LMFDB - Embedded newform 1296.4.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,4,Mod(1,1296)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1296.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1296.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:	$76.4664753674$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.72153.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 3x + 12 $ x^4 - x^3 - 8x^2 + 3x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 72)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.15756$ of defining polynomial
Character	$\chi$	$=$	1296.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-17.3189 q^{5} +2.37353 q^{7} +O(q^{10})$	$q-17.3189 q^{5} +2.37353 q^{7} -52.2165 q^{11} +13.6993 q^{13} +82.9217 q^{17} +126.803 q^{19} -54.3661 q^{23} +174.944 q^{25} +212.850 q^{29} +224.122 q^{31} -41.1068 q^{35} -32.2229 q^{37} -500.662 q^{41} +12.8254 q^{43} +209.541 q^{47} -337.366 q^{49} -371.213 q^{53} +904.331 q^{55} +5.81848 q^{59} -604.032 q^{61} -237.256 q^{65} -754.066 q^{67} +43.4780 q^{71} +671.029 q^{73} -123.937 q^{77} +649.202 q^{79} +134.526 q^{83} -1436.11 q^{85} +206.076 q^{89} +32.5156 q^{91} -2196.09 q^{95} -1452.84 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 5 q^{5} + 3 q^{7}+O(q^{10}) $ 4 * q - 5 * q^5 + 3 * q^7	$ 4 q - 5 q^{5} + 3 q^{7} - 16 q^{11} - 29 q^{13} + 17 q^{17} + 109 q^{19} - 37 q^{23} - 97 q^{25} - 3 q^{29} + 331 q^{31} - 171 q^{35} - 366 q^{37} - 378 q^{41} + 506 q^{43} - 171 q^{47} - 829 q^{49} - 410 q^{53} + 1163 q^{55} - 616 q^{59} - 1331 q^{61} - 815 q^{65} + 1162 q^{67} - 344 q^{71} - 1307 q^{73} - 741 q^{77} + 1853 q^{79} - 1421 q^{83} - 2074 q^{85} - 816 q^{89} + 1995 q^{91} - 1292 q^{95} - 2506 q^{97}+O(q^{100}) $ 4 * q - 5 * q^5 + 3 * q^7 - 16 * q^11 - 29 * q^13 + 17 * q^17 + 109 * q^19 - 37 * q^23 - 97 * q^25 - 3 * q^29 + 331 * q^31 - 171 * q^35 - 366 * q^37 - 378 * q^41 + 506 * q^43 - 171 * q^47 - 829 * q^49 - 410 * q^53 + 1163 * q^55 - 616 * q^59 - 1331 * q^61 - 815 * q^65 + 1162 * q^67 - 344 * q^71 - 1307 * q^73 - 741 * q^77 + 1853 * q^79 - 1421 * q^83 - 2074 * q^85 - 816 * q^89 + 1995 * q^91 - 1292 * q^95 - 2506 * 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
$5$	−17.3189	−1.54905	−0.774524	−	0.632545i	$-0.782010\pi$
$5$	−17.3189	−1.54905	−0.774524	+	0.632545i	$0.782010\pi$
$6$	0	0
$7$	2.37353	0.128158	0.0640792	−	0.997945i	$-0.479589\pi$
$7$	2.37353	0.128158	0.0640792	+	0.997945i	$0.479589\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−52.2165	−1.43126	−0.715630	−	0.698480i	$-0.753860\pi$
$11$	−52.2165	−1.43126	−0.715630	+	0.698480i	$0.753860\pi$
$12$	0	0
$13$	13.6993	0.292269	0.146135	−	0.989265i	$-0.453317\pi$
$13$	13.6993	0.292269	0.146135	+	0.989265i	$0.453317\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	82.9217	1.18303	0.591514	−	0.806295i	$-0.298531\pi$
$17$	82.9217	1.18303	0.591514	+	0.806295i	$0.298531\pi$
$18$	0	0
$19$	126.803	1.53108	0.765542	−	0.643386i	$-0.222471\pi$
$19$	126.803	1.53108	0.765542	+	0.643386i	$0.222471\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−54.3661	−0.492875	−0.246437	−	0.969159i	$-0.579260\pi$
$23$	−54.3661	−0.492875	−0.246437	+	0.969159i	$0.579260\pi$
$24$	0	0
$25$	174.944	1.39955
$26$	0	0
$27$	0	0
$28$	0	0
$29$	212.850	1.36294	0.681471	−	0.731845i	$-0.261340\pi$
$29$	212.850	1.36294	0.681471	+	0.731845i	$0.261340\pi$
$30$	0	0
$31$	224.122	1.29850	0.649250	−	0.760575i	$-0.275083\pi$
$31$	224.122	1.29850	0.649250	+	0.760575i	$0.275083\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−41.1068	−0.198523
$36$	0	0
$37$	−32.2229	−0.143173	−0.0715866	−	0.997434i	$-0.522806\pi$
$37$	−32.2229	−0.143173	−0.0715866	+	0.997434i	$0.522806\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−500.662	−1.90708	−0.953540	−	0.301265i	$-0.902591\pi$
$41$	−500.662	−1.90708	−0.953540	+	0.301265i	$0.902591\pi$
$42$	0	0
$43$	12.8254	0.0454850	0.0227425	−	0.999741i	$-0.492760\pi$
$43$	12.8254	0.0454850	0.0227425	+	0.999741i	$0.492760\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	209.541	0.650311	0.325156	−	0.945661i	$-0.394583\pi$
$47$	209.541	0.650311	0.325156	+	0.945661i	$0.394583\pi$
$48$	0	0
$49$	−337.366	−0.983575
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−371.213	−0.962075	−0.481038	−	0.876700i	$-0.659740\pi$
$53$	−371.213	−0.962075	−0.481038	+	0.876700i	$0.659740\pi$
$54$	0	0
$55$	904.331	2.21709
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.81848	0.0128390	0.00641950	−	0.999979i	$-0.497957\pi$
$59$	5.81848	0.0128390	0.00641950	+	0.999979i	$0.497957\pi$
$60$	0	0
$61$	−604.032	−1.26784	−0.633921	−	0.773398i	$-0.718556\pi$
$61$	−604.032	−1.26784	−0.633921	+	0.773398i	$0.718556\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−237.256	−0.452739
$66$	0	0
$67$	−754.066	−1.37498	−0.687491	−	0.726193i	$-0.741288\pi$
$67$	−754.066	−1.37498	−0.687491	+	0.726193i	$0.741288\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	43.4780	0.0726744	0.0363372	−	0.999340i	$-0.488431\pi$
$71$	43.4780	0.0726744	0.0363372	+	0.999340i	$0.488431\pi$
$72$	0	0
$73$	671.029	1.07586	0.537931	−	0.842989i	$-0.319206\pi$
$73$	671.029	1.07586	0.537931	+	0.842989i	$0.319206\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−123.937	−0.183428
$78$	0	0
$79$	649.202	0.924569	0.462284	−	0.886732i	$-0.347030\pi$
$79$	649.202	0.924569	0.462284	+	0.886732i	$0.347030\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	134.526	0.177905	0.0889526	−	0.996036i	$-0.471648\pi$
$83$	134.526	0.177905	0.0889526	+	0.996036i	$0.471648\pi$
$84$	0	0
$85$	−1436.11	−1.83257
$86$	0	0
$87$	0	0
$88$	0	0
$89$	206.076	0.245439	0.122719	−	0.992441i	$-0.460838\pi$
$89$	206.076	0.245439	0.122719	+	0.992441i	$0.460838\pi$
$90$	0	0
$91$	32.5156	0.0374567
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2196.09	−2.37172
$96$	0	0
$97$	−1452.84	−1.52076	−0.760379	−	0.649480i	$-0.774987\pi$
$97$	−1452.84	−1.52076	−0.760379	+	0.649480i	$0.774987\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−203.960	−0.200938	−0.100469	−	0.994940i	$-0.532034\pi$
$101$	−203.960	−0.200938	−0.100469	+	0.994940i	$0.532034\pi$
$102$	0	0
$103$	1555.55	1.48808	0.744042	−	0.668133i	$-0.232906\pi$
$103$	1555.55	1.48808	0.744042	+	0.668133i	$0.232906\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−584.436	−0.528033	−0.264016	−	0.964518i	$-0.585047\pi$
$107$	−584.436	−0.528033	−0.264016	+	0.964518i	$0.585047\pi$
$108$	0	0
$109$	−782.671	−0.687764	−0.343882	−	0.939013i	$-0.611742\pi$
$109$	−782.671	−0.687764	−0.343882	+	0.939013i	$0.611742\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	870.249	0.724479	0.362239	−	0.932085i	$-0.382012\pi$
$113$	870.249	0.724479	0.362239	+	0.932085i	$0.382012\pi$
$114$	0	0
$115$	941.561	0.763487
$116$	0	0
$117$	0	0
$118$	0	0
$119$	196.817	0.151615
$120$	0	0
$121$	1395.56	1.04851
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−864.968	−0.618921
$126$	0	0
$127$	−994.050	−0.694548	−0.347274	−	0.937764i	$-0.612893\pi$
$127$	−994.050	−0.694548	−0.347274	+	0.937764i	$0.612893\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−802.079	−0.534946	−0.267473	−	0.963565i	$-0.586189\pi$
$131$	−802.079	−0.534946	−0.267473	+	0.963565i	$0.586189\pi$
$132$	0	0
$133$	300.970	0.196221
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1319.87	−0.823096	−0.411548	−	0.911388i	$-0.635012\pi$
$137$	−1319.87	−0.823096	−0.411548	+	0.911388i	$0.635012\pi$
$138$	0	0
$139$	−526.793	−0.321453	−0.160727	−	0.986999i	$-0.551384\pi$
$139$	−526.793	−0.321453	−0.160727	+	0.986999i	$0.551384\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−715.329	−0.418313
$144$	0	0
$145$	−3686.33	−2.11126
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3077.75	−1.69221	−0.846105	−	0.533017i	$-0.821058\pi$
$149$	−3077.75	−1.69221	−0.846105	+	0.533017i	$0.821058\pi$
$150$	0	0
$151$	−82.1909	−0.0442953	−0.0221477	−	0.999755i	$-0.507050\pi$
$151$	−82.1909	−0.0442953	−0.0221477	+	0.999755i	$0.507050\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3881.54	−2.01144
$156$	0	0
$157$	−2384.18	−1.21196	−0.605981	−	0.795479i	$-0.707219\pi$
$157$	−2384.18	−1.21196	−0.605981	+	0.795479i	$0.707219\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−129.039	−0.0631660
$162$	0	0
$163$	−1226.77	−0.589496	−0.294748	−	0.955575i	$-0.595236\pi$
$163$	−1226.77	−0.589496	−0.294748	+	0.955575i	$0.595236\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3896.40	1.80547	0.902733	−	0.430202i	$-0.141558\pi$
$167$	3896.40	1.80547	0.902733	+	0.430202i	$0.141558\pi$
$168$	0	0
$169$	−2009.33	−0.914579
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−146.404	−0.0643403	−0.0321701	−	0.999482i	$-0.510242\pi$
$173$	−146.404	−0.0643403	−0.0321701	+	0.999482i	$0.510242\pi$
$174$	0	0
$175$	415.233	0.179364
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1540.84	−0.643396	−0.321698	−	0.946842i	$-0.604254\pi$
$179$	−1540.84	−0.643396	−0.321698	+	0.946842i	$0.604254\pi$
$180$	0	0
$181$	1173.94	0.482090	0.241045	−	0.970514i	$-0.422510\pi$
$181$	1173.94	0.482090	0.241045	+	0.970514i	$0.422510\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	558.065	0.221782
$186$	0	0
$187$	−4329.88	−1.69322
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2193.64	−0.831026	−0.415513	−	0.909587i	$-0.636398\pi$
$191$	−2193.64	−0.831026	−0.415513	+	0.909587i	$0.636398\pi$
$192$	0	0
$193$	2463.31	0.918721	0.459360	−	0.888250i	$-0.348079\pi$
$193$	2463.31	0.918721	0.459360	+	0.888250i	$0.348079\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3593.52	−1.29963	−0.649816	−	0.760091i	$-0.725154\pi$
$197$	−3593.52	−1.29963	−0.649816	+	0.760091i	$0.725154\pi$
$198$	0	0
$199$	4599.36	1.63839	0.819197	−	0.573513i	$-0.194420\pi$
$199$	4599.36	1.63839	0.819197	+	0.573513i	$0.194420\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	505.206	0.174672
$204$	0	0
$205$	8670.91	2.95416
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−6621.20	−2.19138
$210$	0	0
$211$	2332.24	0.760938	0.380469	−	0.924794i	$-0.375763\pi$
$211$	2332.24	0.760938	0.380469	+	0.924794i	$0.375763\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−222.122	−0.0704584
$216$	0	0
$217$	531.959	0.166414
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1135.97	0.345763
$222$	0	0
$223$	4057.10	1.21831	0.609156	−	0.793050i	$-0.291508\pi$
$223$	4057.10	1.21831	0.609156	+	0.793050i	$0.291508\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1297.25	0.379303	0.189651	−	0.981852i	$-0.439264\pi$
$227$	1297.25	0.379303	0.189651	+	0.981852i	$0.439264\pi$
$228$	0	0
$229$	3111.96	0.898010	0.449005	−	0.893529i	$-0.351779\pi$
$229$	3111.96	0.898010	0.449005	+	0.893529i	$0.351779\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2034.74	−0.572103	−0.286052	−	0.958214i	$-0.592343\pi$
$233$	−2034.74	−0.572103	−0.286052	+	0.958214i	$0.592343\pi$
$234$	0	0
$235$	−3629.01	−1.00736
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1468.46	0.397434	0.198717	−	0.980057i	$-0.436323\pi$
$239$	1468.46	0.397434	0.198717	+	0.980057i	$0.436323\pi$
$240$	0	0
$241$	659.163	0.176184	0.0880922	−	0.996112i	$-0.471923\pi$
$241$	659.163	0.176184	0.0880922	+	0.996112i	$0.471923\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	5842.81	1.52361
$246$	0	0
$247$	1737.11	0.447489
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5509.13	−1.38539	−0.692696	−	0.721230i	$-0.743577\pi$
$251$	−5509.13	−1.38539	−0.692696	+	0.721230i	$0.743577\pi$
$252$	0	0
$253$	2838.81	0.705432
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3667.63	−0.890197	−0.445098	−	0.895482i	$-0.646831\pi$
$257$	−3667.63	−0.890197	−0.445098	+	0.895482i	$0.646831\pi$
$258$	0	0
$259$	−76.4819	−0.0183489
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4992.77	−1.17060	−0.585299	−	0.810818i	$-0.699023\pi$
$263$	−4992.77	−1.17060	−0.585299	+	0.810818i	$0.699023\pi$
$264$	0	0
$265$	6428.99	1.49030
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−88.5142	−0.0200625	−0.0100312	−	0.999950i	$-0.503193\pi$
$269$	−88.5142	−0.0200625	−0.0100312	+	0.999950i	$0.503193\pi$
$270$	0	0
$271$	−5226.56	−1.17155	−0.585777	−	0.810472i	$-0.699211\pi$
$271$	−5226.56	−1.17155	−0.585777	+	0.810472i	$0.699211\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−9134.94	−2.00312
$276$	0	0
$277$	−2112.25	−0.458170	−0.229085	−	0.973406i	$-0.573573\pi$
$277$	−2112.25	−0.458170	−0.229085	+	0.973406i	$0.573573\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1725.14	−0.366239	−0.183120	−	0.983091i	$-0.558620\pi$
$281$	−1725.14	−0.366239	−0.183120	+	0.983091i	$0.558620\pi$
$282$	0	0
$283$	−3341.79	−0.701940	−0.350970	−	0.936387i	$-0.614148\pi$
$283$	−3341.79	−0.701940	−0.350970	+	0.936387i	$0.614148\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1188.33	−0.244408
$288$	0	0
$289$	1963.01	0.399554
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5258.42	1.04846	0.524232	−	0.851575i	$-0.324352\pi$
$293$	5258.42	1.04846	0.524232	+	0.851575i	$0.324352\pi$
$294$	0	0
$295$	−100.770	−0.0198882
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−744.778	−0.144052
$300$	0	0
$301$	30.4414	0.00582928
$302$	0	0
$303$	0	0
$304$	0	0
$305$	10461.2	1.96395
$306$	0	0
$307$	−2582.56	−0.480112	−0.240056	−	0.970759i	$-0.577166\pi$
$307$	−2582.56	−0.480112	−0.240056	+	0.970759i	$0.577166\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−5088.53	−0.927794	−0.463897	−	0.885889i	$-0.653549\pi$
$311$	−5088.53	−0.927794	−0.463897	+	0.885889i	$0.653549\pi$
$312$	0	0
$313$	−2931.06	−0.529308	−0.264654	−	0.964343i	$-0.585258\pi$
$313$	−2931.06	−0.529308	−0.264654	+	0.964343i	$0.585258\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3599.57	−0.637766	−0.318883	−	0.947794i	$-0.603308\pi$
$317$	−3599.57	−0.637766	−0.318883	+	0.947794i	$0.603308\pi$
$318$	0	0
$319$	−11114.3	−1.95073
$320$	0	0
$321$	0	0
$322$	0	0
$323$	10514.7	1.81131
$324$	0	0
$325$	2396.60	0.409045
$326$	0	0
$327$	0	0
$328$	0	0
$329$	497.350	0.0833428
$330$	0	0
$331$	−3677.81	−0.610727	−0.305364	−	0.952236i	$-0.598778\pi$
$331$	−3677.81	−0.610727	−0.305364	+	0.952236i	$0.598778\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	13059.6	2.12991
$336$	0	0
$337$	−5556.04	−0.898091	−0.449046	−	0.893509i	$-0.648236\pi$
$337$	−5556.04	−0.898091	−0.449046	+	0.893509i	$0.648236\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−11702.9	−1.85849
$342$	0	0
$343$	−1614.87	−0.254212
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3069.12	−0.474810	−0.237405	−	0.971411i	$-0.576297\pi$
$347$	−3069.12	−0.474810	−0.237405	+	0.971411i	$0.576297\pi$
$348$	0	0
$349$	−1191.43	−0.182739	−0.0913694	−	0.995817i	$-0.529124\pi$
$349$	−1191.43	−0.182739	−0.0913694	+	0.995817i	$0.529124\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6431.00	−0.969653	−0.484827	−	0.874610i	$-0.661117\pi$
$353$	−6431.00	−0.969653	−0.484827	+	0.874610i	$0.661117\pi$
$354$	0	0
$355$	−752.989	−0.112576
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4236.73	0.622858	0.311429	−	0.950270i	$-0.399192\pi$
$359$	4236.73	0.622858	0.311429	+	0.950270i	$0.399192\pi$
$360$	0	0
$361$	9219.99	1.34422
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−11621.5	−1.66656
$366$	0	0
$367$	−7372.78	−1.04865	−0.524327	−	0.851517i	$-0.675683\pi$
$367$	−7372.78	−1.04865	−0.524327	+	0.851517i	$0.675683\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−881.083	−0.123298
$372$	0	0
$373$	−11985.0	−1.66370	−0.831852	−	0.554998i	$-0.812719\pi$
$373$	−11985.0	−1.66370	−0.831852	+	0.554998i	$0.812719\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2915.90	0.398346
$378$	0	0
$379$	3872.02	0.524782	0.262391	−	0.964962i	$-0.415489\pi$
$379$	3872.02	0.524782	0.262391	+	0.964962i	$0.415489\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7242.66	0.966274	0.483137	−	0.875545i	$-0.339497\pi$
$383$	7242.66	0.966274	0.483137	+	0.875545i	$0.339497\pi$
$384$	0	0
$385$	2146.45	0.284139
$386$	0	0
$387$	0	0
$388$	0	0
$389$	9732.33	1.26851	0.634253	−	0.773126i	$-0.281308\pi$
$389$	9732.33	1.26851	0.634253	+	0.773126i	$0.281308\pi$
$390$	0	0
$391$	−4508.13	−0.583085
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−11243.5	−1.43220
$396$	0	0
$397$	−1393.66	−0.176186	−0.0880930	−	0.996112i	$-0.528077\pi$
$397$	−1393.66	−0.176186	−0.0880930	+	0.996112i	$0.528077\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1225.66	0.152634	0.0763171	−	0.997084i	$-0.475684\pi$
$401$	1225.66	0.152634	0.0763171	+	0.997084i	$0.475684\pi$
$402$	0	0
$403$	3070.31	0.379512
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1682.57	0.204918
$408$	0	0
$409$	8162.83	0.986860	0.493430	−	0.869785i	$-0.335743\pi$
$409$	8162.83	0.986860	0.493430	+	0.869785i	$0.335743\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	13.8103	0.00164543
$414$	0	0
$415$	−2329.84	−0.275584
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2821.60	0.328984	0.164492	−	0.986378i	$-0.447402\pi$
$419$	2821.60	0.328984	0.164492	+	0.986378i	$0.447402\pi$
$420$	0	0
$421$	−4639.90	−0.537138	−0.268569	−	0.963260i	$-0.586551\pi$
$421$	−4639.90	−0.537138	−0.268569	+	0.963260i	$0.586551\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	14506.6	1.65571
$426$	0	0
$427$	−1433.69	−0.162485
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5998.72	0.670413	0.335207	−	0.942145i	$-0.391194\pi$
$431$	5998.72	0.670413	0.335207	+	0.942145i	$0.391194\pi$
$432$	0	0
$433$	−10187.5	−1.13067	−0.565334	−	0.824862i	$-0.691253\pi$
$433$	−10187.5	−1.13067	−0.565334	+	0.824862i	$0.691253\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6893.79	−0.754633
$438$	0	0
$439$	−15745.4	−1.71182	−0.855910	−	0.517125i	$-0.827002\pi$
$439$	−15745.4	−1.71182	−0.855910	+	0.517125i	$0.827002\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2433.66	0.261009	0.130504	−	0.991448i	$-0.458340\pi$
$443$	2433.66	0.261009	0.130504	+	0.991448i	$0.458340\pi$
$444$	0	0
$445$	−3569.01	−0.380197
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−7394.75	−0.777238	−0.388619	−	0.921399i	$-0.627048\pi$
$449$	−7394.75	−0.777238	−0.388619	+	0.921399i	$0.627048\pi$
$450$	0	0
$451$	26142.8	2.72953
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−563.134	−0.0580223
$456$	0	0
$457$	2129.91	0.218015	0.109008	−	0.994041i	$-0.465233\pi$
$457$	2129.91	0.218015	0.109008	+	0.994041i	$0.465233\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8313.78	0.839938	0.419969	−	0.907538i	$-0.362041\pi$
$461$	8313.78	0.839938	0.419969	+	0.907538i	$0.362041\pi$
$462$	0	0
$463$	1845.59	0.185253	0.0926263	−	0.995701i	$-0.470474\pi$
$463$	1845.59	0.185253	0.0926263	+	0.995701i	$0.470474\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−15653.2	−1.55106	−0.775529	−	0.631312i	$-0.782517\pi$
$467$	−15653.2	−1.55106	−0.775529	+	0.631312i	$0.782517\pi$
$468$	0	0
$469$	−1789.79	−0.176215
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−669.697	−0.0651009
$474$	0	0
$475$	22183.4	2.14283
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1973.95	0.188292	0.0941462	−	0.995558i	$-0.469988\pi$
$479$	1973.95	0.188292	0.0941462	+	0.995558i	$0.469988\pi$
$480$	0	0
$481$	−441.431	−0.0418451
$482$	0	0
$483$	0	0
$484$	0	0
$485$	25161.6	2.35573
$486$	0	0
$487$	1753.49	0.163158	0.0815792	−	0.996667i	$-0.474004\pi$
$487$	1753.49	0.163158	0.0815792	+	0.996667i	$0.474004\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	6222.18	0.571900	0.285950	−	0.958245i	$-0.407691\pi$
$491$	6222.18	0.571900	0.285950	+	0.958245i	$0.407691\pi$
$492$	0	0
$493$	17649.9	1.61240
$494$	0	0
$495$	0	0
$496$	0	0
$497$	103.196	0.00931383
$498$	0	0
$499$	17293.8	1.55145	0.775727	−	0.631069i	$-0.217384\pi$
$499$	17293.8	1.55145	0.775727	+	0.631069i	$0.217384\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	20578.6	1.82416	0.912082	−	0.410007i	$-0.134474\pi$
$503$	20578.6	1.82416	0.912082	+	0.410007i	$0.134474\pi$
$504$	0	0
$505$	3532.35	0.311263
$506$	0	0
$507$	0	0
$508$	0	0
$509$	18555.1	1.61580	0.807900	−	0.589319i	$-0.200604\pi$
$509$	18555.1	1.61580	0.807900	+	0.589319i	$0.200604\pi$
$510$	0	0
$511$	1592.70	0.137881
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−26940.3	−2.30511
$516$	0	0
$517$	−10941.5	−0.930764
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5321.51	0.447485	0.223742	−	0.974648i	$-0.428173\pi$
$521$	5321.51	0.447485	0.223742	+	0.974648i	$0.428173\pi$
$522$	0	0
$523$	−7766.94	−0.649378	−0.324689	−	0.945821i	$-0.605260\pi$
$523$	−7766.94	−0.649378	−0.324689	+	0.945821i	$0.605260\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	18584.6	1.53616
$528$	0	0
$529$	−9211.32	−0.757074
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−6858.72	−0.557381
$534$	0	0
$535$	10121.8	0.817948
$536$	0	0
$537$	0	0
$538$	0	0
$539$	17616.1	1.40775
$540$	0	0
$541$	−1292.25	−0.102695	−0.0513476	−	0.998681i	$-0.516352\pi$
$541$	−1292.25	−0.102695	−0.0513476	+	0.998681i	$0.516352\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	13555.0	1.06538
$546$	0	0
$547$	−7532.95	−0.588822	−0.294411	−	0.955679i	$-0.595124\pi$
$547$	−7532.95	−0.588822	−0.294411	+	0.955679i	$0.595124\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	26990.1	2.08678
$552$	0	0
$553$	1540.90	0.118491
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9856.22	0.749769	0.374884	−	0.927072i	$-0.377682\pi$
$557$	9856.22	0.749769	0.374884	+	0.927072i	$0.377682\pi$
$558$	0	0
$559$	175.699	0.0132939
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−10282.2	−0.769704	−0.384852	−	0.922978i	$-0.625747\pi$
$563$	−10282.2	−0.769704	−0.384852	+	0.922978i	$0.625747\pi$
$564$	0	0
$565$	−15071.7	−1.12225
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1322.64	0.0974483	0.0487241	−	0.998812i	$-0.484484\pi$
$569$	1322.64	0.0974483	0.0487241	+	0.998812i	$0.484484\pi$
$570$	0	0
$571$	4254.58	0.311819	0.155909	−	0.987771i	$-0.450169\pi$
$571$	4254.58	0.311819	0.155909	+	0.987771i	$0.450169\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−9511.01	−0.689803
$576$	0	0
$577$	−13268.4	−0.957312	−0.478656	−	0.878003i	$-0.658876\pi$
$577$	−13268.4	−0.957312	−0.478656	+	0.878003i	$0.658876\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	319.301	0.0228000
$582$	0	0
$583$	19383.4	1.37698
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4710.50	0.331215	0.165608	−	0.986192i	$-0.447041\pi$
$587$	4710.50	0.331215	0.165608	+	0.986192i	$0.447041\pi$
$588$	0	0
$589$	28419.3	1.98811
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−16303.6	−1.12902	−0.564510	−	0.825426i	$-0.690935\pi$
$593$	−16303.6	−1.12902	−0.564510	+	0.825426i	$0.690935\pi$
$594$	0	0
$595$	−3408.65	−0.234859
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−10558.9	−0.720240	−0.360120	−	0.932906i	$-0.617264\pi$
$599$	−10558.9	−0.720240	−0.360120	+	0.932906i	$0.617264\pi$
$600$	0	0
$601$	−1872.35	−0.127080	−0.0635399	−	0.997979i	$-0.520239\pi$
$601$	−1872.35	−0.127080	−0.0635399	+	0.997979i	$0.520239\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−24169.5	−1.62418
$606$	0	0
$607$	23071.7	1.54276	0.771378	−	0.636377i	$-0.219568\pi$
$607$	23071.7	1.54276	0.771378	+	0.636377i	$0.219568\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2870.56	0.190066
$612$	0	0
$613$	−24292.0	−1.60056	−0.800281	−	0.599625i	$-0.795317\pi$
$613$	−24292.0	−1.60056	−0.800281	+	0.599625i	$0.795317\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7773.95	0.507240	0.253620	−	0.967304i	$-0.418379\pi$
$617$	7773.95	0.507240	0.253620	+	0.967304i	$0.418379\pi$
$618$	0	0
$619$	30575.1	1.98533	0.992664	−	0.120905i	$-0.0385795\pi$
$619$	30575.1	1.98533	0.992664	+	0.120905i	$0.0385795\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	489.128	0.0314550
$624$	0	0
$625$	−6887.68	−0.440811
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−2671.98	−0.169378
$630$	0	0
$631$	−14474.9	−0.913209	−0.456605	−	0.889670i	$-0.650935\pi$
$631$	−14474.9	−0.913209	−0.456605	+	0.889670i	$0.650935\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	17215.8	1.07589
$636$	0	0
$637$	−4621.68	−0.287469
$638$	0	0
$639$	0	0
$640$	0	0
$641$	13683.8	0.843179	0.421589	−	0.906787i	$-0.361472\pi$
$641$	13683.8	0.843179	0.421589	+	0.906787i	$0.361472\pi$
$642$	0	0
$643$	−16015.5	−0.982255	−0.491128	−	0.871088i	$-0.663415\pi$
$643$	−16015.5	−0.982255	−0.491128	+	0.871088i	$0.663415\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12224.1	0.742783	0.371392	−	0.928476i	$-0.378881\pi$
$647$	12224.1	0.742783	0.371392	+	0.928476i	$0.378881\pi$
$648$	0	0
$649$	−303.821	−0.0183760
$650$	0	0
$651$	0	0
$652$	0	0
$653$	16129.9	0.966636	0.483318	−	0.875445i	$-0.339431\pi$
$653$	16129.9	0.966636	0.483318	+	0.875445i	$0.339431\pi$
$654$	0	0
$655$	13891.1	0.828657
$656$	0	0
$657$	0	0
$658$	0	0
$659$	7158.63	0.423157	0.211579	−	0.977361i	$-0.432140\pi$
$659$	7158.63	0.423157	0.211579	+	0.977361i	$0.432140\pi$
$660$	0	0
$661$	9520.06	0.560193	0.280096	−	0.959972i	$-0.409634\pi$
$661$	9520.06	0.560193	0.280096	+	0.959972i	$0.409634\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5212.46	−0.303956
$666$	0	0
$667$	−11571.9	−0.671760
$668$	0	0
$669$	0	0
$670$	0	0
$671$	31540.4	1.81461
$672$	0	0
$673$	−7603.64	−0.435511	−0.217755	−	0.976003i	$-0.569874\pi$
$673$	−7603.64	−0.435511	−0.217755	+	0.976003i	$0.569874\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4951.96	0.281122	0.140561	−	0.990072i	$-0.455109\pi$
$677$	4951.96	0.281122	0.140561	+	0.990072i	$0.455109\pi$
$678$	0	0
$679$	−3448.35	−0.194898
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−28231.4	−1.58162	−0.790809	−	0.612063i	$-0.790340\pi$
$683$	−28231.4	−1.58162	−0.790809	+	0.612063i	$0.790340\pi$
$684$	0	0
$685$	22858.7	1.27502
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−5085.35	−0.281185
$690$	0	0
$691$	−25510.3	−1.40442	−0.702212	−	0.711968i	$-0.747804\pi$
$691$	−25510.3	−1.40442	−0.702212	+	0.711968i	$0.747804\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	9123.47	0.497947
$696$	0	0
$697$	−41515.8	−2.25613
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−17524.1	−0.944191	−0.472095	−	0.881547i	$-0.656502\pi$
$701$	−17524.1	−0.944191	−0.472095	+	0.881547i	$0.656502\pi$
$702$	0	0
$703$	−4085.96	−0.219210
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−484.103	−0.0257519
$708$	0	0
$709$	−34981.6	−1.85298	−0.926490	−	0.376319i	$-0.877190\pi$
$709$	−34981.6	−1.85298	−0.926490	+	0.376319i	$0.877190\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−12184.6	−0.639998
$714$	0	0
$715$	12388.7	0.647987
$716$	0	0
$717$	0	0
$718$	0	0
$719$	30485.5	1.58125	0.790623	−	0.612303i	$-0.209757\pi$
$719$	30485.5	1.58125	0.790623	+	0.612303i	$0.209757\pi$
$720$	0	0
$721$	3692.13	0.190710
$722$	0	0
$723$	0	0
$724$	0	0
$725$	37236.8	1.90751
$726$	0	0
$727$	−26697.8	−1.36199	−0.680994	−	0.732289i	$-0.738452\pi$
$727$	−26697.8	−1.36199	−0.680994	+	0.732289i	$0.738452\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1063.50	0.0538100
$732$	0	0
$733$	30516.9	1.53775	0.768874	−	0.639401i	$-0.220818\pi$
$733$	30516.9	1.53775	0.768874	+	0.639401i	$0.220818\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	39374.7	1.96796
$738$	0	0
$739$	20020.0	0.996544	0.498272	−	0.867021i	$-0.333968\pi$
$739$	20020.0	0.996544	0.498272	+	0.867021i	$0.333968\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	16034.7	0.791731	0.395866	−	0.918308i	$-0.370445\pi$
$743$	16034.7	0.791731	0.395866	+	0.918308i	$0.370445\pi$
$744$	0	0
$745$	53303.2	2.62131
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1387.17	−0.0676718
$750$	0	0
$751$	−24725.1	−1.20137	−0.600686	−	0.799485i	$-0.705106\pi$
$751$	−24725.1	−1.20137	−0.600686	+	0.799485i	$0.705106\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1423.45	0.0686156
$756$	0	0
$757$	−13567.6	−0.651419	−0.325709	−	0.945470i	$-0.605603\pi$
$757$	−13567.6	−0.651419	−0.325709	+	0.945470i	$0.605603\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2217.39	0.105625	0.0528124	−	0.998604i	$-0.483181\pi$
$761$	2217.39	0.105625	0.0528124	+	0.998604i	$0.483181\pi$
$762$	0	0
$763$	−1857.69	−0.0881427
$764$	0	0
$765$	0	0
$766$	0	0
$767$	79.7091	0.00375245
$768$	0	0
$769$	−34899.0	−1.63653	−0.818263	−	0.574844i	$-0.805063\pi$
$769$	−34899.0	−1.63653	−0.818263	+	0.574844i	$0.805063\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	10244.1	0.476656	0.238328	−	0.971185i	$-0.423401\pi$
$773$	10244.1	0.476656	0.238328	+	0.971185i	$0.423401\pi$
$774$	0	0
$775$	39208.7	1.81732
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−63485.5	−2.91990
$780$	0	0
$781$	−2270.27	−0.104016
$782$	0	0
$783$	0	0
$784$	0	0
$785$	41291.3	1.87739
$786$	0	0
$787$	10861.5	0.491959	0.245979	−	0.969275i	$-0.420890\pi$
$787$	10861.5	0.491959	0.245979	+	0.969275i	$0.420890\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2065.56	0.0928480
$792$	0	0
$793$	−8274.81	−0.370551
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−35604.8	−1.58242	−0.791209	−	0.611546i	$-0.790548\pi$
$797$	−35604.8	−1.58242	−0.791209	+	0.611546i	$0.790548\pi$
$798$	0	0
$799$	17375.5	0.769336
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−35038.7	−1.53984
$804$	0	0
$805$	2234.82	0.0978472
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−13929.4	−0.605355	−0.302678	−	0.953093i	$-0.597881\pi$
$809$	−13929.4	−0.605355	−0.302678	+	0.953093i	$0.597881\pi$
$810$	0	0
$811$	−41996.6	−1.81837	−0.909187	−	0.416388i	$-0.863296\pi$
$811$	−41996.6	−1.81837	−0.909187	+	0.416388i	$0.863296\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	21246.2	0.913157
$816$	0	0
$817$	1626.30	0.0696414
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−21080.1	−0.896104	−0.448052	−	0.894007i	$-0.647882\pi$
$821$	−21080.1	−0.896104	−0.448052	+	0.894007i	$0.647882\pi$
$822$	0	0
$823$	−34365.1	−1.45552	−0.727760	−	0.685832i	$-0.759439\pi$
$823$	−34365.1	−1.45552	−0.727760	+	0.685832i	$0.759439\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−32925.7	−1.38445	−0.692225	−	0.721682i	$-0.743369\pi$
$827$	−32925.7	−1.38445	−0.692225	+	0.721682i	$0.743369\pi$
$828$	0	0
$829$	21195.7	0.888008	0.444004	−	0.896025i	$-0.353558\pi$
$829$	21195.7	0.888008	0.444004	+	0.896025i	$0.353558\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−27975.0	−1.16360
$834$	0	0
$835$	−67481.4	−2.79675
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−23736.8	−0.976740	−0.488370	−	0.872637i	$-0.662408\pi$
$839$	−23736.8	−0.976740	−0.488370	+	0.872637i	$0.662408\pi$
$840$	0	0
$841$	20916.3	0.857613
$842$	0	0
$843$	0	0
$844$	0	0
$845$	34799.3	1.41673
$846$	0	0
$847$	3312.40	0.134375
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1751.83	0.0705665
$852$	0	0
$853$	21928.5	0.880208	0.440104	−	0.897947i	$-0.354942\pi$
$853$	21928.5	0.880208	0.440104	+	0.897947i	$0.354942\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	40332.7	1.60763	0.803814	−	0.594881i	$-0.202801\pi$
$857$	40332.7	1.60763	0.803814	+	0.594881i	$0.202801\pi$
$858$	0	0
$859$	−22945.1	−0.911382	−0.455691	−	0.890138i	$-0.650608\pi$
$859$	−22945.1	−0.911382	−0.455691	+	0.890138i	$0.650608\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−40453.2	−1.59565	−0.797824	−	0.602890i	$-0.794016\pi$
$863$	−40453.2	−1.59565	−0.797824	+	0.602890i	$0.794016\pi$
$864$	0	0
$865$	2535.55	0.0996662
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−33899.1	−1.32330
$870$	0	0
$871$	−10330.2	−0.401865
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2053.02	−0.0793198
$876$	0	0
$877$	4561.41	0.175631	0.0878153	−	0.996137i	$-0.472011\pi$
$877$	4561.41	0.175631	0.0878153	+	0.996137i	$0.472011\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−31553.9	−1.20667	−0.603336	−	0.797487i	$-0.706162\pi$
$881$	−31553.9	−1.20667	−0.603336	+	0.797487i	$0.706162\pi$
$882$	0	0
$883$	−22726.0	−0.866128	−0.433064	−	0.901363i	$-0.642568\pi$
$883$	−22726.0	−0.866128	−0.433064	+	0.901363i	$0.642568\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1634.36	0.0618675	0.0309337	−	0.999521i	$-0.490152\pi$
$887$	1634.36	0.0618675	0.0309337	+	0.999521i	$0.490152\pi$
$888$	0	0
$889$	−2359.40	−0.0890121
$890$	0	0
$891$	0	0
$892$	0	0
$893$	26570.4	0.995681
$894$	0	0
$895$	26685.6	0.996651
$896$	0	0
$897$	0	0
$898$	0	0
$899$	47704.5	1.76978
$900$	0	0
$901$	−30781.6	−1.13816
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−20331.3	−0.746781
$906$	0	0
$907$	6535.33	0.239253	0.119626	−	0.992819i	$-0.461830\pi$
$907$	6535.33	0.239253	0.119626	+	0.992819i	$0.461830\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	46086.7	1.67609	0.838046	−	0.545600i	$-0.183698\pi$
$911$	46086.7	1.67609	0.838046	+	0.545600i	$0.183698\pi$
$912$	0	0
$913$	−7024.47	−0.254629
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1903.75	−0.0685578
$918$	0	0
$919$	−1857.94	−0.0666898	−0.0333449	−	0.999444i	$-0.510616\pi$
$919$	−1857.94	−0.0666898	−0.0333449	+	0.999444i	$0.510616\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	595.617	0.0212405
$924$	0	0
$925$	−5637.19	−0.200378
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−24301.6	−0.858246	−0.429123	−	0.903246i	$-0.641177\pi$
$929$	−24301.6	−0.858246	−0.429123	+	0.903246i	$0.641177\pi$
$930$	0	0
$931$	−42779.1	−1.50594
$932$	0	0
$933$	0	0
$934$	0	0
$935$	74988.7	2.62288
$936$	0	0
$937$	52069.9	1.81542	0.907710	−	0.419597i	$-0.137829\pi$
$937$	52069.9	1.81542	0.907710	+	0.419597i	$0.137829\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−15709.0	−0.544206	−0.272103	−	0.962268i	$-0.587719\pi$
$941$	−15709.0	−0.544206	−0.272103	+	0.962268i	$0.587719\pi$
$942$	0	0
$943$	27219.1	0.939952
$944$	0	0
$945$	0	0
$946$	0	0
$947$	52488.7	1.80111	0.900556	−	0.434740i	$-0.143160\pi$
$947$	52488.7	1.80111	0.900556	+	0.434740i	$0.143160\pi$
$948$	0	0
$949$	9192.62	0.314441
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2447.37	−0.0831880	−0.0415940	−	0.999135i	$-0.513244\pi$
$953$	−2447.37	−0.0831880	−0.0415940	+	0.999135i	$0.513244\pi$
$954$	0	0
$955$	37991.3	1.28730
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3132.75	−0.105487
$960$	0	0
$961$	20439.7	0.686103
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−42661.8	−1.42314
$966$	0	0
$967$	29392.2	0.977447	0.488723	−	0.872439i	$-0.337463\pi$
$967$	29392.2	0.977447	0.488723	+	0.872439i	$0.337463\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−32763.9	−1.08285	−0.541423	−	0.840751i	$-0.682114\pi$
$971$	−32763.9	−1.08285	−0.541423	+	0.840751i	$0.682114\pi$
$972$	0	0
$973$	−1250.36	−0.0411969
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−669.381	−0.0219195	−0.0109598	−	0.999940i	$-0.503489\pi$
$977$	−669.381	−0.0219195	−0.0109598	+	0.999940i	$0.503489\pi$
$978$	0	0
$979$	−10760.6	−0.351287
$980$	0	0
$981$	0	0
$982$	0	0
$983$	23791.0	0.771938	0.385969	−	0.922512i	$-0.373867\pi$
$983$	23791.0	0.771938	0.385969	+	0.922512i	$0.373867\pi$
$984$	0	0
$985$	62235.7	2.01319
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−697.267	−0.0224184
$990$	0	0
$991$	−7069.41	−0.226607	−0.113303	−	0.993560i	$-0.536143\pi$
$991$	−7069.41	−0.226607	−0.113303	+	0.993560i	$0.536143\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−79655.8	−2.53795
$996$	0	0
$997$	34324.7	1.09034	0.545172	−	0.838324i	$-0.316464\pi$
$997$	34324.7	1.09034	0.545172	+	0.838324i	$0.316464\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.4.a.y.1.1		4
3.2	odd	2		1296.4.a.ba.1.4		4
4.3	odd	2		648.4.a.h.1.1		4
9.2	odd	6		144.4.i.e.49.3		8
9.4	even	3		432.4.i.e.289.4		8
9.5	odd	6		144.4.i.e.97.3		8
9.7	even	3		432.4.i.e.145.4		8
12.11	even	2		648.4.a.i.1.4		4
36.7	odd	6		216.4.i.a.145.4		8
36.11	even	6		72.4.i.a.49.2	yes	8
36.23	even	6		72.4.i.a.25.2	✓	8
36.31	odd	6		216.4.i.a.73.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.4.i.a.25.2	✓	8	36.23	even	6
72.4.i.a.49.2	yes	8	36.11	even	6
144.4.i.e.49.3		8	9.2	odd	6
144.4.i.e.97.3		8	9.5	odd	6
216.4.i.a.73.4		8	36.31	odd	6
216.4.i.a.145.4		8	36.7	odd	6
432.4.i.e.145.4		8	9.7	even	3
432.4.i.e.289.4		8	9.4	even	3
648.4.a.h.1.1		4	4.3	odd	2
648.4.a.i.1.4		4	12.11	even	2
1296.4.a.y.1.1		4	1.1	even	1	trivial
1296.4.a.ba.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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1296.a (trivial)

\(f(q)\)	\(=\)	\(q-17.3189 q^{5} +2.37353 q^{7} +O(q^{10})\)	\(q-17.3189 q^{5} +2.37353 q^{7} -52.2165 q^{11} +13.6993 q^{13} +82.9217 q^{17} +126.803 q^{19} -54.3661 q^{23} +174.944 q^{25} +212.850 q^{29} +224.122 q^{31} -41.1068 q^{35} -32.2229 q^{37} -500.662 q^{41} +12.8254 q^{43} +209.541 q^{47} -337.366 q^{49} -371.213 q^{53} +904.331 q^{55} +5.81848 q^{59} -604.032 q^{61} -237.256 q^{65} -754.066 q^{67} +43.4780 q^{71} +671.029 q^{73} -123.937 q^{77} +649.202 q^{79} +134.526 q^{83} -1436.11 q^{85} +206.076 q^{89} +32.5156 q^{91} -2196.09 q^{95} -1452.84 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 5 q^{5} + 3 q^{7}+O(q^{10}) \) 4 * q - 5 * q^5 + 3 * q^7	\( 4 q - 5 q^{5} + 3 q^{7} - 16 q^{11} - 29 q^{13} + 17 q^{17} + 109 q^{19} - 37 q^{23} - 97 q^{25} - 3 q^{29} + 331 q^{31} - 171 q^{35} - 366 q^{37} - 378 q^{41} + 506 q^{43} - 171 q^{47} - 829 q^{49} - 410 q^{53} + 1163 q^{55} - 616 q^{59} - 1331 q^{61} - 815 q^{65} + 1162 q^{67} - 344 q^{71} - 1307 q^{73} - 741 q^{77} + 1853 q^{79} - 1421 q^{83} - 2074 q^{85} - 816 q^{89} + 1995 q^{91} - 1292 q^{95} - 2506 q^{97}+O(q^{100}) \) 4 * q - 5 * q^5 + 3 * q^7 - 16 * q^11 - 29 * q^13 + 17 * q^17 + 109 * q^19 - 37 * q^23 - 97 * q^25 - 3 * q^29 + 331 * q^31 - 171 * q^35 - 366 * q^37 - 378 * q^41 + 506 * q^43 - 171 * q^47 - 829 * q^49 - 410 * q^53 + 1163 * q^55 - 616 * q^59 - 1331 * q^61 - 815 * q^65 + 1162 * q^67 - 344 * q^71 - 1307 * q^73 - 741 * q^77 + 1853 * q^79 - 1421 * q^83 - 2074 * q^85 - 816 * q^89 + 1995 * q^91 - 1292 * q^95 - 2506 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more