	521.725	0.116947	0.0584734	+	0.998289i	$0.481377\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−159.890	−0.0350608
$276$	0	0
$277$	−2017.18	−0.437548	−0.218774	−	0.975776i	$-0.570206\pi$
$277$	−2017.18	−0.437548	−0.218774	+	0.975776i	$0.570206\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3532.33	0.749898	0.374949	−	0.927045i	$-0.377660\pi$
$281$	3532.33	0.749898	0.374949	+	0.927045i	$0.377660\pi$
$282$	0	0
$283$	1839.14	0.386309	0.193155	−	0.981168i	$-0.438128\pi$
$283$	1839.14	0.386309	0.193155	+	0.981168i	$0.438128\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3934.95	−0.809313
$288$	0	0
$289$	−1759.64	−0.358159
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−8569.30	−1.70861	−0.854307	−	0.519769i	$-0.826018\pi$
$293$	−8569.30	−1.70861	−0.854307	+	0.519769i	$0.826018\pi$
$294$	0	0
$295$	−3278.80	−0.647116
$296$	0	0
$297$	0	0
$298$	0	0
$299$	805.373	0.155772
$300$	0	0
$301$	−6108.61	−1.16975
$302$	0	0
$303$	0	0
$304$	0	0
$305$	243.716	0.0457546
$306$	0	0
$307$	−4574.84	−0.850488	−0.425244	−	0.905079i	$-0.639812\pi$
$307$	−4574.84	−0.850488	−0.425244	+	0.905079i	$0.639812\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2963.60	0.540355	0.270177	−	0.962811i	$-0.412918\pi$
$311$	2963.60	0.540355	0.270177	+	0.962811i	$0.412918\pi$
$312$	0	0
$313$	2131.23	0.384869	0.192435	−	0.981310i	$-0.438362\pi$
$313$	2131.23	0.384869	0.192435	+	0.981310i	$0.438362\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8492.05	−1.50461	−0.752305	−	0.658815i	$-0.771058\pi$
$317$	−8492.05	−1.50461	−0.752305	+	0.658815i	$0.771058\pi$
$318$	0	0
$319$	−868.753	−0.152479
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4656.25	−0.802107
$324$	0	0
$325$	1963.41	0.335109
$326$	0	0
$327$	0	0
$328$	0	0
$329$	13483.3	2.25944
$330$	0	0
$331$	6073.88	1.00861	0.504306	−	0.863525i	$-0.331748\pi$
$331$	6073.88	1.00861	0.504306	+	0.863525i	$0.331748\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1793.78	0.292551
$336$	0	0
$337$	−6616.84	−1.06956	−0.534781	−	0.844991i	$-0.679606\pi$
$337$	−6616.84	−1.06956	−0.534781	+	0.844991i	$0.679606\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−282.806	−0.0449114
$342$	0	0
$343$	−3971.92	−0.625257
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6346.58	−0.981850	−0.490925	−	0.871202i	$-0.663341\pi$
$347$	−6346.58	−0.981850	−0.490925	+	0.871202i	$0.663341\pi$
$348$	0	0
$349$	4000.58	0.613599	0.306799	−	0.951774i	$-0.400742\pi$
$349$	4000.58	0.613599	0.306799	+	0.951774i	$0.400742\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−952.711	−0.143648	−0.0718239	−	0.997417i	$-0.522882\pi$
$353$	−952.711	−0.143648	−0.0718239	+	0.997417i	$0.522882\pi$
$354$	0	0
$355$	486.504	0.0727350
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3226.49	−0.474339	−0.237169	−	0.971468i	$-0.576220\pi$
$359$	−3226.49	−0.474339	−0.237169	+	0.971468i	$0.576220\pi$
$360$	0	0
$361$	16.4016	0.00239125
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3086.09	−0.442557
$366$	0	0
$367$	9864.26	1.40303	0.701513	−	0.712657i	$-0.252508\pi$
$367$	9864.26	1.40303	0.701513	+	0.712657i	$0.252508\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−6.54276	−0.000915589 0
$372$	0	0
$373$	−4388.19	−0.609147	−0.304573	−	0.952489i	$-0.598514\pi$
$373$	−4388.19	−0.609147	−0.304573	+	0.952489i	$0.598514\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	10668.1	1.45738
$378$	0	0
$379$	−4973.16	−0.674022	−0.337011	−	0.941501i	$-0.609416\pi$
$379$	−4973.16	−0.674022	−0.337011	+	0.941501i	$0.609416\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4210.82	0.561783	0.280892	−	0.959739i	$-0.409370\pi$
$383$	4210.82	0.561783	0.280892	+	0.959739i	$0.409370\pi$
$384$	0	0
$385$	−722.268	−0.0956109
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1520.60	−0.198194	−0.0990972	−	0.995078i	$-0.531595\pi$
$389$	−1520.60	−0.198194	−0.0990972	+	0.995078i	$0.531595\pi$
$390$	0	0
$391$	−575.855	−0.0744814
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2243.64	0.285797
$396$	0	0
$397$	8353.74	1.05608	0.528038	−	0.849221i	$-0.322928\pi$
$397$	8353.74	1.05608	0.528038	+	0.849221i	$0.322928\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3496.50	0.435429	0.217714	−	0.976012i	$-0.430140\pi$
$401$	3496.50	0.435429	0.217714	+	0.976012i	$0.430140\pi$
$402$	0	0
$403$	3472.78	0.429260
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−822.655	−0.100190
$408$	0	0
$409$	−1233.74	−0.149155	−0.0745777	−	0.997215i	$-0.523761\pi$
$409$	−1233.74	−0.149155	−0.0745777	+	0.997215i	$0.523761\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−14811.3	−1.76468
$414$	0	0
$415$	−3618.42	−0.428003
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−13879.9	−1.61832	−0.809159	−	0.587590i	$-0.800077\pi$
$419$	−13879.9	−1.61832	−0.809159	+	0.587590i	$0.800077\pi$
$420$	0	0
$421$	9607.40	1.11220	0.556100	−	0.831116i	$-0.312297\pi$
$421$	9607.40	1.11220	0.556100	+	0.831116i	$0.312297\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1403.87	−0.160230
$426$	0	0
$427$	1100.94	0.124773
$428$	0	0
$429$	0	0
$430$	0	0
$431$	13349.8	1.49196	0.745981	−	0.665968i	$-0.231981\pi$
$431$	13349.8	1.49196	0.745981	+	0.665968i	$0.231981\pi$
$432$	0	0
$433$	7546.88	0.837598	0.418799	−	0.908079i	$-0.362451\pi$
$433$	7546.88	0.837598	0.418799	+	0.908079i	$0.362451\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	850.305	0.0930792
$438$	0	0
$439$	9884.96	1.07468	0.537339	−	0.843367i	$-0.319430\pi$
$439$	9884.96	1.07468	0.537339	+	0.843367i	$0.319430\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4358.12	−0.467405	−0.233703	−	0.972308i	$-0.575084\pi$
$443$	−4358.12	−0.467405	−0.233703	+	0.972308i	$0.575084\pi$
$444$	0	0
$445$	564.176	0.0601000
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4821.91	−0.506816	−0.253408	−	0.967360i	$-0.581551\pi$
$449$	−4821.91	−0.506816	−0.253408	+	0.967360i	$0.581551\pi$
$450$	0	0
$451$	1114.23	0.116335
$452$	0	0
$453$	0	0
$454$	0	0
$455$	8869.28	0.913842
$456$	0	0
$457$	7952.00	0.813958	0.406979	−	0.913438i	$-0.366582\pi$
$457$	7952.00	0.813958	0.406979	+	0.913438i	$0.366582\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	14851.4	1.50043	0.750215	−	0.661194i	$-0.229950\pi$
$461$	14851.4	1.50043	0.750215	+	0.661194i	$0.229950\pi$
$462$	0	0
$463$	−6129.34	−0.615237	−0.307618	−	0.951510i	$-0.599532\pi$
$463$	−6129.34	−0.615237	−0.307618	+	0.951510i	$0.599532\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8441.69	0.836477	0.418238	−	0.908337i	$-0.362648\pi$
$467$	8441.69	0.836477	0.418238	+	0.908337i	$0.362648\pi$
$468$	0	0
$469$	8102.99	0.797785
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1729.72	0.168145
$474$	0	0
$475$	2072.95	0.200239
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−18616.3	−1.77579	−0.887894	−	0.460049i	$-0.847832\pi$
$479$	−18616.3	−1.77579	−0.887894	+	0.460049i	$0.847832\pi$
$480$	0	0
$481$	10102.0	0.957614
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1983.72	−0.185724
$486$	0	0
$487$	−16711.5	−1.55497	−0.777484	−	0.628903i	$-0.783504\pi$
$487$	−16711.5	−1.55497	−0.777484	+	0.628903i	$0.783504\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	2982.33	0.274115	0.137058	−	0.990563i	$-0.456235\pi$
$491$	2982.33	0.274115	0.137058	+	0.990563i	$0.456235\pi$
$492$	0	0
$493$	−7627.85	−0.696838
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2197.67	0.198348
$498$	0	0
$499$	−10128.7	−0.908662	−0.454331	−	0.890833i	$-0.650122\pi$
$499$	−10128.7	−0.908662	−0.454331	+	0.890833i	$0.650122\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−20086.0	−1.78050	−0.890249	−	0.455475i	$-0.849470\pi$
$503$	−20086.0	−1.78050	−0.890249	+	0.455475i	$0.849470\pi$
$504$	0	0
$505$	−9081.67	−0.800255
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−14570.7	−1.26883	−0.634416	−	0.772992i	$-0.718759\pi$
$509$	−14570.7	−1.26883	−0.634416	+	0.772992i	$0.718759\pi$
$510$	0	0
$511$	−13940.7	−1.20685
$512$	0	0
$513$	0	0
$514$	0	0
$515$	7335.74	0.627672
$516$	0	0
$517$	−3817.94	−0.324783
$518$	0	0
$519$	0	0
$520$	0	0
$521$	16023.8	1.34743	0.673717	−	0.738989i	$-0.264697\pi$
$521$	16023.8	1.34743	0.673717	+	0.738989i	$0.264697\pi$
$522$	0	0
$523$	−18506.3	−1.54727	−0.773636	−	0.633631i	$-0.781564\pi$
$523$	−18506.3	−1.54727	−0.773636	+	0.633631i	$0.781564\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2483.10	−0.205247
$528$	0	0
$529$	−12061.8	−0.991357
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−13682.4	−1.11192
$534$	0	0
$535$	6083.88	0.491643
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1069.00	−0.0854266
$540$	0	0
$541$	−20604.1	−1.63741	−0.818704	−	0.574216i	$-0.805307\pi$
$541$	−20604.1	−1.63741	−0.818704	+	0.574216i	$0.805307\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−6051.99	−0.475668
$546$	0	0
$547$	−18362.0	−1.43529	−0.717646	−	0.696408i	$-0.754780\pi$
$547$	−18362.0	−1.43529	−0.717646	+	0.696408i	$0.754780\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	11263.3	0.870837
$552$	0	0
$553$	10135.2	0.779369
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6329.02	0.481452	0.240726	−	0.970593i	$-0.422614\pi$
$557$	6329.02	0.481452	0.240726	+	0.970593i	$0.422614\pi$
$558$	0	0
$559$	−21240.6	−1.60712
$560$	0	0
$561$	0	0
$562$	0	0
$563$	9929.10	0.743271	0.371635	−	0.928379i	$-0.378797\pi$
$563$	9929.10	0.743271	0.371635	+	0.928379i	$0.378797\pi$
$564$	0	0
$565$	11912.1	0.886987
$566$	0	0
$567$	0	0
$568$	0	0
$569$	12881.6	0.949074	0.474537	−	0.880236i	$-0.342615\pi$
$569$	12881.6	0.949074	0.474537	+	0.880236i	$0.342615\pi$
$570$	0	0
$571$	−16052.6	−1.17650	−0.588251	−	0.808679i	$-0.700183\pi$
$571$	−16052.6	−1.17650	−0.588251	+	0.808679i	$0.700183\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	256.369	0.0185936
$576$	0	0
$577$	−21368.2	−1.54172	−0.770858	−	0.637007i	$-0.780172\pi$
$577$	−21368.2	−1.54172	−0.770858	+	0.637007i	$0.780172\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−16345.4	−1.16716
$582$	0	0
$583$	1.85266	0.000131611 0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−13913.6	−0.978321	−0.489160	−	0.872194i	$-0.662697\pi$
$587$	−13913.6	−0.978321	−0.489160	+	0.872194i	$0.662697\pi$
$588$	0	0
$589$	3666.53	0.256497
$590$	0	0
$591$	0	0
$592$	0	0
$593$	5244.77	0.363199	0.181600	−	0.983373i	$-0.441873\pi$
$593$	5244.77	0.363199	0.181600	+	0.983373i	$0.441873\pi$
$594$	0	0
$595$	−6341.68	−0.436947
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−3140.90	−0.214246	−0.107123	−	0.994246i	$-0.534164\pi$
$599$	−3140.90	−0.214246	−0.107123	+	0.994246i	$0.534164\pi$
$600$	0	0
$601$	−22736.7	−1.54317	−0.771587	−	0.636124i	$-0.780537\pi$
$601$	−22736.7	−1.54317	−0.771587	+	0.636124i	$0.780537\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−6450.48	−0.433470
$606$	0	0
$607$	27.5200	0.00184020	0.000920102	−	1.00000i	$-0.499707\pi$
$607$	27.5200	0.00184020	0.000920102		1.00000i	$0.499707\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	46883.4	3.10425
$612$	0	0
$613$	−10729.4	−0.706944	−0.353472	−	0.935445i	$-0.614999\pi$
$613$	−10729.4	−0.706944	−0.353472	+	0.935445i	$0.614999\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9914.94	0.646937	0.323469	−	0.946239i	$-0.395151\pi$
$617$	9914.94	0.646937	0.323469	+	0.946239i	$0.395151\pi$
$618$	0	0
$619$	6432.47	0.417678	0.208839	−	0.977950i	$-0.433031\pi$
$619$	6432.47	0.417678	0.208839	+	0.977950i	$0.433031\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2548.54	0.163893
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−7223.10	−0.457876
$630$	0	0
$631$	−19656.6	−1.24012	−0.620061	−	0.784554i	$-0.712892\pi$
$631$	−19656.6	−1.24012	−0.620061	+	0.784554i	$0.712892\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	9277.94	0.579817
$636$	0	0
$637$	13127.0	0.816502
$638$	0	0
$639$	0	0
$640$	0	0
$641$	16342.0	1.00698	0.503488	−	0.864002i	$-0.332050\pi$
$641$	16342.0	1.00698	0.503488	+	0.864002i	$0.332050\pi$
$642$	0	0
$643$	3375.38	0.207017	0.103508	−	0.994629i	$-0.466993\pi$
$643$	3375.38	0.207017	0.103508	+	0.994629i	$0.466993\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−13862.2	−0.842319	−0.421160	−	0.906986i	$-0.638377\pi$
$647$	−13862.2	−0.842319	−0.421160	+	0.906986i	$0.638377\pi$
$648$	0	0
$649$	4193.98	0.253664
$650$	0	0
$651$	0	0
$652$	0	0
$653$	14818.2	0.888025	0.444013	−	0.896021i	$-0.353555\pi$
$653$	14818.2	0.888025	0.444013	+	0.896021i	$0.353555\pi$
$654$	0	0
$655$	−1449.21	−0.0864509
$656$	0	0
$657$	0	0
$658$	0	0
$659$	4730.05	0.279600	0.139800	−	0.990180i	$-0.455354\pi$
$659$	4730.05	0.279600	0.139800	+	0.990180i	$0.455354\pi$
$660$	0	0
$661$	10646.7	0.626486	0.313243	−	0.949673i	$-0.398585\pi$
$661$	10646.7	0.626486	0.313243	+	0.949673i	$0.398585\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	9364.10	0.546052
$666$	0	0
$667$	1392.97	0.0808635
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−311.742	−0.0179354
$672$	0	0
$673$	−4179.96	−0.239414	−0.119707	−	0.992809i	$-0.538196\pi$
$673$	−4179.96	−0.239414	−0.119707	+	0.992809i	$0.538196\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−31610.0	−1.79449	−0.897245	−	0.441534i	$-0.854435\pi$
$677$	−31610.0	−1.79449	−0.897245	+	0.441534i	$0.854435\pi$
$678$	0	0
$679$	−8961.00	−0.506468
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−12901.6	−0.722790	−0.361395	−	0.932413i	$-0.617699\pi$
$683$	−12901.6	−0.722790	−0.361395	+	0.932413i	$0.617699\pi$
$684$	0	0
$685$	−5967.61	−0.332862
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−22.7502	−0.00125793
$690$	0	0
$691$	−5650.46	−0.311076	−0.155538	−	0.987830i	$-0.549711\pi$
$691$	−5650.46	−0.311076	−0.155538	+	0.987830i	$0.549711\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3545.31	0.193498
$696$	0	0
$697$	9783.16	0.531655
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−17485.6	−0.942112	−0.471056	−	0.882103i	$-0.656127\pi$
$701$	−17485.6	−0.942112	−0.471056	+	0.882103i	$0.656127\pi$
$702$	0	0
$703$	10665.6	0.572207
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−41024.4	−2.18230
$708$	0	0
$709$	−21485.3	−1.13808	−0.569039	−	0.822310i	$-0.692685\pi$
$709$	−21485.3	−1.13808	−0.569039	+	0.822310i	$0.692685\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	453.453	0.0238176
$714$	0	0
$715$	−2511.44	−0.131360
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−3342.83	−0.173389	−0.0866944	−	0.996235i	$-0.527630\pi$
$719$	−3342.83	−0.173389	−0.0866944	+	0.996235i	$0.527630\pi$
$720$	0	0
$721$	33137.6	1.71166
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3395.90	0.173959
$726$	0	0
$727$	−29934.3	−1.52710	−0.763549	−	0.645750i	$-0.776545\pi$
$727$	−29934.3	−1.52710	−0.763549	+	0.645750i	$0.776545\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	15187.4	0.768434
$732$	0	0
$733$	8732.45	0.440028	0.220014	−	0.975497i	$-0.429390\pi$
$733$	8732.45	0.440028	0.220014	+	0.975497i	$0.429390\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2294.45	−0.114677
$738$	0	0
$739$	−25004.3	−1.24465	−0.622327	−	0.782758i	$-0.713812\pi$
$739$	−25004.3	−1.24465	−0.622327	+	0.782758i	$0.713812\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	29841.4	1.47345	0.736727	−	0.676190i	$-0.236370\pi$
$743$	29841.4	1.47345	0.736727	+	0.676190i	$0.236370\pi$
$744$	0	0
$745$	−8291.89	−0.407773
$746$	0	0
$747$	0	0
$748$	0	0
$749$	27482.6	1.34071
$750$	0	0
$751$	16131.9	0.783836	0.391918	−	0.920000i	$-0.371812\pi$
$751$	16131.9	0.783836	0.391918	+	0.920000i	$0.371812\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−5902.80	−0.284536
$756$	0	0
$757$	9728.98	0.467115	0.233557	−	0.972343i	$-0.424963\pi$
$757$	9728.98	0.467115	0.233557	+	0.972343i	$0.424963\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−12912.5	−0.615082	−0.307541	−	0.951535i	$-0.599506\pi$
$761$	−12912.5	−0.615082	−0.307541	+	0.951535i	$0.599506\pi$
$762$	0	0
$763$	−27338.6	−1.29715
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−51501.0	−2.42450
$768$	0	0
$769$	32798.5	1.53803	0.769014	−	0.639232i	$-0.220747\pi$
$769$	32798.5	1.53803	0.769014	+	0.639232i	$0.220747\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−3174.82	−0.147724	−0.0738618	−	0.997268i	$-0.523532\pi$
$773$	−3174.82	−0.147724	−0.0738618	+	0.997268i	$0.523532\pi$
$774$	0	0
$775$	1105.47	0.0512382
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−14445.8	−0.664408
$780$	0	0
$781$	−622.297	−0.0285115
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3419.77	0.155487
$786$	0	0
$787$	18315.6	0.829582	0.414791	−	0.909917i	$-0.363855\pi$
$787$	18315.6	0.829582	0.414791	+	0.909917i	$0.363855\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	53810.5	2.41881
$792$	0	0
$793$	3828.12	0.171426
$794$	0	0
$795$	0	0
$796$	0	0
$797$	32597.9	1.44878	0.724389	−	0.689391i	$-0.242122\pi$
$797$	32597.9	1.44878	0.724389	+	0.689391i	$0.242122\pi$
$798$	0	0
$799$	−33522.4	−1.48428
$800$	0	0
$801$	0	0
$802$	0	0
$803$	3947.48	0.173479
$804$	0	0
$805$	1158.09	0.0507048
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−11753.1	−0.510774	−0.255387	−	0.966839i	$-0.582203\pi$
$809$	−11753.1	−0.510774	−0.255387	+	0.966839i	$0.582203\pi$
$810$	0	0
$811$	−35501.2	−1.53713	−0.768567	−	0.639769i	$-0.779030\pi$
$811$	−35501.2	−1.53713	−0.768567	+	0.639769i	$0.779030\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	12263.4	0.527076
$816$	0	0
$817$	−22425.6	−0.960310
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−25528.6	−1.08520	−0.542602	−	0.839990i	$-0.682561\pi$
$821$	−25528.6	−1.08520	−0.542602	+	0.839990i	$0.682561\pi$
$822$	0	0
$823$	−30522.5	−1.29277	−0.646384	−	0.763013i	$-0.723719\pi$
$823$	−30522.5	−1.29277	−0.646384	+	0.763013i	$0.723719\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−43126.4	−1.81336	−0.906682	−	0.421814i	$-0.861394\pi$
$827$	−43126.4	−1.81336	−0.906682	+	0.421814i	$0.861394\pi$
$828$	0	0
$829$	−19469.8	−0.815699	−0.407849	−	0.913049i	$-0.633721\pi$
$829$	−19469.8	−0.815699	−0.407849	+	0.913049i	$0.633721\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−9386.04	−0.390404
$834$	0	0
$835$	−14278.2	−0.591759
$836$	0	0
$837$	0	0
$838$	0	0
$839$	21999.5	0.905252	0.452626	−	0.891700i	$-0.350487\pi$
$839$	21999.5	0.905252	0.452626	+	0.891700i	$0.350487\pi$
$840$	0	0
$841$	−5937.56	−0.243452
$842$	0	0
$843$	0	0
$844$	0	0
$845$	19854.9	0.808317
$846$	0	0
$847$	−29138.6	−1.18207
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1319.05	0.0531335
$852$	0	0
$853$	−4233.65	−0.169938	−0.0849691	−	0.996384i	$-0.527079\pi$
$853$	−4233.65	−0.169938	−0.0849691	+	0.996384i	$0.527079\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−26604.8	−1.06045	−0.530223	−	0.847858i	$-0.677892\pi$
$857$	−26604.8	−1.06045	−0.530223	+	0.847858i	$0.677892\pi$
$858$	0	0
$859$	32033.5	1.27237	0.636187	−	0.771535i	$-0.280511\pi$
$859$	32033.5	1.27237	0.636187	+	0.771535i	$0.280511\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−34015.1	−1.34170	−0.670851	−	0.741592i	$-0.734071\pi$
$863$	−34015.1	−1.34170	−0.670851	+	0.741592i	$0.734071\pi$
$864$	0	0
$865$	17981.7	0.706818
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2869.89	−0.112030
$870$	0	0
$871$	28175.4	1.09608
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2823.30	0.109080
$876$	0	0
$877$	34479.6	1.32759	0.663793	−	0.747916i	$-0.268946\pi$
$877$	34479.6	1.32759	0.663793	+	0.747916i	$0.268946\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−33877.5	−1.29553	−0.647765	−	0.761841i	$-0.724296\pi$
$881$	−33877.5	−1.29553	−0.647765	+	0.761841i	$0.724296\pi$
$882$	0	0
$883$	−3423.93	−0.130492	−0.0652460	−	0.997869i	$-0.520783\pi$
$883$	−3423.93	−0.130492	−0.0652460	+	0.997869i	$0.520783\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−38158.8	−1.44447	−0.722237	−	0.691646i	$-0.756886\pi$
$887$	−38158.8	−1.44447	−0.722237	+	0.691646i	$0.756886\pi$
$888$	0	0
$889$	41911.0	1.58116
$890$	0	0
$891$	0	0
$892$	0	0
$893$	49499.1	1.85490
$894$	0	0
$895$	6407.91	0.239322
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6006.51	0.222834
$900$	0	0
$901$	16.2668	0.000601470 0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	11926.9	0.438083
$906$	0	0
$907$	5349.09	0.195826	0.0979128	−	0.995195i	$-0.468783\pi$
$907$	5349.09	0.195826	0.0979128	+	0.995195i	$0.468783\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	53491.9	1.94541	0.972703	−	0.232054i	$-0.0745446\pi$
$911$	53491.9	1.94541	0.972703	+	0.232054i	$0.0745446\pi$
$912$	0	0
$913$	4628.39	0.167774
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6546.49	−0.235752
$918$	0	0
$919$	30116.2	1.08100	0.540502	−	0.841343i	$-0.318234\pi$
$919$	30116.2	1.08100	0.540502	+	0.841343i	$0.318234\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	7641.65	0.272511
$924$	0	0
$925$	3215.71	0.114305
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−3791.12	−0.133889	−0.0669444	−	0.997757i	$-0.521325\pi$
$929$	−3791.12	−0.133889	−0.0669444	+	0.997757i	$0.521325\pi$
$930$	0	0
$931$	13859.4	0.487888
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1795.72	0.0628088
$936$	0	0
$937$	−38027.5	−1.32583	−0.662916	−	0.748693i	$-0.730682\pi$
$937$	−38027.5	−1.32583	−0.662916	+	0.748693i	$0.730682\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	25378.5	0.879187	0.439594	−	0.898197i	$-0.355122\pi$
$941$	25378.5	0.879187	0.439594	+	0.898197i	$0.355122\pi$
$942$	0	0
$943$	−1786.56	−0.0616951
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−18643.7	−0.639744	−0.319872	−	0.947461i	$-0.603640\pi$
$947$	−18643.7	−0.639744	−0.319872	+	0.947461i	$0.603640\pi$
$948$	0	0
$949$	−48474.1	−1.65810
$950$	0	0
$951$	0	0
$952$	0	0
$953$	38558.2	1.31062	0.655311	−	0.755360i	$-0.272538\pi$
$953$	38558.2	1.31062	0.655311	+	0.755360i	$0.272538\pi$
$954$	0	0
$955$	4645.85	0.157420
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−26957.4	−0.907716
$960$	0	0
$961$	−27835.7	−0.934366
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−10634.9	−0.354767
$966$	0	0
$967$	33635.3	1.11855	0.559275	−	0.828982i	$-0.311080\pi$
$967$	33635.3	1.11855	0.559275	+	0.828982i	$0.311080\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	37902.0	1.25266	0.626329	−	0.779559i	$-0.284557\pi$
$971$	37902.0	1.25266	0.626329	+	0.779559i	$0.284557\pi$
$972$	0	0
$973$	16015.2	0.527670
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−50034.8	−1.63844	−0.819219	−	0.573481i	$-0.805593\pi$
$977$	−50034.8	−1.63844	−0.819219	+	0.573481i	$0.805593\pi$
$978$	0	0
$979$	−721.649	−0.0235587
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−3864.79	−0.125400	−0.0626998	−	0.998032i	$-0.519971\pi$
$983$	−3864.79	−0.125400	−0.0626998	+	0.998032i	$0.519971\pi$
$984$	0	0
$985$	−8278.64	−0.267796
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2773.46	−0.0891717
$990$	0	0
$991$	33515.9	1.07434	0.537168	−	0.843476i	$-0.319494\pi$
$991$	33515.9	1.07434	0.537168	+	0.843476i	$0.319494\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	25046.2	0.798009
$996$	0	0
$997$	−21937.2	−0.696848	−0.348424	−	0.937337i	$-0.613283\pi$
$997$	−21937.2	−0.696848	−0.348424	+	0.937337i	$0.613283\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.4.a.bk.1.3	yes	3
3.2	odd	2		1440.4.a.bi.1.3	yes	3
4.3	odd	2		1440.4.a.bj.1.1	yes	3
12.11	even	2		1440.4.a.bh.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1440.4.a.bh.1.1	✓	3	12.11	even	2
1440.4.a.bi.1.3	yes	3	3.2	odd	2
1440.4.a.bj.1.1	yes	3	4.3	odd	2
1440.4.a.bk.1.3	yes	3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1440.a (trivial)

\(f(q)\)	\(=\)	\(q+5.00000 q^{5} +22.5864 q^{7} -6.39560 q^{11} +78.5364 q^{13} -56.1548 q^{17} +82.9180 q^{19} +10.2548 q^{23} +25.0000 q^{25} +135.836 q^{29} +44.2188 q^{31} +112.932 q^{35} +128.628 q^{37} -174.218 q^{41} -270.455 q^{43} +596.964 q^{47} +167.146 q^{49} -0.289677 q^{53} -31.9780 q^{55} -655.760 q^{59} +48.7433 q^{61} +392.682 q^{65} +358.755 q^{67} +97.3007 q^{71} -617.219 q^{73} -144.454 q^{77} +448.728 q^{79} -723.684 q^{83} -280.774 q^{85} +112.835 q^{89} +1773.86 q^{91} +414.590 q^{95} -396.743 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 15 q^{5} + 14 q^{7} + 22 q^{11} + 8 q^{13} + 34 q^{17} - 4 q^{19} + 176 q^{23} + 75 q^{25} - 98 q^{29} + 88 q^{31} + 70 q^{35} + 284 q^{37} + 8 q^{41} + 504 q^{43} + 280 q^{47} - 409 q^{49} + 150 q^{53}+ \cdots - 1394 q^{97}+O(q^{100}) \) 3 * q + 15 * q^5 + 14 * q^7 + 22 * q^11 + 8 * q^13 + 34 * q^17 - 4 * q^19 + 176 * q^23 + 75 * q^25 - 98 * q^29 + 88 * q^31 + 70 * q^35 + 284 * q^37 + 8 * q^41 + 504 * q^43 + 280 * q^47 - 409 * q^49 + 150 * q^53 + 110 * q^55 + 350 * q^59 + 350 * q^61 + 40 * q^65 + 804 * q^67 + 500 * q^71 - 486 * q^73 - 788 * q^77 + 1592 * q^79 - 684 * q^83 + 170 * q^85 + 668 * q^89 + 1920 * q^91 - 20 * q^95 - 1394 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more