LMFDB - Embedded newform 1521.4.a.w.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1521.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:	$89.7419051187$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.1362828.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 23x^{2} + 48 $ x^4 - 23*x^2 + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$4.54739$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.54739 q^{2} +12.6788 q^{4} +12.9118 q^{5} -16.7289 q^{7} +21.2762 q^{8} +O(q^{10})$	$q+4.54739 q^{2} +12.6788 q^{4} +12.9118 q^{5} -16.7289 q^{7} +21.2762 q^{8} +58.7151 q^{10} +24.9280 q^{11} -76.0727 q^{14} -4.67878 q^{16} +134.145 q^{17} +14.9376 q^{19} +163.706 q^{20} +113.358 q^{22} -72.0000 q^{23} +41.7151 q^{25} -212.101 q^{28} +206.145 q^{29} +249.142 q^{31} -191.486 q^{32} +610.012 q^{34} -216.000 q^{35} -293.955 q^{37} +67.9273 q^{38} +274.715 q^{40} +250.506 q^{41} +432.145 q^{43} +316.057 q^{44} -327.412 q^{46} -159.889 q^{47} -63.1454 q^{49} +189.695 q^{50} +194.581 q^{53} +321.866 q^{55} -355.927 q^{56} +937.424 q^{58} +232.647 q^{59} -185.006 q^{61} +1132.94 q^{62} -833.333 q^{64} +39.4393 q^{67} +1700.80 q^{68} -982.237 q^{70} +920.460 q^{71} +549.078 q^{73} -1336.73 q^{74} +189.391 q^{76} -417.018 q^{77} +933.140 q^{79} -60.4116 q^{80} +1139.15 q^{82} -1095.38 q^{83} +1732.06 q^{85} +1965.13 q^{86} +530.375 q^{88} -532.114 q^{89} -912.872 q^{92} -727.079 q^{94} +192.872 q^{95} +362.661 q^{97} -287.147 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 14 q^{4}+O(q^{10}) $ 4 * q + 14 * q^4	$ 4 q + 14 q^{4} + 88 q^{10} - 84 q^{14} + 18 q^{16} + 96 q^{17} + 380 q^{22} - 288 q^{23} + 20 q^{25} + 384 q^{29} - 864 q^{35} + 492 q^{38} + 952 q^{40} + 1288 q^{43} + 188 q^{49} - 984 q^{53} - 328 q^{55} - 1644 q^{56} + 288 q^{61} + 1668 q^{62} - 1314 q^{64} + 4380 q^{68} - 3144 q^{74} + 1416 q^{77} + 4320 q^{79} + 3088 q^{82} - 1036 q^{88} - 1008 q^{92} - 1660 q^{94} - 1872 q^{95}+O(q^{100}) $ 4 * q + 14 * q^4 + 88 * q^10 - 84 * q^14 + 18 * q^16 + 96 * q^17 + 380 * q^22 - 288 * q^23 + 20 * q^25 + 384 * q^29 - 864 * q^35 + 492 * q^38 + 952 * q^40 + 1288 * q^43 + 188 * q^49 - 984 * q^53 - 328 * q^55 - 1644 * q^56 + 288 * q^61 + 1668 * q^62 - 1314 * q^64 + 4380 * q^68 - 3144 * q^74 + 1416 * q^77 + 4320 * q^79 + 3088 * q^82 - 1036 * q^88 - 1008 * q^92 - 1660 * q^94 - 1872 * q^95

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 $

$2$	4.54739	1.60775	0.803873	−	0.594801i	$-0.202769\pi$
$2$	4.54739	1.60775	0.803873	+	0.594801i	$0.202769\pi$
$3$	0	0
$3$	0	0
$4$	12.6788	1.58485
$5$	12.9118	1.15487	0.577434	−	0.816437i	$-0.304054\pi$
$5$	12.9118	1.15487	0.577434	+	0.816437i	$0.304054\pi$
$6$	0	0
$7$	−16.7289	−0.903273	−0.451637	−	0.892202i	$-0.649160\pi$
$7$	−16.7289	−0.903273	−0.451637	+	0.892202i	$0.649160\pi$
$8$	21.2762	0.940286
$9$	0	0
$10$	58.7151	1.85674
$11$	24.9280	0.683280	0.341640	−	0.939831i	$-0.389018\pi$
$11$	24.9280	0.683280	0.341640	+	0.939831i	$0.389018\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−76.0727	−1.45223
$15$	0	0
$16$	−4.67878	−0.0731059
$17$	134.145	1.91383	0.956913	−	0.290376i	$-0.0937804\pi$
$17$	134.145	1.91383	0.956913	+	0.290376i	$0.0937804\pi$
$18$	0	0
$19$	14.9376	0.180365	0.0901824	−	0.995925i	$-0.471255\pi$
$19$	14.9376	0.180365	0.0901824	+	0.995925i	$0.471255\pi$
$20$	163.706	1.83029
$21$	0	0
$22$	113.358	1.09854
$23$	−72.0000	−0.652741	−0.326370	−	0.945242i	$-0.605826\pi$
$23$	−72.0000	−0.652741	−0.326370	+	0.945242i	$0.605826\pi$
$24$	0	0
$25$	41.7151	0.333721
$26$	0	0
$27$	0	0
$28$	−212.101	−1.43155
$29$	206.145	1.32001	0.660004	−	0.751262i	$-0.270555\pi$
$29$	206.145	1.32001	0.660004	+	0.751262i	$0.270555\pi$
$30$	0	0
$31$	249.142	1.44346	0.721728	−	0.692176i	$-0.243348\pi$
$31$	249.142	1.44346	0.721728	+	0.692176i	$0.243348\pi$
$32$	−191.486	−1.05782
$33$	0	0
$34$	610.012	3.07694
$35$	−216.000	−1.04316
$36$	0	0
$37$	−293.955	−1.30610	−0.653052	−	0.757313i	$-0.726512\pi$
$37$	−293.955	−1.30610	−0.653052	+	0.757313i	$0.726512\pi$
$38$	67.9273	0.289981
$39$	0	0
$40$	274.715	1.08591
$41$	250.506	0.954208	0.477104	−	0.878847i	$-0.341686\pi$
$41$	250.506	0.954208	0.477104	+	0.878847i	$0.341686\pi$
$42$	0	0
$43$	432.145	1.53259	0.766297	−	0.642486i	$-0.222097\pi$
$43$	432.145	1.53259	0.766297	+	0.642486i	$0.222097\pi$
$44$	316.057	1.08290
$45$	0	0
$46$	−327.412	−1.04944
$47$	−159.889	−0.496217	−0.248109	−	0.968732i	$-0.579809\pi$
$47$	−159.889	−0.496217	−0.248109	+	0.968732i	$0.579809\pi$
$48$	0	0
$49$	−63.1454	−0.184097
$50$	189.695	0.536539
$51$	0	0
$52$	0	0
$53$	194.581	0.504298	0.252149	−	0.967688i	$-0.418863\pi$
$53$	194.581	0.504298	0.252149	+	0.967688i	$0.418863\pi$
$54$	0	0
$55$	321.866	0.789099
$56$	−355.927	−0.849336
$57$	0	0
$58$	937.424	2.12224
$59$	232.647	0.513358	0.256679	−	0.966497i	$-0.417372\pi$
$59$	232.647	0.513358	0.256679	+	0.966497i	$0.417372\pi$
$60$	0	0
$61$	−185.006	−0.388321	−0.194160	−	0.980970i	$-0.562198\pi$
$61$	−185.006	−0.388321	−0.194160	+	0.980970i	$0.562198\pi$
$62$	1132.94	2.32071
$63$	0	0
$64$	−833.333	−1.62760
$65$	0	0
$66$	0	0
$67$	39.4393	0.0719145	0.0359573	−	0.999353i	$-0.488552\pi$
$67$	39.4393	0.0719145	0.0359573	+	0.999353i	$0.488552\pi$
$68$	1700.80	3.03312
$69$	0	0
$70$	−982.237	−1.67714
$71$	920.460	1.53857	0.769286	−	0.638905i	$-0.220612\pi$
$71$	920.460	1.53857	0.769286	+	0.638905i	$0.220612\pi$
$72$	0	0
$73$	549.078	0.880338	0.440169	−	0.897915i	$-0.354918\pi$
$73$	549.078	0.880338	0.440169	+	0.897915i	$0.354918\pi$
$74$	−1336.73	−2.09988
$75$	0	0
$76$	189.391	0.285851
$77$	−417.018	−0.617189
$78$	0	0
$79$	933.140	1.32894	0.664471	−	0.747314i	$-0.268657\pi$
$79$	933.140	1.32894	0.664471	+	0.747314i	$0.268657\pi$
$80$	−60.4116	−0.0844277
$81$	0	0
$82$	1139.15	1.53412
$83$	−1095.38	−1.44860	−0.724301	−	0.689484i	$-0.757837\pi$
$83$	−1095.38	−1.44860	−0.724301	+	0.689484i	$0.757837\pi$
$84$	0	0
$85$	1732.06	2.21022
$86$	1965.13	2.46402
$87$	0	0
$88$	530.375	0.642479
$89$	−532.114	−0.633753	−0.316876	−	0.948467i	$-0.602634\pi$
$89$	−532.114	−0.633753	−0.316876	+	0.948467i	$0.602634\pi$
$90$	0	0
$91$	0	0
$92$	−912.872	−1.03449
$93$	0	0
$94$	−727.079	−0.797792
$95$	192.872	0.208298
$96$	0	0
$97$	362.661	0.379615	0.189808	−	0.981821i	$-0.439214\pi$
$97$	362.661	0.379615	0.189808	+	0.981821i	$0.439214\pi$
$98$	−287.147	−0.295982
$99$	0	0
$100$	528.897	0.528897
$101$	−1490.58	−1.46850	−0.734249	−	0.678880i	$-0.762466\pi$
$101$	−1490.58	−1.46850	−0.734249	+	0.678880i	$0.762466\pi$
$102$	0	0
$103$	628.436	0.601181	0.300591	−	0.953753i	$-0.402816\pi$
$103$	628.436	0.601181	0.300591	+	0.953753i	$0.402816\pi$
$104$	0	0
$105$	0	0
$106$	884.838	0.810784
$107$	−477.454	−0.431376	−0.215688	−	0.976462i	$-0.569199\pi$
$107$	−477.454	−0.431376	−0.215688	+	0.976462i	$0.569199\pi$
$108$	0	0
$109$	378.207	0.332345	0.166173	−	0.986097i	$-0.446859\pi$
$109$	378.207	0.332345	0.166173	+	0.986097i	$0.446859\pi$
$110$	1463.65	1.26867
$111$	0	0
$112$	78.2706	0.0660346
$113$	−13.2732	−0.0110499	−0.00552495	−	0.999985i	$-0.501759\pi$
$113$	−13.2732	−0.0110499	−0.00552495	+	0.999985i	$0.501759\pi$
$114$	0	0
$115$	−929.651	−0.753830
$116$	2613.67	2.09201
$117$	0	0
$118$	1057.94	0.825349
$119$	−2244.10	−1.72871
$120$	0	0
$121$	−709.593	−0.533128
$122$	−841.294	−0.624321
$123$	0	0
$124$	3158.81	2.28766
$125$	−1075.36	−0.769465
$126$	0	0
$127$	145.988	0.102003	0.0510015	−	0.998699i	$-0.483759\pi$
$127$	145.988	0.102003	0.0510015	+	0.998699i	$0.483759\pi$
$128$	−2257.60	−1.55895
$129$	0	0
$130$	0	0
$131$	−317.163	−0.211532	−0.105766	−	0.994391i	$-0.533729\pi$
$131$	−317.163	−0.211532	−0.105766	+	0.994391i	$0.533729\pi$
$132$	0	0
$133$	−249.890	−0.162919
$134$	179.346	0.115620
$135$	0	0
$136$	2854.11	1.79954
$137$	443.149	0.276356	0.138178	−	0.990407i	$-0.455875\pi$
$137$	443.149	0.276356	0.138178	+	0.990407i	$0.455875\pi$
$138$	0	0
$139$	785.018	0.479024	0.239512	−	0.970893i	$-0.423013\pi$
$139$	785.018	0.479024	0.239512	+	0.970893i	$0.423013\pi$
$140$	−2738.62	−1.65325
$141$	0	0
$142$	4185.69	2.47363
$143$	0	0
$144$	0	0
$145$	2661.71	1.52444
$146$	2496.87	1.41536
$147$	0	0
$148$	−3726.99	−2.06997
$149$	135.420	0.0744566	0.0372283	−	0.999307i	$-0.488147\pi$
$149$	135.420	0.0744566	0.0372283	+	0.999307i	$0.488147\pi$
$150$	0	0
$151$	2373.74	1.27929	0.639643	−	0.768672i	$-0.279082\pi$
$151$	2373.74	1.27929	0.639643	+	0.768672i	$0.279082\pi$
$152$	317.817	0.169594
$153$	0	0
$154$	−1896.34	−0.992283
$155$	3216.87	1.66700
$156$	0	0
$157$	−1166.73	−0.593089	−0.296544	−	0.955019i	$-0.595834\pi$
$157$	−1166.73	−0.593089	−0.296544	+	0.955019i	$0.595834\pi$
$158$	4243.35	2.13660
$159$	0	0
$160$	−2472.44	−1.22165
$161$	1204.48	0.589603
$162$	0	0
$163$	2309.19	1.10963	0.554815	−	0.831974i	$-0.312789\pi$
$163$	2309.19	1.10963	0.554815	+	0.831974i	$0.312789\pi$
$164$	3176.12	1.51227
$165$	0	0
$166$	−4981.14	−2.32898
$167$	−600.788	−0.278386	−0.139193	−	0.990265i	$-0.544451\pi$
$167$	−600.788	−0.278386	−0.139193	+	0.990265i	$0.544451\pi$
$168$	0	0
$169$	0	0
$170$	7876.36	3.55347
$171$	0	0
$172$	5479.08	2.42893
$173$	−3430.36	−1.50755	−0.753773	−	0.657135i	$-0.771768\pi$
$173$	−3430.36	−1.50755	−0.753773	+	0.657135i	$0.771768\pi$
$174$	0	0
$175$	−697.846	−0.301441
$176$	−116.633	−0.0499519
$177$	0	0
$178$	−2419.73	−1.01891
$179$	−978.837	−0.408725	−0.204362	−	0.978895i	$-0.565512\pi$
$179$	−978.837	−0.408725	−0.204362	+	0.978895i	$0.565512\pi$
$180$	0	0
$181$	−3839.09	−1.57656	−0.788279	−	0.615318i	$-0.789028\pi$
$181$	−3839.09	−1.57656	−0.788279	+	0.615318i	$0.789028\pi$
$182$	0	0
$183$	0	0
$184$	−1531.89	−0.613763
$185$	−3795.49	−1.50838
$186$	0	0
$187$	3343.98	1.30768
$188$	−2027.20	−0.786429
$189$	0	0
$190$	877.065	0.334890
$191$	487.709	0.184761	0.0923806	−	0.995724i	$-0.470552\pi$
$191$	487.709	0.184761	0.0923806	+	0.995724i	$0.470552\pi$
$192$	0	0
$193$	−4245.61	−1.58345	−0.791725	−	0.610878i	$-0.790817\pi$
$193$	−4245.61	−1.58345	−0.791725	+	0.610878i	$0.790817\pi$
$194$	1649.16	0.610325
$195$	0	0
$196$	−800.606	−0.291766
$197$	−2712.71	−0.981079	−0.490539	−	0.871419i	$-0.663200\pi$
$197$	−2712.71	−0.981079	−0.490539	+	0.871419i	$0.663200\pi$
$198$	0	0
$199$	3116.90	1.11031	0.555153	−	0.831748i	$-0.312660\pi$
$199$	3116.90	1.11031	0.555153	+	0.831748i	$0.312660\pi$
$200$	887.541	0.313793
$201$	0	0
$202$	−6778.26	−2.36097
$203$	−3448.58	−1.19233
$204$	0	0
$205$	3234.49	1.10198
$206$	2857.75	0.966546
$207$	0	0
$208$	0	0
$209$	372.366	0.123240
$210$	0	0
$211$	−1051.22	−0.342981	−0.171491	−	0.985186i	$-0.554858\pi$
$211$	−1051.22	−0.342981	−0.171491	+	0.985186i	$0.554858\pi$
$212$	2467.06	0.799236
$213$	0	0
$214$	−2171.17	−0.693542
$215$	5579.78	1.76994
$216$	0	0
$217$	−4167.85	−1.30384
$218$	1719.85	0.534327
$219$	0	0
$220$	4080.87	1.25060
$221$	0	0
$222$	0	0
$223$	−5496.12	−1.65044	−0.825218	−	0.564814i	$-0.808948\pi$
$223$	−5496.12	−1.65044	−0.825218	+	0.564814i	$0.808948\pi$
$224$	3203.35	0.955502
$225$	0	0
$226$	−60.3585	−0.0177654
$227$	−921.570	−0.269457	−0.134729	−	0.990883i	$-0.543016\pi$
$227$	−921.570	−0.269457	−0.134729	+	0.990883i	$0.543016\pi$
$228$	0	0
$229$	−192.941	−0.0556764	−0.0278382	−	0.999612i	$-0.508862\pi$
$229$	−192.941	−0.0556764	−0.0278382	+	0.999612i	$0.508862\pi$
$230$	−4227.49	−1.21197
$231$	0	0
$232$	4386.00	1.24119
$233$	−913.779	−0.256926	−0.128463	−	0.991714i	$-0.541004\pi$
$233$	−913.779	−0.256926	−0.128463	+	0.991714i	$0.541004\pi$
$234$	0	0
$235$	−2064.46	−0.573066
$236$	2949.68	0.813594
$237$	0	0
$238$	−10204.8	−2.77932
$239$	−1976.86	−0.535032	−0.267516	−	0.963553i	$-0.586203\pi$
$239$	−1976.86	−0.535032	−0.267516	+	0.963553i	$0.586203\pi$
$240$	0	0
$241$	−3904.45	−1.04360	−0.521800	−	0.853068i	$-0.674739\pi$
$241$	−3904.45	−1.04360	−0.521800	+	0.853068i	$0.674739\pi$
$242$	−3226.80	−0.857134
$243$	0	0
$244$	−2345.65	−0.615429
$245$	−815.322	−0.212608
$246$	0	0
$247$	0	0
$248$	5300.80	1.35726
$249$	0	0
$250$	−4890.08	−1.23710
$251$	942.035	0.236895	0.118448	−	0.992960i	$-0.462208\pi$
$251$	942.035	0.236895	0.118448	+	0.992960i	$0.462208\pi$
$252$	0	0
$253$	−1794.82	−0.446005
$254$	663.866	0.163995
$255$	0	0
$256$	−3599.54	−0.878794
$257$	812.616	0.197236	0.0986179	−	0.995125i	$-0.468558\pi$
$257$	812.616	0.197236	0.0986179	+	0.995125i	$0.468558\pi$
$258$	0	0
$259$	4917.52	1.17977
$260$	0	0
$261$	0	0
$262$	−1442.26	−0.340089
$263$	2608.29	0.611536	0.305768	−	0.952106i	$-0.401087\pi$
$263$	2608.29	0.611536	0.305768	+	0.952106i	$0.401087\pi$
$264$	0	0
$265$	2512.40	0.582398
$266$	−1136.35	−0.261932
$267$	0	0
$268$	500.042	0.113974
$269$	4791.02	1.08592	0.542962	−	0.839757i	$-0.317303\pi$
$269$	4791.02	1.08592	0.542962	+	0.839757i	$0.317303\pi$
$270$	0	0
$271$	3663.62	0.821214	0.410607	−	0.911812i	$-0.365317\pi$
$271$	3663.62	0.821214	0.410607	+	0.911812i	$0.365317\pi$
$272$	−627.637	−0.139912
$273$	0	0
$274$	2015.17	0.444311
$275$	1039.88	0.228025
$276$	0	0
$277$	−624.326	−0.135423	−0.0677114	−	0.997705i	$-0.521570\pi$
$277$	−624.326	−0.135423	−0.0677114	+	0.997705i	$0.521570\pi$
$278$	3569.78	0.770149
$279$	0	0
$280$	−4595.67	−0.980871
$281$	5535.12	1.17508	0.587540	−	0.809195i	$-0.300096\pi$
$281$	5535.12	1.17508	0.587540	+	0.809195i	$0.300096\pi$
$282$	0	0
$283$	175.151	0.0367903	0.0183952	−	0.999831i	$-0.494144\pi$
$283$	175.151	0.0367903	0.0183952	+	0.999831i	$0.494144\pi$
$284$	11670.3	2.43840
$285$	0	0
$286$	0	0
$287$	−4190.69	−0.861911
$288$	0	0
$289$	13082.0	2.66273
$290$	12103.8	2.45091
$291$	0	0
$292$	6961.64	1.39520
$293$	−7774.33	−1.55011	−0.775054	−	0.631895i	$-0.782277\pi$
$293$	−7774.33	−1.55011	−0.775054	+	0.631895i	$0.782277\pi$
$294$	0	0
$295$	3003.90	0.592861
$296$	−6254.25	−1.22811
$297$	0	0
$298$	615.808	0.119707
$299$	0	0
$300$	0	0
$301$	−7229.30	−1.38435
$302$	10794.3	2.05677
$303$	0	0
$304$	−69.8899	−0.0131857
$305$	−2388.76	−0.448459
$306$	0	0
$307$	−8022.85	−1.49149	−0.745746	−	0.666230i	$-0.767907\pi$
$307$	−8022.85	−1.49149	−0.745746	+	0.666230i	$0.767907\pi$
$308$	−5287.27	−0.978150
$309$	0	0
$310$	14628.4	2.68012
$311$	9264.87	1.68927	0.844635	−	0.535343i	$-0.179818\pi$
$311$	9264.87	1.68927	0.844635	+	0.535343i	$0.179818\pi$
$312$	0	0
$313$	7423.57	1.34059	0.670296	−	0.742094i	$-0.266167\pi$
$313$	7423.57	1.34059	0.670296	+	0.742094i	$0.266167\pi$
$314$	−5305.56	−0.953536
$315$	0	0
$316$	11831.1	2.10617
$317$	2641.04	0.467935	0.233968	−	0.972244i	$-0.424829\pi$
$317$	2641.04	0.467935	0.233968	+	0.972244i	$0.424829\pi$
$318$	0	0
$319$	5138.80	0.901936
$320$	−10759.8	−1.87967
$321$	0	0
$322$	5477.23	0.947932
$323$	2003.82	0.345187
$324$	0	0
$325$	0	0
$326$	10500.8	1.78400
$327$	0	0
$328$	5329.84	0.897229
$329$	2674.76	0.448220
$330$	0	0
$331$	−10779.5	−1.79001	−0.895007	−	0.446052i	$-0.852830\pi$
$331$	−10779.5	−1.79001	−0.895007	+	0.446052i	$0.852830\pi$
$332$	−13888.1	−2.29581
$333$	0	0
$334$	−2732.02	−0.447573
$335$	509.233	0.0830518
$336$	0	0
$337$	−313.465	−0.0506693	−0.0253346	−	0.999679i	$-0.508065\pi$
$337$	−313.465	−0.0506693	−0.0253346	+	0.999679i	$0.508065\pi$
$338$	0	0
$339$	0	0
$340$	21960.4	3.50286
$341$	6210.61	0.986286
$342$	0	0
$343$	6794.35	1.06956
$344$	9194.43	1.44108
$345$	0	0
$346$	−15599.2	−2.42375
$347$	2849.23	0.440792	0.220396	−	0.975410i	$-0.429265\pi$
$347$	2849.23	0.440792	0.220396	+	0.975410i	$0.429265\pi$
$348$	0	0
$349$	−6466.94	−0.991883	−0.495941	−	0.868356i	$-0.665177\pi$
$349$	−6466.94	−0.991883	−0.495941	+	0.868356i	$0.665177\pi$
$350$	−3173.38	−0.484641
$351$	0	0
$352$	−4773.38	−0.722789
$353$	−2773.10	−0.418122	−0.209061	−	0.977903i	$-0.567041\pi$
$353$	−2773.10	−0.418122	−0.209061	+	0.977903i	$0.567041\pi$
$354$	0	0
$355$	11884.8	1.77685
$356$	−6746.56	−1.00440
$357$	0	0
$358$	−4451.16	−0.657126
$359$	−1467.11	−0.215685	−0.107843	−	0.994168i	$-0.534394\pi$
$359$	−1467.11	−0.215685	−0.107843	+	0.994168i	$0.534394\pi$
$360$	0	0
$361$	−6635.87	−0.967469
$362$	−17457.8	−2.53471
$363$	0	0
$364$	0	0
$365$	7089.59	1.01667
$366$	0	0
$367$	4648.22	0.661130	0.330565	−	0.943783i	$-0.392761\pi$
$367$	4648.22	0.661130	0.330565	+	0.943783i	$0.392761\pi$
$368$	336.872	0.0477192
$369$	0	0
$370$	−17259.6	−2.42509
$371$	−3255.12	−0.455519
$372$	0	0
$373$	1763.72	0.244831	0.122416	−	0.992479i	$-0.460936\pi$
$373$	1763.72	0.244831	0.122416	+	0.992479i	$0.460936\pi$
$374$	15206.4	2.10242
$375$	0	0
$376$	−3401.84	−0.466586
$377$	0	0
$378$	0	0
$379$	−1930.47	−0.261640	−0.130820	−	0.991406i	$-0.541761\pi$
$379$	−1930.47	−0.261640	−0.130820	+	0.991406i	$0.541761\pi$
$380$	2445.38	0.330120
$381$	0	0
$382$	2217.81	0.297049
$383$	−8845.93	−1.18017	−0.590086	−	0.807340i	$-0.700906\pi$
$383$	−8845.93	−1.18017	−0.590086	+	0.807340i	$0.700906\pi$
$384$	0	0
$385$	−5384.46	−0.712772
$386$	−19306.5	−2.54579
$387$	0	0
$388$	4598.10	0.601632
$389$	1598.08	0.208292	0.104146	−	0.994562i	$-0.466789\pi$
$389$	1598.08	0.208292	0.104146	+	0.994562i	$0.466789\pi$
$390$	0	0
$391$	−9658.47	−1.24923
$392$	−1343.50	−0.173104
$393$	0	0
$394$	−12335.8	−1.57733
$395$	12048.5	1.53475
$396$	0	0
$397$	−3578.82	−0.452433	−0.226217	−	0.974077i	$-0.572636\pi$
$397$	−3578.82	−0.452433	−0.226217	+	0.974077i	$0.572636\pi$
$398$	14173.7	1.78509
$399$	0	0
$400$	−195.176	−0.0243970
$401$	−3485.99	−0.434120	−0.217060	−	0.976158i	$-0.569647\pi$
$401$	−3485.99	−0.434120	−0.217060	+	0.976158i	$0.569647\pi$
$402$	0	0
$403$	0	0
$404$	−18898.8	−2.32735
$405$	0	0
$406$	−15682.0	−1.91696
$407$	−7327.71	−0.892435
$408$	0	0
$409$	−14709.1	−1.77828	−0.889139	−	0.457637i	$-0.848696\pi$
$409$	−14709.1	−1.77828	−0.889139	+	0.457637i	$0.848696\pi$
$410$	14708.5	1.77171
$411$	0	0
$412$	7967.80	0.952780
$413$	−3891.92	−0.463702
$414$	0	0
$415$	−14143.4	−1.67294
$416$	0	0
$417$	0	0
$418$	1693.29	0.198138
$419$	−3709.01	−0.432451	−0.216226	−	0.976343i	$-0.569375\pi$
$419$	−3709.01	−0.432451	−0.216226	+	0.976343i	$0.569375\pi$
$420$	0	0
$421$	−794.029	−0.0919207	−0.0459603	−	0.998943i	$-0.514635\pi$
$421$	−794.029	−0.0919207	−0.0459603	+	0.998943i	$0.514635\pi$
$422$	−4780.32	−0.551427
$423$	0	0
$424$	4139.96	0.474185
$425$	5595.89	0.638684
$426$	0	0
$427$	3094.94	0.350760
$428$	−6053.53	−0.683664
$429$	0	0
$430$	25373.5	2.84562
$431$	2891.52	0.323155	0.161577	−	0.986860i	$-0.448342\pi$
$431$	2891.52	0.323155	0.161577	+	0.986860i	$0.448342\pi$
$432$	0	0
$433$	5560.94	0.617186	0.308593	−	0.951194i	$-0.400142\pi$
$433$	5560.94	0.617186	0.308593	+	0.951194i	$0.400142\pi$
$434$	−18952.9	−2.09624
$435$	0	0
$436$	4795.20	0.526717
$437$	−1075.51	−0.117731
$438$	0	0
$439$	15127.2	1.64460	0.822302	−	0.569051i	$-0.192689\pi$
$439$	15127.2	1.64460	0.822302	+	0.569051i	$0.192689\pi$
$440$	6848.11	0.741979
$441$	0	0
$442$	0	0
$443$	−2357.89	−0.252883	−0.126441	−	0.991974i	$-0.540356\pi$
$443$	−2357.89	−0.252883	−0.126441	+	0.991974i	$0.540356\pi$
$444$	0	0
$445$	−6870.56	−0.731901
$446$	−24993.0	−2.65348
$447$	0	0
$448$	13940.7	1.47017
$449$	−7165.06	−0.753096	−0.376548	−	0.926397i	$-0.622889\pi$
$449$	−7165.06	−0.753096	−0.376548	+	0.926397i	$0.622889\pi$
$450$	0	0
$451$	6244.63	0.651992
$452$	−168.288	−0.0175124
$453$	0	0
$454$	−4190.74	−0.433219
$455$	0	0
$456$	0	0
$457$	−8020.96	−0.821017	−0.410508	−	0.911857i	$-0.634649\pi$
$457$	−8020.96	−0.821017	−0.410508	+	0.911857i	$0.634649\pi$
$458$	−877.378	−0.0895135
$459$	0	0
$460$	−11786.8	−1.19471
$461$	−4146.59	−0.418928	−0.209464	−	0.977816i	$-0.567172\pi$
$461$	−4146.59	−0.418928	−0.209464	+	0.977816i	$0.567172\pi$
$462$	0	0
$463$	−7118.21	−0.714495	−0.357248	−	0.934010i	$-0.616285\pi$
$463$	−7118.21	−0.714495	−0.357248	+	0.934010i	$0.616285\pi$
$464$	−964.509	−0.0965004
$465$	0	0
$466$	−4155.31	−0.413071
$467$	2128.22	0.210883	0.105441	−	0.994426i	$-0.466374\pi$
$467$	2128.22	0.210883	0.105441	+	0.994426i	$0.466374\pi$
$468$	0	0
$469$	−659.774	−0.0649585
$470$	−9387.91	−0.921344
$471$	0	0
$472$	4949.86	0.482703
$473$	10772.5	1.04719
$474$	0	0
$475$	623.125	0.0601915
$476$	−28452.4	−2.73974
$477$	0	0
$478$	−8989.57	−0.860196
$479$	−3715.30	−0.354397	−0.177199	−	0.984175i	$-0.556704\pi$
$479$	−3715.30	−0.354397	−0.177199	+	0.984175i	$0.556704\pi$
$480$	0	0
$481$	0	0
$482$	−17755.0	−1.67784
$483$	0	0
$484$	−8996.77	−0.844926
$485$	4682.62	0.438405
$486$	0	0
$487$	8139.28	0.757343	0.378671	−	0.925531i	$-0.376381\pi$
$487$	8139.28	0.757343	0.378671	+	0.925531i	$0.376381\pi$
$488$	−3936.23	−0.365133
$489$	0	0
$490$	−3707.59	−0.341820
$491$	−18081.7	−1.66194	−0.830972	−	0.556315i	$-0.812215\pi$
$491$	−18081.7	−1.66194	−0.830972	+	0.556315i	$0.812215\pi$
$492$	0	0
$493$	27653.4	2.52626
$494$	0	0
$495$	0	0
$496$	−1165.68	−0.105525
$497$	−15398.3	−1.38975
$498$	0	0
$499$	−11031.5	−0.989659	−0.494829	−	0.868990i	$-0.664769\pi$
$499$	−11031.5	−0.989659	−0.494829	+	0.868990i	$0.664769\pi$
$500$	−13634.2	−1.21948
$501$	0	0
$502$	4283.80	0.380868
$503$	−8016.14	−0.710581	−0.355290	−	0.934756i	$-0.615618\pi$
$503$	−8016.14	−0.710581	−0.355290	+	0.934756i	$0.615618\pi$
$504$	0	0
$505$	−19246.1	−1.69592
$506$	−8161.74	−0.717063
$507$	0	0
$508$	1850.95	0.161659
$509$	20173.9	1.75676	0.878382	−	0.477959i	$-0.158623\pi$
$509$	20173.9	1.75676	0.878382	+	0.477959i	$0.158623\pi$
$510$	0	0
$511$	−9185.44	−0.795186
$512$	1692.30	0.146074
$513$	0	0
$514$	3695.29	0.317105
$515$	8114.25	0.694285
$516$	0	0
$517$	−3985.72	−0.339056
$518$	22361.9	1.89677
$519$	0	0
$520$	0	0
$521$	−9746.95	−0.819619	−0.409810	−	0.912171i	$-0.634405\pi$
$521$	−9746.95	−0.819619	−0.409810	+	0.912171i	$0.634405\pi$
$522$	0	0
$523$	−18929.3	−1.58264	−0.791320	−	0.611402i	$-0.790606\pi$
$523$	−18929.3	−1.58264	−0.791320	+	0.611402i	$0.790606\pi$
$524$	−4021.24	−0.335245
$525$	0	0
$526$	11860.9	0.983195
$527$	33421.2	2.76252
$528$	0	0
$529$	−6983.00	−0.573929
$530$	11424.9	0.936349
$531$	0	0
$532$	−3168.30	−0.258201
$533$	0	0
$534$	0	0
$535$	−6164.80	−0.498182
$536$	839.120	0.0676203
$537$	0	0
$538$	21786.6	1.74589
$539$	−1574.09	−0.125790
$540$	0	0
$541$	11366.8	0.903321	0.451661	−	0.892190i	$-0.350832\pi$
$541$	11366.8	0.903321	0.451661	+	0.892190i	$0.350832\pi$
$542$	16659.9	1.32030
$543$	0	0
$544$	−25687.0	−2.02449
$545$	4883.34	0.383815
$546$	0	0
$547$	17495.4	1.36755	0.683775	−	0.729693i	$-0.260337\pi$
$547$	17495.4	1.36755	0.683775	+	0.729693i	$0.260337\pi$
$548$	5618.59	0.437983
$549$	0	0
$550$	4728.72	0.366606
$551$	3079.33	0.238083
$552$	0	0
$553$	−15610.4	−1.20040
$554$	−2839.05	−0.217725
$555$	0	0
$556$	9953.06	0.759180
$557$	−11873.1	−0.903192	−0.451596	−	0.892223i	$-0.649145\pi$
$557$	−11873.1	−0.903192	−0.451596	+	0.892223i	$0.649145\pi$
$558$	0	0
$559$	0	0
$560$	1010.62	0.0762613
$561$	0	0
$562$	25170.4	1.88923
$563$	2829.31	0.211796	0.105898	−	0.994377i	$-0.466228\pi$
$563$	2829.31	0.211796	0.105898	+	0.994377i	$0.466228\pi$
$564$	0	0
$565$	−171.381	−0.0127612
$566$	796.481	0.0591495
$567$	0	0
$568$	19583.9	1.44670
$569$	−16136.8	−1.18891	−0.594453	−	0.804130i	$-0.702632\pi$
$569$	−16136.8	−1.18891	−0.594453	+	0.804130i	$0.702632\pi$
$570$	0	0
$571$	−17840.5	−1.30754	−0.653769	−	0.756695i	$-0.726813\pi$
$571$	−17840.5	−1.30754	−0.653769	+	0.756695i	$0.726813\pi$
$572$	0	0
$573$	0	0
$574$	−19056.7	−1.38573
$575$	−3003.49	−0.217833
$576$	0	0
$577$	−8516.17	−0.614442	−0.307221	−	0.951638i	$-0.599399\pi$
$577$	−8516.17	−0.614442	−0.307221	+	0.951638i	$0.599399\pi$
$578$	59488.9	4.28099
$579$	0	0
$580$	33747.3	2.41600
$581$	18324.5	1.30848
$582$	0	0
$583$	4850.53	0.344577
$584$	11682.3	0.827770
$585$	0	0
$586$	−35352.9	−2.49218
$587$	−20688.3	−1.45468	−0.727340	−	0.686277i	$-0.759244\pi$
$587$	−20688.3	−1.45468	−0.727340	+	0.686277i	$0.759244\pi$
$588$	0	0
$589$	3721.59	0.260349
$590$	13659.9	0.953169
$591$	0	0
$592$	1375.35	0.0954839
$593$	11435.9	0.791933	0.395966	−	0.918265i	$-0.370410\pi$
$593$	11435.9	0.791933	0.395966	+	0.918265i	$0.370410\pi$
$594$	0	0
$595$	−28975.4	−1.99643
$596$	1716.96	0.118002
$597$	0	0
$598$	0	0
$599$	−1260.80	−0.0860016	−0.0430008	−	0.999075i	$-0.513692\pi$
$599$	−1260.80	−0.0860016	−0.0430008	+	0.999075i	$0.513692\pi$
$600$	0	0
$601$	6261.10	0.424951	0.212476	−	0.977166i	$-0.431847\pi$
$601$	6261.10	0.424951	0.212476	+	0.977166i	$0.431847\pi$
$602$	−32874.5	−2.22569
$603$	0	0
$604$	30096.1	2.02747
$605$	−9162.14	−0.615692
$606$	0	0
$607$	3230.33	0.216005	0.108003	−	0.994151i	$-0.465555\pi$
$607$	3230.33	0.216005	0.108003	+	0.994151i	$0.465555\pi$
$608$	−2860.35	−0.190794
$609$	0	0
$610$	−10862.6	−0.721009
$611$	0	0
$612$	0	0
$613$	−14868.5	−0.979660	−0.489830	−	0.871818i	$-0.662941\pi$
$613$	−14868.5	−0.979660	−0.489830	+	0.871818i	$0.662941\pi$
$614$	−36483.0	−2.39794
$615$	0	0
$616$	−8872.57	−0.580334
$617$	−19952.8	−1.30190	−0.650949	−	0.759121i	$-0.725629\pi$
$617$	−19952.8	−1.30190	−0.650949	+	0.759121i	$0.725629\pi$
$618$	0	0
$619$	−8316.48	−0.540012	−0.270006	−	0.962859i	$-0.587026\pi$
$619$	−8316.48	−0.540012	−0.270006	+	0.962859i	$0.587026\pi$
$620$	40786.0	2.64194
$621$	0	0
$622$	42131.0	2.71592
$623$	8901.66	0.572452
$624$	0	0
$625$	−19099.2	−1.22235
$626$	33757.9	2.15533
$627$	0	0
$628$	−14792.7	−0.939955
$629$	−39432.6	−2.49965
$630$	0	0
$631$	12605.9	0.795299	0.397649	−	0.917537i	$-0.369826\pi$
$631$	12605.9	0.795299	0.397649	+	0.917537i	$0.369826\pi$
$632$	19853.7	1.24959
$633$	0	0
$634$	12009.8	0.752321
$635$	1884.98	0.117800
$636$	0	0
$637$	0	0
$638$	23368.1	1.45008
$639$	0	0
$640$	−29149.8	−1.80038
$641$	9224.04	0.568374	0.284187	−	0.958769i	$-0.408276\pi$
$641$	9224.04	0.568374	0.284187	+	0.958769i	$0.408276\pi$
$642$	0	0
$643$	−4439.16	−0.272260	−0.136130	−	0.990691i	$-0.543467\pi$
$643$	−4439.16	−0.272260	−0.136130	+	0.990691i	$0.543467\pi$
$644$	15271.3	0.934431
$645$	0	0
$646$	9112.13	0.554972
$647$	9601.15	0.583401	0.291700	−	0.956510i	$-0.405779\pi$
$647$	9601.15	0.583401	0.291700	+	0.956510i	$0.405779\pi$
$648$	0	0
$649$	5799.44	0.350767
$650$	0	0
$651$	0	0
$652$	29277.7	1.75859
$653$	−27112.8	−1.62482	−0.812410	−	0.583087i	$-0.801845\pi$
$653$	−27112.8	−1.62482	−0.812410	+	0.583087i	$0.801845\pi$
$654$	0	0
$655$	−4095.15	−0.244291
$656$	−1172.06	−0.0697583
$657$	0	0
$658$	12163.2	0.720624
$659$	−5587.26	−0.330271	−0.165136	−	0.986271i	$-0.552806\pi$
$659$	−5587.26	−0.330271	−0.165136	+	0.986271i	$0.552806\pi$
$660$	0	0
$661$	3060.13	0.180069	0.0900343	−	0.995939i	$-0.471302\pi$
$661$	3060.13	0.180069	0.0900343	+	0.995939i	$0.471302\pi$
$662$	−49018.6	−2.87789
$663$	0	0
$664$	−23305.6	−1.36210
$665$	−3226.53	−0.188150
$666$	0	0
$667$	−14842.5	−0.861623
$668$	−7617.26	−0.441199
$669$	0	0
$670$	2315.68	0.133526
$671$	−4611.83	−0.265332
$672$	0	0
$673$	4121.55	0.236069	0.118034	−	0.993010i	$-0.462341\pi$
$673$	4121.55	0.236069	0.118034	+	0.993010i	$0.462341\pi$
$674$	−1425.45	−0.0814633
$675$	0	0
$676$	0	0
$677$	−22889.5	−1.29943	−0.649715	−	0.760178i	$-0.725112\pi$
$677$	−22889.5	−1.29943	−0.649715	+	0.760178i	$0.725112\pi$
$678$	0	0
$679$	−6066.91	−0.342896
$680$	36851.8	2.07824
$681$	0	0
$682$	28242.1	1.58570
$683$	−19297.9	−1.08113	−0.540566	−	0.841301i	$-0.681790\pi$
$683$	−19297.9	−1.08113	−0.540566	+	0.841301i	$0.681790\pi$
$684$	0	0
$685$	5721.87	0.319155
$686$	30896.6	1.71959
$687$	0	0
$688$	−2021.91	−0.112042
$689$	0	0
$690$	0	0
$691$	30317.8	1.66910	0.834548	−	0.550935i	$-0.185729\pi$
$691$	30317.8	1.66910	0.834548	+	0.550935i	$0.185729\pi$
$692$	−43492.8	−2.38923
$693$	0	0
$694$	12956.6	0.708682
$695$	10136.0	0.553210
$696$	0	0
$697$	33604.3	1.82619
$698$	−29407.7	−1.59470
$699$	0	0
$700$	−8847.84	−0.477738
$701$	−9606.16	−0.517574	−0.258787	−	0.965934i	$-0.583323\pi$
$701$	−9606.16	−0.517574	−0.258787	+	0.965934i	$0.583323\pi$
$702$	0	0
$703$	−4390.99	−0.235575
$704$	−20773.4	−1.11211
$705$	0	0
$706$	−12610.4	−0.672235
$707$	24935.7	1.32646
$708$	0	0
$709$	23398.8	1.23944	0.619718	−	0.784825i	$-0.287247\pi$
$709$	23398.8	1.23944	0.619718	+	0.784825i	$0.287247\pi$
$710$	54044.9	2.85672
$711$	0	0
$712$	−11321.4	−0.595909
$713$	−17938.2	−0.942203
$714$	0	0
$715$	0	0
$716$	−12410.5	−0.647766
$717$	0	0
$718$	−6671.51	−0.346767
$719$	23588.7	1.22352	0.611758	−	0.791045i	$-0.290463\pi$
$719$	23588.7	1.22352	0.611758	+	0.791045i	$0.290463\pi$
$720$	0	0
$721$	−10513.0	−0.543031
$722$	−30175.9	−1.55544
$723$	0	0
$724$	−48674.9	−2.49861
$725$	8599.38	0.440514
$726$	0	0
$727$	15733.5	0.802644	0.401322	−	0.915937i	$-0.368551\pi$
$727$	15733.5	0.802644	0.401322	+	0.915937i	$0.368551\pi$
$728$	0	0
$729$	0	0
$730$	32239.2	1.63455
$731$	57970.3	2.93312
$732$	0	0
$733$	17297.1	0.871598	0.435799	−	0.900044i	$-0.356466\pi$
$733$	17297.1	0.871598	0.435799	+	0.900044i	$0.356466\pi$
$734$	21137.3	1.06293
$735$	0	0
$736$	13787.0	0.690484
$737$	983.144	0.0491378
$738$	0	0
$739$	−38749.2	−1.92884	−0.964419	−	0.264377i	$-0.914834\pi$
$739$	−38749.2	−1.92884	−0.964419	+	0.264377i	$0.914834\pi$
$740$	−48122.2	−2.39055
$741$	0	0
$742$	−14802.3	−0.732359
$743$	−3139.76	−0.155029	−0.0775144	−	0.996991i	$-0.524698\pi$
$743$	−3139.76	−0.155029	−0.0775144	+	0.996991i	$0.524698\pi$
$744$	0	0
$745$	1748.52	0.0859876
$746$	8020.33	0.393626
$747$	0	0
$748$	42397.6	2.07247
$749$	7987.25	0.389650
$750$	0	0
$751$	40628.6	1.97411	0.987055	−	0.160380i	$-0.0512721\pi$
$751$	40628.6	1.97411	0.987055	+	0.160380i	$0.0512721\pi$
$752$	748.086	0.0362764
$753$	0	0
$754$	0	0
$755$	30649.3	1.47741
$756$	0	0
$757$	24004.9	1.15254	0.576271	−	0.817259i	$-0.304507\pi$
$757$	24004.9	1.15254	0.576271	+	0.817259i	$0.304507\pi$
$758$	−8778.62	−0.420651
$759$	0	0
$760$	4103.60	0.195859
$761$	29540.1	1.40713	0.703566	−	0.710630i	$-0.251590\pi$
$761$	29540.1	1.40713	0.703566	+	0.710630i	$0.251590\pi$
$762$	0	0
$763$	−6326.97	−0.300199
$764$	6183.56	0.292818
$765$	0	0
$766$	−40225.9	−1.89742
$767$	0	0
$768$	0	0
$769$	−7585.63	−0.355715	−0.177857	−	0.984056i	$-0.556917\pi$
$769$	−7585.63	−0.355715	−0.177857	+	0.984056i	$0.556917\pi$
$770$	−24485.2	−1.14596
$771$	0	0
$772$	−53829.2	−2.50953
$773$	3284.29	0.152817	0.0764086	−	0.997077i	$-0.475655\pi$
$773$	3284.29	0.152817	0.0764086	+	0.997077i	$0.475655\pi$
$774$	0	0
$775$	10393.0	0.481712
$776$	7716.07	0.356947
$777$	0	0
$778$	7267.08	0.334881
$779$	3741.98	0.172106
$780$	0	0
$781$	22945.3	1.05128
$782$	−43920.8	−2.00845
$783$	0	0
$784$	295.443	0.0134586
$785$	−15064.6	−0.684939
$786$	0	0
$787$	41624.1	1.88531	0.942654	−	0.333772i	$-0.108321\pi$
$787$	41624.1	1.88531	0.942654	+	0.333772i	$0.108321\pi$
$788$	−34393.8	−1.55486
$789$	0	0
$790$	54789.4	2.46749
$791$	222.046	0.00998108
$792$	0	0
$793$	0	0
$794$	−16274.3	−0.727398
$795$	0	0
$796$	39518.4	1.75967
$797$	30333.3	1.34813	0.674066	−	0.738671i	$-0.264546\pi$
$797$	30333.3	1.34813	0.674066	+	0.738671i	$0.264546\pi$
$798$	0	0
$799$	−21448.4	−0.949674
$800$	−7987.87	−0.353017
$801$	0	0
$802$	−15852.2	−0.697954
$803$	13687.4	0.601518
$804$	0	0
$805$	15552.0	0.680914
$806$	0	0
$807$	0	0
$808$	−31714.0	−1.38081
$809$	24853.9	1.08012	0.540060	−	0.841627i	$-0.318402\pi$
$809$	24853.9	1.08012	0.540060	+	0.841627i	$0.318402\pi$
$810$	0	0
$811$	17383.5	0.752674	0.376337	−	0.926483i	$-0.377184\pi$
$811$	17383.5	0.752674	0.376337	+	0.926483i	$0.377184\pi$
$812$	−43723.7	−1.88966
$813$	0	0
$814$	−33322.0	−1.43481
$815$	29815.8	1.28148
$816$	0	0
$817$	6455.23	0.276426
$818$	−66887.8	−2.85902
$819$	0	0
$820$	41009.4	1.74648
$821$	−31169.4	−1.32499	−0.662496	−	0.749065i	$-0.730503\pi$
$821$	−31169.4	−1.32499	−0.662496	+	0.749065i	$0.730503\pi$
$822$	0	0
$823$	5512.79	0.233492	0.116746	−	0.993162i	$-0.462754\pi$
$823$	5512.79	0.233492	0.116746	+	0.993162i	$0.462754\pi$
$824$	13370.8	0.565282
$825$	0	0
$826$	−17698.1	−0.745516
$827$	−13335.8	−0.560738	−0.280369	−	0.959892i	$-0.590457\pi$
$827$	−13335.8	−0.560738	−0.280369	+	0.959892i	$0.590457\pi$
$828$	0	0
$829$	−28338.5	−1.18726	−0.593628	−	0.804739i	$-0.702305\pi$
$829$	−28338.5	−1.18726	−0.593628	+	0.804739i	$0.702305\pi$
$830$	−64315.5	−2.68967
$831$	0	0
$832$	0	0
$833$	−8470.66	−0.352330
$834$	0	0
$835$	−7757.27	−0.321499
$836$	4721.15	0.195316
$837$	0	0
$838$	−16866.3	−0.695272
$839$	−27149.5	−1.11717	−0.558585	−	0.829447i	$-0.688656\pi$
$839$	−27149.5	−1.11717	−0.558585	+	0.829447i	$0.688656\pi$
$840$	0	0
$841$	18106.9	0.742421
$842$	−3610.76	−0.147785
$843$	0	0
$844$	−13328.2	−0.543573
$845$	0	0
$846$	0	0
$847$	11870.7	0.481560
$848$	−910.404	−0.0368672
$849$	0	0
$850$	25446.7	1.02684
$851$	21164.7	0.852547
$852$	0	0
$853$	−7978.22	−0.320245	−0.160123	−	0.987097i	$-0.551189\pi$
$853$	−7978.22	−0.320245	−0.160123	+	0.987097i	$0.551189\pi$
$854$	14073.9	0.563933
$855$	0	0
$856$	−10158.4	−0.405616
$857$	13614.4	0.542657	0.271329	−	0.962487i	$-0.412537\pi$
$857$	13614.4	0.542657	0.271329	+	0.962487i	$0.412537\pi$
$858$	0	0
$859$	−35007.7	−1.39051	−0.695255	−	0.718763i	$-0.744709\pi$
$859$	−35007.7	−1.39051	−0.695255	+	0.718763i	$0.744709\pi$
$860$	70744.8	2.80509
$861$	0	0
$862$	13148.9	0.519551
$863$	−24461.5	−0.964867	−0.482434	−	0.875933i	$-0.660247\pi$
$863$	−24461.5	−0.964867	−0.482434	+	0.875933i	$0.660247\pi$
$864$	0	0
$865$	−44292.2	−1.74102
$866$	25287.8	0.992278
$867$	0	0
$868$	−52843.3	−2.06638
$869$	23261.3	0.908040
$870$	0	0
$871$	0	0
$872$	8046.82	0.312500
$873$	0	0
$874$	−4890.77	−0.189282
$875$	17989.5	0.695037
$876$	0	0
$877$	43121.6	1.66034	0.830168	−	0.557514i	$-0.188245\pi$
$877$	43121.6	1.66034	0.830168	+	0.557514i	$0.188245\pi$
$878$	68789.3	2.64411
$879$	0	0
$880$	−1505.94	−0.0576878
$881$	40824.5	1.56120	0.780598	−	0.625034i	$-0.214915\pi$
$881$	40824.5	1.56120	0.780598	+	0.625034i	$0.214915\pi$
$882$	0	0
$883$	8262.14	0.314885	0.157442	−	0.987528i	$-0.449675\pi$
$883$	8262.14	0.314885	0.157442	+	0.987528i	$0.449675\pi$
$884$	0	0
$885$	0	0
$886$	−10722.3	−0.406571
$887$	40858.6	1.54667	0.773336	−	0.633997i	$-0.218587\pi$
$887$	40858.6	1.54667	0.773336	+	0.633997i	$0.218587\pi$
$888$	0	0
$889$	−2442.22	−0.0921365
$890$	−31243.2	−1.17671
$891$	0	0
$892$	−69684.1	−2.61569
$893$	−2388.37	−0.0895001
$894$	0	0
$895$	−12638.6	−0.472023
$896$	37767.1	1.40816
$897$	0	0
$898$	−32582.3	−1.21079
$899$	51359.4	1.90537
$900$	0	0
$901$	26102.2	0.965139
$902$	28396.8	1.04824
$903$	0	0
$904$	−282.404	−0.0103901
$905$	−49569.6	−1.82072
$906$	0	0
$907$	−32729.3	−1.19819	−0.599095	−	0.800678i	$-0.704473\pi$
$907$	−32729.3	−1.19819	−0.599095	+	0.800678i	$0.704473\pi$
$908$	−11684.4	−0.427048
$909$	0	0
$910$	0	0
$911$	−11065.9	−0.402447	−0.201223	−	0.979545i	$-0.564492\pi$
$911$	−11065.9	−0.402447	−0.201223	+	0.979545i	$0.564492\pi$
$912$	0	0
$913$	−27305.7	−0.989801
$914$	−36474.5	−1.31999
$915$	0	0
$916$	−2446.26	−0.0882386
$917$	5305.77	0.191071
$918$	0	0
$919$	50682.2	1.81921	0.909604	−	0.415477i	$-0.136385\pi$
$919$	50682.2	1.81921	0.909604	+	0.415477i	$0.136385\pi$
$920$	−19779.5	−0.708816
$921$	0	0
$922$	−18856.2	−0.673531
$923$	0	0
$924$	0	0
$925$	−12262.3	−0.435874
$926$	−32369.3	−1.14873
$927$	0	0
$928$	−39474.0	−1.39633
$929$	41045.5	1.44958	0.724790	−	0.688970i	$-0.241937\pi$
$929$	41045.5	1.44958	0.724790	+	0.688970i	$0.241937\pi$
$930$	0	0
$931$	−943.243	−0.0332047
$932$	−11585.6	−0.407188
$933$	0	0
$934$	9677.86	0.339046
$935$	43176.9	1.51020
$936$	0	0
$937$	−788.985	−0.0275080	−0.0137540	−	0.999905i	$-0.504378\pi$
$937$	−788.985	−0.0275080	−0.0137540	+	0.999905i	$0.504378\pi$
$938$	−3000.25	−0.104437
$939$	0	0
$940$	−26174.8	−0.908222
$941$	−25676.2	−0.889499	−0.444750	−	0.895655i	$-0.646707\pi$
$941$	−25676.2	−0.889499	−0.444750	+	0.895655i	$0.646707\pi$
$942$	0	0
$943$	−18036.5	−0.622851
$944$	−1088.51	−0.0375295
$945$	0	0
$946$	48986.9	1.68362
$947$	−679.352	−0.0233115	−0.0116557	−	0.999932i	$-0.503710\pi$
$947$	−679.352	−0.0233115	−0.0116557	+	0.999932i	$0.503710\pi$
$948$	0	0
$949$	0	0
$950$	2833.60	0.0967726
$951$	0	0
$952$	−47746.0	−1.62548
$953$	20238.4	0.687917	0.343958	−	0.938985i	$-0.388232\pi$
$953$	20238.4	0.687917	0.343958	+	0.938985i	$0.388232\pi$
$954$	0	0
$955$	6297.21	0.213375
$956$	−25064.2	−0.847944
$957$	0	0
$958$	−16894.9	−0.569781
$959$	−7413.38	−0.249625
$960$	0	0
$961$	32280.6	1.08357
$962$	0	0
$963$	0	0
$964$	−49503.6	−1.65395
$965$	−54818.6	−1.82868
$966$	0	0
$967$	6161.63	0.204906	0.102453	−	0.994738i	$-0.467331\pi$
$967$	6161.63	0.204906	0.102453	+	0.994738i	$0.467331\pi$
$968$	−15097.5	−0.501293
$969$	0	0
$970$	21293.7	0.704845
$971$	48569.5	1.60522	0.802610	−	0.596504i	$-0.203444\pi$
$971$	48569.5	1.60522	0.802610	+	0.596504i	$0.203444\pi$
$972$	0	0
$973$	−13132.4	−0.432689
$974$	37012.5	1.21761
$975$	0	0
$976$	865.602	0.0283886
$977$	−44595.8	−1.46033	−0.730167	−	0.683269i	$-0.760558\pi$
$977$	−44595.8	−1.46033	−0.730167	+	0.683269i	$0.760558\pi$
$978$	0	0
$979$	−13264.6	−0.433031
$980$	−10337.3	−0.336951
$981$	0	0
$982$	−82224.4	−2.67198
$983$	−29599.0	−0.960389	−0.480195	−	0.877162i	$-0.659434\pi$
$983$	−29599.0	−0.960389	−0.480195	+	0.877162i	$0.659434\pi$
$984$	0	0
$985$	−35026.0	−1.13302
$986$	125751.	4.06159
$987$	0	0
$988$	0	0
$989$	−31114.5	−1.00039
$990$	0	0
$991$	5718.45	0.183302	0.0916511	−	0.995791i	$-0.470786\pi$
$991$	5718.45	0.183302	0.0916511	+	0.995791i	$0.470786\pi$
$992$	−47707.2	−1.52692
$993$	0	0
$994$	−70021.9	−2.23437
$995$	40244.8	1.28226
$996$	0	0
$997$	−1491.44	−0.0473766	−0.0236883	−	0.999719i	$-0.507541\pi$
$997$	−1491.44	−0.0473766	−0.0236883	+	0.999719i	$0.507541\pi$
$998$	−50164.8	−1.59112
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.w.1.4		4
3.2	odd	2		507.4.a.l.1.1		4
13.5	odd	4		117.4.b.e.64.1		4
13.8	odd	4		117.4.b.e.64.4		4
13.12	even	2	inner	1521.4.a.w.1.1		4
39.5	even	4		39.4.b.b.25.4	yes	4
39.8	even	4		39.4.b.b.25.1	✓	4
39.38	odd	2		507.4.a.l.1.4		4
156.47	odd	4		624.4.c.c.337.1		4
156.83	odd	4		624.4.c.c.337.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.b.b.25.1	✓	4	39.8	even	4
39.4.b.b.25.4	yes	4	39.5	even	4
117.4.b.e.64.1		4	13.5	odd	4
117.4.b.e.64.4		4	13.8	odd	4
507.4.a.l.1.1		4	3.2	odd	2
507.4.a.l.1.4		4	39.38	odd	2
624.4.c.c.337.1		4	156.47	odd	4
624.4.c.c.337.4		4	156.83	odd	4
1521.4.a.w.1.1		4	13.12	even	2	inner
1521.4.a.w.1.4		4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q+4.54739 q^{2} +12.6788 q^{4} +12.9118 q^{5} -16.7289 q^{7} +21.2762 q^{8} +O(q^{10})\)	\(q+4.54739 q^{2} +12.6788 q^{4} +12.9118 q^{5} -16.7289 q^{7} +21.2762 q^{8} +58.7151 q^{10} +24.9280 q^{11} -76.0727 q^{14} -4.67878 q^{16} +134.145 q^{17} +14.9376 q^{19} +163.706 q^{20} +113.358 q^{22} -72.0000 q^{23} +41.7151 q^{25} -212.101 q^{28} +206.145 q^{29} +249.142 q^{31} -191.486 q^{32} +610.012 q^{34} -216.000 q^{35} -293.955 q^{37} +67.9273 q^{38} +274.715 q^{40} +250.506 q^{41} +432.145 q^{43} +316.057 q^{44} -327.412 q^{46} -159.889 q^{47} -63.1454 q^{49} +189.695 q^{50} +194.581 q^{53} +321.866 q^{55} -355.927 q^{56} +937.424 q^{58} +232.647 q^{59} -185.006 q^{61} +1132.94 q^{62} -833.333 q^{64} +39.4393 q^{67} +1700.80 q^{68} -982.237 q^{70} +920.460 q^{71} +549.078 q^{73} -1336.73 q^{74} +189.391 q^{76} -417.018 q^{77} +933.140 q^{79} -60.4116 q^{80} +1139.15 q^{82} -1095.38 q^{83} +1732.06 q^{85} +1965.13 q^{86} +530.375 q^{88} -532.114 q^{89} -912.872 q^{92} -727.079 q^{94} +192.872 q^{95} +362.661 q^{97} -287.147 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 14 q^{4}+O(q^{10}) \) 4 * q + 14 * q^4	\( 4 q + 14 q^{4} + 88 q^{10} - 84 q^{14} + 18 q^{16} + 96 q^{17} + 380 q^{22} - 288 q^{23} + 20 q^{25} + 384 q^{29} - 864 q^{35} + 492 q^{38} + 952 q^{40} + 1288 q^{43} + 188 q^{49} - 984 q^{53} - 328 q^{55} - 1644 q^{56} + 288 q^{61} + 1668 q^{62} - 1314 q^{64} + 4380 q^{68} - 3144 q^{74} + 1416 q^{77} + 4320 q^{79} + 3088 q^{82} - 1036 q^{88} - 1008 q^{92} - 1660 q^{94} - 1872 q^{95}+O(q^{100}) \) 4 * q + 14 * q^4 + 88 * q^10 - 84 * q^14 + 18 * q^16 + 96 * q^17 + 380 * q^22 - 288 * q^23 + 20 * q^25 + 384 * q^29 - 864 * q^35 + 492 * q^38 + 952 * q^40 + 1288 * q^43 + 188 * q^49 - 984 * q^53 - 328 * q^55 - 1644 * q^56 + 288 * q^61 + 1668 * q^62 - 1314 * q^64 + 4380 * q^68 - 3144 * q^74 + 1416 * q^77 + 4320 * q^79 + 3088 * q^82 - 1036 * q^88 - 1008 * q^92 - 1660 * q^94 - 1872 * q^95

Introduction

L-functions

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Database

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