LMFDB - Embedded newform 1521.4.a.bf.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1521,4,Mod(1,1521)] 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("1521.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [9,-6,0,44,-33,0,83,-87,0,-54,-85,0,0,-158,0,216,-178] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$89.7419051187$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 56x^{7} - 27x^{6} + 945x^{5} + 763x^{4} - 4139x^{3} - 2478x^{2} + 63x + 27 $ x^9 - 56x^7 - 27x^6 + 945x^5 + 763x^4 - 4139x^3 - 2478x^2 + 63*x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	no (minimal twist has level 507)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.100291$ 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)

$f(q)$	$=$	$q-2.34727 q^{2} -2.49032 q^{4} -15.3991 q^{5} -10.1317 q^{7} +24.6236 q^{8} +36.1458 q^{10} -15.0669 q^{11} +23.7819 q^{14} -37.8757 q^{16} -90.8352 q^{17} +114.640 q^{19} +38.3486 q^{20} +35.3661 q^{22} -75.7635 q^{23} +112.132 q^{25} +25.2313 q^{28} -214.817 q^{29} -284.476 q^{31} -108.084 q^{32} +213.215 q^{34} +156.019 q^{35} +358.878 q^{37} -269.091 q^{38} -379.181 q^{40} -313.154 q^{41} -296.702 q^{43} +37.5214 q^{44} +177.837 q^{46} +316.691 q^{47} -240.348 q^{49} -263.203 q^{50} -163.911 q^{53} +232.016 q^{55} -249.480 q^{56} +504.233 q^{58} -254.149 q^{59} -935.247 q^{61} +667.742 q^{62} +556.709 q^{64} -240.494 q^{67} +226.209 q^{68} -366.220 q^{70} -947.455 q^{71} -430.712 q^{73} -842.384 q^{74} -285.490 q^{76} +152.654 q^{77} -496.620 q^{79} +583.251 q^{80} +735.058 q^{82} -392.527 q^{83} +1398.78 q^{85} +696.439 q^{86} -371.002 q^{88} -979.895 q^{89} +188.675 q^{92} -743.360 q^{94} -1765.35 q^{95} -553.356 q^{97} +564.162 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 6 q^{2} + 44 q^{4} - 33 q^{5} + 83 q^{7} - 87 q^{8} - 54 q^{10} - 85 q^{11} - 158 q^{14} + 216 q^{16} - 178 q^{17} + 352 q^{19} - 402 q^{20} - 630 q^{22} - 150 q^{23} - 20 q^{25} + 940 q^{28} + 97 q^{29}+ \cdots - 1457 q^{98}+O(q^{100}) $ 9 * q - 6 * q^2 + 44 * q^4 - 33 * q^5 + 83 * q^7 - 87 * q^8 - 54 * q^10 - 85 * q^11 - 158 * q^14 + 216 * q^16 - 178 * q^17 + 352 * q^19 - 402 * q^20 - 630 * q^22 - 150 * q^23 - 20 * q^25 + 940 * q^28 + 97 * q^29 + 717 * q^31 - 707 * q^32 + 632 * q^34 + 418 * q^35 + 1108 * q^37 + 660 * q^38 - 1506 * q^40 - 334 * q^41 + 242 * q^43 + 307 * q^44 + 979 * q^46 + 184 * q^47 - 38 * q^49 + 2031 * q^50 + 151 * q^53 + 2064 * q^55 - 2276 * q^56 + 1161 * q^58 - 537 * q^59 - 1340 * q^61 - 347 * q^62 + 893 * q^64 + 2308 * q^67 - 2785 * q^68 - 1420 * q^70 - 96 * q^71 + 2505 * q^73 + 1191 * q^74 + 2409 * q^76 + 2142 * q^77 - 1591 * q^79 + 2671 * q^80 + 1517 * q^82 - 1539 * q^83 + 4296 * q^85 + 3763 * q^86 - 3716 * q^88 + 592 * q^89 - 515 * q^92 - 692 * q^94 - 4158 * q^95 + 1445 * q^97 - 1457 * q^98

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 $

$2$	−2.34727	−0.829886	−0.414943	−	0.909848i	$-0.636198\pi$
$2$	−2.34727	−0.829886	−0.414943	+	0.909848i	$0.636198\pi$
$3$	0	0
$3$	0	0
$4$	−2.49032	−0.311290
$5$	−15.3991	−1.37734	−0.688668	−	0.725077i	$-0.741804\pi$
$5$	−15.3991	−1.37734	−0.688668	+	0.725077i	$0.741804\pi$
$6$	0	0
$7$	−10.1317	−0.547062	−0.273531	−	0.961863i	$-0.588192\pi$
$7$	−10.1317	−0.547062	−0.273531	+	0.961863i	$0.588192\pi$
$8$	24.6236	1.08822
$9$	0	0
$10$	36.1458	1.14303
$11$	−15.0669	−0.412985	−0.206493	−	0.978448i	$-0.566205\pi$
$11$	−15.0669	−0.412985	−0.206493	+	0.978448i	$0.566205\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	23.7819	0.453999
$15$	0	0
$16$	−37.8757	−0.591809
$17$	−90.8352	−1.29593	−0.647964	−	0.761671i	$-0.724379\pi$
$17$	−90.8352	−1.29593	−0.647964	+	0.761671i	$0.724379\pi$
$18$	0	0
$19$	114.640	1.38422	0.692110	−	0.721792i	$-0.256681\pi$
$19$	114.640	1.38422	0.692110	+	0.721792i	$0.256681\pi$
$20$	38.3486	0.428751
$21$	0	0
$22$	35.3661	0.342731
$23$	−75.7635	−0.686860	−0.343430	−	0.939178i	$-0.611589\pi$
$23$	−75.7635	−0.686860	−0.343430	+	0.939178i	$0.611589\pi$
$24$	0	0
$25$	112.132	0.897052
$26$	0	0
$27$	0	0
$28$	25.2313	0.170295
$29$	−214.817	−1.37553	−0.687767	−	0.725931i	$-0.741409\pi$
$29$	−214.817	−1.37553	−0.687767	+	0.725931i	$0.741409\pi$
$30$	0	0
$31$	−284.476	−1.64817	−0.824087	−	0.566463i	$-0.808311\pi$
$31$	−284.476	−1.64817	−0.824087	+	0.566463i	$0.808311\pi$
$32$	−108.084	−0.597087
$33$	0	0
$34$	213.215	1.07547
$35$	156.019	0.753488
$36$	0	0
$37$	358.878	1.59457	0.797286	−	0.603602i	$-0.206268\pi$
$37$	358.878	1.59457	0.797286	+	0.603602i	$0.206268\pi$
$38$	−269.091	−1.14874
$39$	0	0
$40$	−379.181	−1.49884
$41$	−313.154	−1.19284	−0.596421	−	0.802672i	$-0.703411\pi$
$41$	−313.154	−1.19284	−0.596421	+	0.802672i	$0.703411\pi$
$42$	0	0
$43$	−296.702	−1.05225	−0.526123	−	0.850409i	$-0.676355\pi$
$43$	−296.702	−1.05225	−0.526123	+	0.850409i	$0.676355\pi$
$44$	37.5214	0.128558
$45$	0	0
$46$	177.837	0.570015
$47$	316.691	0.982854	0.491427	−	0.870919i	$-0.336475\pi$
$47$	316.691	0.982854	0.491427	+	0.870919i	$0.336475\pi$
$48$	0	0
$49$	−240.348	−0.700723
$50$	−263.203	−0.744451
$51$	0	0
$52$	0	0
$53$	−163.911	−0.424810	−0.212405	−	0.977182i	$-0.568130\pi$
$53$	−163.911	−0.424810	−0.212405	+	0.977182i	$0.568130\pi$
$54$	0	0
$55$	232.016	0.568819
$56$	−249.480	−0.595324
$57$	0	0
$58$	504.233	1.14154
$59$	−254.149	−0.560803	−0.280401	−	0.959883i	$-0.590468\pi$
$59$	−254.149	−0.560803	−0.280401	+	0.959883i	$0.590468\pi$
$60$	0	0
$61$	−935.247	−1.96305	−0.981526	−	0.191330i	$-0.938720\pi$
$61$	−935.247	−1.96305	−0.981526	+	0.191330i	$0.938720\pi$
$62$	667.742	1.36780
$63$	0	0
$64$	556.709	1.08732
$65$	0	0
$66$	0	0
$67$	−240.494	−0.438522	−0.219261	−	0.975666i	$-0.570365\pi$
$67$	−240.494	−0.438522	−0.219261	+	0.975666i	$0.570365\pi$
$68$	226.209	0.403409
$69$	0	0
$70$	−366.220	−0.625309
$71$	−947.455	−1.58369	−0.791847	−	0.610720i	$-0.790880\pi$
$71$	−947.455	−1.58369	−0.791847	+	0.610720i	$0.790880\pi$
$72$	0	0
$73$	−430.712	−0.690562	−0.345281	−	0.938499i	$-0.612216\pi$
$73$	−430.712	−0.690562	−0.345281	+	0.938499i	$0.612216\pi$
$74$	−842.384	−1.32331
$75$	0	0
$76$	−285.490	−0.430894
$77$	152.654	0.225929
$78$	0	0
$79$	−496.620	−0.707268	−0.353634	−	0.935384i	$-0.615054\pi$
$79$	−496.620	−0.707268	−0.353634	+	0.935384i	$0.615054\pi$
$80$	583.251	0.815119
$81$	0	0
$82$	735.058	0.989922
$83$	−392.527	−0.519102	−0.259551	−	0.965729i	$-0.583575\pi$
$83$	−392.527	−0.519102	−0.259551	+	0.965729i	$0.583575\pi$
$84$	0	0
$85$	1398.78	1.78493
$86$	696.439	0.873244
$87$	0	0
$88$	−371.002	−0.449419
$89$	−979.895	−1.16706	−0.583532	−	0.812090i	$-0.698330\pi$
$89$	−979.895	−1.16706	−0.583532	+	0.812090i	$0.698330\pi$
$90$	0	0
$91$	0	0
$92$	188.675	0.213813
$93$	0	0
$94$	−743.360	−0.815656
$95$	−1765.35	−1.90654
$96$	0	0
$97$	−553.356	−0.579225	−0.289613	−	0.957144i	$-0.593526\pi$
$97$	−553.356	−0.579225	−0.289613	+	0.957144i	$0.593526\pi$
$98$	564.162	0.581520
$99$	0	0
$100$	−279.243	−0.279243
$101$	763.951	0.752633	0.376317	−	0.926491i	$-0.377190\pi$
$101$	763.951	0.752633	0.376317	+	0.926491i	$0.377190\pi$
$102$	0	0
$103$	182.518	0.174602	0.0873010	−	0.996182i	$-0.472176\pi$
$103$	182.518	0.174602	0.0873010	+	0.996182i	$0.472176\pi$
$104$	0	0
$105$	0	0
$106$	384.744	0.352544
$107$	−183.699	−0.165971	−0.0829855	−	0.996551i	$-0.526446\pi$
$107$	−183.699	−0.165971	−0.0829855	+	0.996551i	$0.526446\pi$
$108$	0	0
$109$	1774.42	1.55925	0.779627	−	0.626244i	$-0.215408\pi$
$109$	1774.42	1.55925	0.779627	+	0.626244i	$0.215408\pi$
$110$	−544.605	−0.472055
$111$	0	0
$112$	383.747	0.323756
$113$	−417.288	−0.347391	−0.173695	−	0.984799i	$-0.555571\pi$
$113$	−417.288	−0.347391	−0.173695	+	0.984799i	$0.555571\pi$
$114$	0	0
$115$	1166.69	0.946037
$116$	534.963	0.428190
$117$	0	0
$118$	596.556	0.465402
$119$	920.318	0.708953
$120$	0	0
$121$	−1103.99	−0.829443
$122$	2195.28	1.62911
$123$	0	0
$124$	708.436	0.513060
$125$	198.162	0.141793
$126$	0	0
$127$	−1951.69	−1.36366	−0.681828	−	0.731513i	$-0.738815\pi$
$127$	−1951.69	−1.36366	−0.681828	+	0.731513i	$0.738815\pi$
$128$	−442.072	−0.305266
$129$	0	0
$130$	0	0
$131$	1475.61	0.984159	0.492080	−	0.870550i	$-0.336237\pi$
$131$	1475.61	0.984159	0.492080	+	0.870550i	$0.336237\pi$
$132$	0	0
$133$	−1161.50	−0.757255
$134$	564.503	0.363923
$135$	0	0
$136$	−2236.69	−1.41026
$137$	1900.71	1.18532	0.592660	−	0.805453i	$-0.298078\pi$
$137$	1900.71	1.18532	0.592660	+	0.805453i	$0.298078\pi$
$138$	0	0
$139$	−2326.76	−1.41981	−0.709903	−	0.704299i	$-0.751261\pi$
$139$	−2326.76	−1.41981	−0.709903	+	0.704299i	$0.751261\pi$
$140$	−388.538	−0.234553
$141$	0	0
$142$	2223.93	1.31428
$143$	0	0
$144$	0	0
$145$	3307.98	1.89457
$146$	1011.00	0.573088
$147$	0	0
$148$	−893.721	−0.496374
$149$	1370.92	0.753761	0.376881	−	0.926262i	$-0.376997\pi$
$149$	1370.92	0.753761	0.376881	+	0.926262i	$0.376997\pi$
$150$	0	0
$151$	1177.57	0.634628	0.317314	−	0.948320i	$-0.397219\pi$
$151$	1177.57	0.634628	0.317314	+	0.948320i	$0.397219\pi$
$152$	2822.85	1.50634
$153$	0	0
$154$	−358.320	−0.187495
$155$	4380.67	2.27009
$156$	0	0
$157$	1621.57	0.824302	0.412151	−	0.911116i	$-0.364778\pi$
$157$	1621.57	0.824302	0.412151	+	0.911116i	$0.364778\pi$
$158$	1165.70	0.586951
$159$	0	0
$160$	1664.40	0.822389
$161$	767.616	0.375755
$162$	0	0
$163$	−133.130	−0.0639727	−0.0319864	−	0.999488i	$-0.510183\pi$
$163$	−133.130	−0.0639727	−0.0319864	+	0.999488i	$0.510183\pi$
$164$	779.854	0.371320
$165$	0	0
$166$	921.368	0.430795
$167$	490.724	0.227385	0.113693	−	0.993516i	$-0.463732\pi$
$167$	490.724	0.227385	0.113693	+	0.993516i	$0.463732\pi$
$168$	0	0
$169$	0	0
$170$	−3283.31	−1.48129
$171$	0	0
$172$	738.882	0.327554
$173$	−2008.13	−0.882518	−0.441259	−	0.897380i	$-0.645468\pi$
$173$	−2008.13	−0.882518	−0.441259	+	0.897380i	$0.645468\pi$
$174$	0	0
$175$	−1136.09	−0.490743
$176$	570.670	0.244408
$177$	0	0
$178$	2300.08	0.968529
$179$	2152.60	0.898843	0.449421	−	0.893320i	$-0.351630\pi$
$179$	2152.60	0.898843	0.449421	+	0.893320i	$0.351630\pi$
$180$	0	0
$181$	834.690	0.342774	0.171387	−	0.985204i	$-0.445175\pi$
$181$	834.690	0.342774	0.171387	+	0.985204i	$0.445175\pi$
$182$	0	0
$183$	0	0
$184$	−1865.57	−0.747455
$185$	−5526.39	−2.19626
$186$	0	0
$187$	1368.60	0.535200
$188$	−788.662	−0.305953
$189$	0	0
$190$	4143.75	1.58221
$191$	−4464.71	−1.69139	−0.845695	−	0.533667i	$-0.820814\pi$
$191$	−4464.71	−1.69139	−0.845695	+	0.533667i	$0.820814\pi$
$192$	0	0
$193$	−3299.84	−1.23071	−0.615357	−	0.788248i	$-0.710988\pi$
$193$	−3299.84	−1.23071	−0.615357	+	0.788248i	$0.710988\pi$
$194$	1298.88	0.480691
$195$	0	0
$196$	598.543	0.218128
$197$	−2973.59	−1.07543	−0.537714	−	0.843127i	$-0.680712\pi$
$197$	−2973.59	−1.07543	−0.537714	+	0.843127i	$0.680712\pi$
$198$	0	0
$199$	−5430.82	−1.93457	−0.967287	−	0.253684i	$-0.918358\pi$
$199$	−5430.82	−1.93457	−0.967287	+	0.253684i	$0.918358\pi$
$200$	2761.08	0.976191
$201$	0	0
$202$	−1793.20	−0.624600
$203$	2176.47	0.752503
$204$	0	0
$205$	4822.29	1.64294
$206$	−428.419	−0.144900
$207$	0	0
$208$	0	0
$209$	−1727.27	−0.571663
$210$	0	0
$211$	1228.25	0.400742	0.200371	−	0.979720i	$-0.435785\pi$
$211$	1228.25	0.400742	0.200371	+	0.979720i	$0.435785\pi$
$212$	408.192	0.132239
$213$	0	0
$214$	431.192	0.137737
$215$	4568.93	1.44930
$216$	0	0
$217$	2882.24	0.901654
$218$	−4165.05	−1.29400
$219$	0	0
$220$	−577.795	−0.177068
$221$	0	0
$222$	0	0
$223$	−685.256	−0.205776	−0.102888	−	0.994693i	$-0.532808\pi$
$223$	−685.256	−0.205776	−0.102888	+	0.994693i	$0.532808\pi$
$224$	1095.08	0.326644
$225$	0	0
$226$	979.487	0.288294
$227$	−287.067	−0.0839354	−0.0419677	−	0.999119i	$-0.513363\pi$
$227$	−287.067	−0.0839354	−0.0419677	+	0.999119i	$0.513363\pi$
$228$	0	0
$229$	−2302.23	−0.664347	−0.332174	−	0.943218i	$-0.607782\pi$
$229$	−2302.23	−0.664347	−0.332174	+	0.943218i	$0.607782\pi$
$230$	−2738.53	−0.785102
$231$	0	0
$232$	−5289.57	−1.49688
$233$	970.620	0.272908	0.136454	−	0.990646i	$-0.456429\pi$
$233$	970.620	0.272908	0.136454	+	0.990646i	$0.456429\pi$
$234$	0	0
$235$	−4876.75	−1.35372
$236$	632.912	0.174572
$237$	0	0
$238$	−2160.24	−0.588350
$239$	5007.40	1.35524	0.677619	−	0.735413i	$-0.263012\pi$
$239$	5007.40	1.35524	0.677619	+	0.735413i	$0.263012\pi$
$240$	0	0
$241$	−540.092	−0.144358	−0.0721792	−	0.997392i	$-0.522995\pi$
$241$	−540.092	−0.144358	−0.0721792	+	0.997392i	$0.522995\pi$
$242$	2591.36	0.688343
$243$	0	0
$244$	2329.07	0.611078
$245$	3701.14	0.965130
$246$	0	0
$247$	0	0
$248$	−7004.83	−1.79358
$249$	0	0
$250$	−465.141	−0.117672
$251$	6087.80	1.53091	0.765455	−	0.643489i	$-0.222514\pi$
$251$	6087.80	1.53091	0.765455	+	0.643489i	$0.222514\pi$
$252$	0	0
$253$	1141.52	0.283663
$254$	4581.14	1.13168
$255$	0	0
$256$	−3416.01	−0.833987
$257$	5096.34	1.23697	0.618484	−	0.785797i	$-0.287747\pi$
$257$	5096.34	1.23697	0.618484	+	0.785797i	$0.287747\pi$
$258$	0	0
$259$	−3636.06	−0.872330
$260$	0	0
$261$	0	0
$262$	−3463.66	−0.816739
$263$	−3405.50	−0.798450	−0.399225	−	0.916853i	$-0.630721\pi$
$263$	−3405.50	−0.798450	−0.399225	+	0.916853i	$0.630721\pi$
$264$	0	0
$265$	2524.08	0.585106
$266$	2726.36	0.628435
$267$	0	0
$268$	598.906	0.136507
$269$	2720.44	0.616611	0.308306	−	0.951287i	$-0.400238\pi$
$269$	2720.44	0.616611	0.308306	+	0.951287i	$0.400238\pi$
$270$	0	0
$271$	6954.27	1.55883	0.779413	−	0.626511i	$-0.215518\pi$
$271$	6954.27	1.55883	0.779413	+	0.626511i	$0.215518\pi$
$272$	3440.45	0.766941
$273$	0	0
$274$	−4461.49	−0.983680
$275$	−1689.47	−0.370470
$276$	0	0
$277$	−6563.96	−1.42379	−0.711895	−	0.702286i	$-0.752163\pi$
$277$	−6563.96	−1.42379	−0.711895	+	0.702286i	$0.752163\pi$
$278$	5461.53	1.17828
$279$	0	0
$280$	3841.76	0.819961
$281$	652.800	0.138586	0.0692932	−	0.997596i	$-0.477926\pi$
$281$	652.800	0.138586	0.0692932	+	0.997596i	$0.477926\pi$
$282$	0	0
$283$	6010.62	1.26252	0.631262	−	0.775570i	$-0.282537\pi$
$283$	6010.62	1.26252	0.631262	+	0.775570i	$0.282537\pi$
$284$	2359.47	0.492988
$285$	0	0
$286$	0	0
$287$	3172.80	0.652558
$288$	0	0
$289$	3338.04	0.679430
$290$	−7764.73	−1.57228
$291$	0	0
$292$	1072.61	0.214965
$293$	1912.72	0.381372	0.190686	−	0.981651i	$-0.438929\pi$
$293$	1912.72	0.381372	0.190686	+	0.981651i	$0.438929\pi$
$294$	0	0
$295$	3913.66	0.772413
$296$	8836.87	1.73525
$297$	0	0
$298$	−3217.93	−0.625535
$299$	0	0
$300$	0	0
$301$	3006.10	0.575644
$302$	−2764.06	−0.526669
$303$	0	0
$304$	−4342.07	−0.819194
$305$	14401.9	2.70378
$306$	0	0
$307$	−1983.07	−0.368665	−0.184332	−	0.982864i	$-0.559012\pi$
$307$	−1983.07	−0.368665	−0.184332	+	0.982864i	$0.559012\pi$
$308$	−380.157	−0.0703294
$309$	0	0
$310$	−10282.6	−1.88391
$311$	−1893.76	−0.345291	−0.172645	−	0.984984i	$-0.555231\pi$
$311$	−1893.76	−0.345291	−0.172645	+	0.984984i	$0.555231\pi$
$312$	0	0
$313$	4574.19	0.826034	0.413017	−	0.910723i	$-0.364475\pi$
$313$	4574.19	0.826034	0.413017	+	0.910723i	$0.364475\pi$
$314$	−3806.26	−0.684076
$315$	0	0
$316$	1236.74	0.220165
$317$	−7594.53	−1.34559	−0.672794	−	0.739830i	$-0.734906\pi$
$317$	−7594.53	−1.34559	−0.672794	+	0.739830i	$0.734906\pi$
$318$	0	0
$319$	3236.62	0.568076
$320$	−8572.81	−1.49761
$321$	0	0
$322$	−1801.80	−0.311834
$323$	−10413.3	−1.79385
$324$	0	0
$325$	0	0
$326$	312.492	0.0530901
$327$	0	0
$328$	−7710.99	−1.29807
$329$	−3208.63	−0.537682
$330$	0	0
$331$	−1738.58	−0.288705	−0.144352	−	0.989526i	$-0.546110\pi$
$331$	−1738.58	−0.288705	−0.144352	+	0.989526i	$0.546110\pi$
$332$	977.519	0.161591
$333$	0	0
$334$	−1151.86	−0.188704
$335$	3703.38	0.603992
$336$	0	0
$337$	−2710.61	−0.438149	−0.219074	−	0.975708i	$-0.570304\pi$
$337$	−2710.61	−0.438149	−0.219074	+	0.975708i	$0.570304\pi$
$338$	0	0
$339$	0	0
$340$	−3483.41	−0.555630
$341$	4286.17	0.680672
$342$	0	0
$343$	5910.33	0.930401
$344$	−7305.87	−1.14508
$345$	0	0
$346$	4713.63	0.732389
$347$	2506.81	0.387817	0.193908	−	0.981020i	$-0.437884\pi$
$347$	2506.81	0.387817	0.193908	+	0.981020i	$0.437884\pi$
$348$	0	0
$349$	7536.48	1.15593	0.577963	−	0.816063i	$-0.303848\pi$
$349$	7536.48	1.15593	0.577963	+	0.816063i	$0.303848\pi$
$350$	2666.70	0.407261
$351$	0	0
$352$	1628.50	0.246588
$353$	8992.88	1.35593	0.677964	−	0.735095i	$-0.262862\pi$
$353$	8992.88	1.35593	0.677964	+	0.735095i	$0.262862\pi$
$354$	0	0
$355$	14589.9	2.18128
$356$	2440.25	0.363295
$357$	0	0
$358$	−5052.74	−0.745937
$359$	4566.64	0.671359	0.335679	−	0.941976i	$-0.391034\pi$
$359$	4566.64	0.671359	0.335679	+	0.941976i	$0.391034\pi$
$360$	0	0
$361$	6283.31	0.916068
$362$	−1959.24	−0.284463
$363$	0	0
$364$	0	0
$365$	6632.57	0.951136
$366$	0	0
$367$	−6449.50	−0.917332	−0.458666	−	0.888609i	$-0.651673\pi$
$367$	−6449.50	−0.917332	−0.458666	+	0.888609i	$0.651673\pi$
$368$	2869.60	0.406490
$369$	0	0
$370$	12971.9	1.82264
$371$	1660.71	0.232398
$372$	0	0
$373$	7648.89	1.06178	0.530891	−	0.847440i	$-0.321857\pi$
$373$	7648.89	1.06178	0.530891	+	0.847440i	$0.321857\pi$
$374$	−3212.49	−0.444154
$375$	0	0
$376$	7798.08	1.06956
$377$	0	0
$378$	0	0
$379$	−10297.8	−1.39567	−0.697837	−	0.716256i	$-0.745854\pi$
$379$	−10297.8	−1.39567	−0.697837	+	0.716256i	$0.745854\pi$
$380$	4396.28	0.593486
$381$	0	0
$382$	10479.9	1.40366
$383$	−10258.0	−1.36856	−0.684282	−	0.729217i	$-0.739884\pi$
$383$	−10258.0	−1.36856	−0.684282	+	0.729217i	$0.739884\pi$
$384$	0	0
$385$	−2350.73	−0.311180
$386$	7745.63	1.02135
$387$	0	0
$388$	1378.03	0.180307
$389$	4771.57	0.621924	0.310962	−	0.950422i	$-0.399349\pi$
$389$	4771.57	0.621924	0.310962	+	0.950422i	$0.399349\pi$
$390$	0	0
$391$	6882.00	0.890122
$392$	−5918.24	−0.762541
$393$	0	0
$394$	6979.82	0.892483
$395$	7647.50	0.974145
$396$	0	0
$397$	2291.22	0.289655	0.144827	−	0.989457i	$-0.453737\pi$
$397$	2291.22	0.289655	0.144827	+	0.989457i	$0.453737\pi$
$398$	12747.6	1.60548
$399$	0	0
$400$	−4247.07	−0.530883
$401$	−7534.63	−0.938308	−0.469154	−	0.883116i	$-0.655441\pi$
$401$	−7534.63	−0.938308	−0.469154	+	0.883116i	$0.655441\pi$
$402$	0	0
$403$	0	0
$404$	−1902.48	−0.234287
$405$	0	0
$406$	−5108.76	−0.624491
$407$	−5407.18	−0.658535
$408$	0	0
$409$	−1517.97	−0.183517	−0.0917587	−	0.995781i	$-0.529249\pi$
$409$	−1517.97	−0.183517	−0.0917587	+	0.995781i	$0.529249\pi$
$410$	−11319.2	−1.36345
$411$	0	0
$412$	−454.528	−0.0543519
$413$	2574.97	0.306794
$414$	0	0
$415$	6044.56	0.714978
$416$	0	0
$417$	0	0
$418$	4054.36	0.474415
$419$	−8080.97	−0.942199	−0.471100	−	0.882080i	$-0.656143\pi$
$419$	−8080.97	−0.942199	−0.471100	+	0.882080i	$0.656143\pi$
$420$	0	0
$421$	8073.13	0.934585	0.467293	−	0.884103i	$-0.345229\pi$
$421$	8073.13	0.934585	0.467293	+	0.884103i	$0.345229\pi$
$422$	−2883.05	−0.332570
$423$	0	0
$424$	−4036.09	−0.462287
$425$	−10185.5	−1.16252
$426$	0	0
$427$	9475.68	1.07391
$428$	457.470	0.0516651
$429$	0	0
$430$	−10724.5	−1.20275
$431$	−2241.39	−0.250496	−0.125248	−	0.992125i	$-0.539973\pi$
$431$	−2241.39	−0.250496	−0.125248	+	0.992125i	$0.539973\pi$
$432$	0	0
$433$	10237.9	1.13626	0.568130	−	0.822939i	$-0.307667\pi$
$433$	10237.9	1.13626	0.568130	+	0.822939i	$0.307667\pi$
$434$	−6765.39	−0.748270
$435$	0	0
$436$	−4418.88	−0.485380
$437$	−8685.52	−0.950766
$438$	0	0
$439$	−5416.69	−0.588894	−0.294447	−	0.955668i	$-0.595135\pi$
$439$	−5416.69	−0.588894	−0.294447	+	0.955668i	$0.595135\pi$
$440$	5713.08	0.619001
$441$	0	0
$442$	0	0
$443$	2537.44	0.272138	0.136069	−	0.990699i	$-0.456553\pi$
$443$	2537.44	0.272138	0.136069	+	0.990699i	$0.456553\pi$
$444$	0	0
$445$	15089.5	1.60744
$446$	1608.48	0.170771
$447$	0	0
$448$	−5640.43	−0.594833
$449$	−7790.61	−0.818846	−0.409423	−	0.912345i	$-0.634270\pi$
$449$	−7790.61	−0.818846	−0.409423	+	0.912345i	$0.634270\pi$
$450$	0	0
$451$	4718.26	0.492626
$452$	1039.18	0.108139
$453$	0	0
$454$	673.825	0.0696568
$455$	0	0
$456$	0	0
$457$	6599.71	0.675539	0.337770	−	0.941229i	$-0.390328\pi$
$457$	6599.71	0.675539	0.337770	+	0.941229i	$0.390328\pi$
$458$	5403.95	0.551332
$459$	0	0
$460$	−2905.43	−0.294492
$461$	−4482.57	−0.452872	−0.226436	−	0.974026i	$-0.572707\pi$
$461$	−4482.57	−0.452872	−0.226436	+	0.974026i	$0.572707\pi$
$462$	0	0
$463$	−3805.86	−0.382016	−0.191008	−	0.981589i	$-0.561176\pi$
$463$	−3805.86	−0.382016	−0.191008	+	0.981589i	$0.561176\pi$
$464$	8136.35	0.814053
$465$	0	0
$466$	−2278.31	−0.226482
$467$	−14778.8	−1.46441	−0.732207	−	0.681082i	$-0.761510\pi$
$467$	−14778.8	−1.46441	−0.732207	+	0.681082i	$0.761510\pi$
$468$	0	0
$469$	2436.62	0.239899
$470$	11447.0	1.12343
$471$	0	0
$472$	−6258.06	−0.610277
$473$	4470.37	0.434562
$474$	0	0
$475$	12854.8	1.24172
$476$	−2291.89	−0.220690
$477$	0	0
$478$	−11753.7	−1.12469
$479$	−11171.2	−1.06560	−0.532801	−	0.846240i	$-0.678861\pi$
$479$	−11171.2	−1.06560	−0.532801	+	0.846240i	$0.678861\pi$
$480$	0	0
$481$	0	0
$482$	1267.74	0.119801
$483$	0	0
$484$	2749.29	0.258197
$485$	8521.18	0.797787
$486$	0	0
$487$	6046.57	0.562621	0.281310	−	0.959617i	$-0.409231\pi$
$487$	6046.57	0.562621	0.281310	+	0.959617i	$0.409231\pi$
$488$	−23029.2	−2.13623
$489$	0	0
$490$	−8687.57	−0.800948
$491$	1035.04	0.0951338	0.0475669	−	0.998868i	$-0.484853\pi$
$491$	1035.04	0.0951338	0.0475669	+	0.998868i	$0.484853\pi$
$492$	0	0
$493$	19512.9	1.78259
$494$	0	0
$495$	0	0
$496$	10774.7	0.975404
$497$	9599.37	0.866379
$498$	0	0
$499$	−11698.3	−1.04947	−0.524736	−	0.851265i	$-0.675836\pi$
$499$	−11698.3	−1.04947	−0.524736	+	0.851265i	$0.675836\pi$
$500$	−493.488	−0.0441389
$501$	0	0
$502$	−14289.7	−1.27048
$503$	13552.0	1.20130	0.600651	−	0.799511i	$-0.294908\pi$
$503$	13552.0	1.20130	0.600651	+	0.799511i	$0.294908\pi$
$504$	0	0
$505$	−11764.1	−1.03663
$506$	−2679.46	−0.235408
$507$	0	0
$508$	4860.33	0.424492
$509$	−5076.46	−0.442064	−0.221032	−	0.975267i	$-0.570942\pi$
$509$	−5076.46	−0.442064	−0.221032	+	0.975267i	$0.570942\pi$
$510$	0	0
$511$	4363.86	0.377781
$512$	11554.9	0.997380
$513$	0	0
$514$	−11962.5	−1.02654
$515$	−2810.60	−0.240486
$516$	0	0
$517$	−4771.55	−0.405904
$518$	8534.81	0.723934
$519$	0	0
$520$	0	0
$521$	8493.73	0.714236	0.357118	−	0.934059i	$-0.383759\pi$
$521$	8493.73	0.714236	0.357118	+	0.934059i	$0.383759\pi$
$522$	0	0
$523$	−1384.53	−0.115758	−0.0578789	−	0.998324i	$-0.518434\pi$
$523$	−1384.53	−0.115758	−0.0578789	+	0.998324i	$0.518434\pi$
$524$	−3674.75	−0.306359
$525$	0	0
$526$	7993.64	0.662622
$527$	25840.4	2.13592
$528$	0	0
$529$	−6426.89	−0.528223
$530$	−5924.71	−0.485571
$531$	0	0
$532$	2892.51	0.235726
$533$	0	0
$534$	0	0
$535$	2828.80	0.228598
$536$	−5921.82	−0.477208
$537$	0	0
$538$	−6385.62	−0.511717
$539$	3621.30	0.289388
$540$	0	0
$541$	5026.83	0.399483	0.199742	−	0.979849i	$-0.435990\pi$
$541$	5026.83	0.399483	0.199742	+	0.979849i	$0.435990\pi$
$542$	−16323.6	−1.29365
$543$	0	0
$544$	9817.87	0.773782
$545$	−27324.5	−2.14762
$546$	0	0
$547$	1540.36	0.120404	0.0602021	−	0.998186i	$-0.480825\pi$
$547$	1540.36	0.120404	0.0602021	+	0.998186i	$0.480825\pi$
$548$	−4733.38	−0.368978
$549$	0	0
$550$	3965.65	0.307447
$551$	−24626.6	−1.90404
$552$	0	0
$553$	5031.63	0.386920
$554$	15407.4	1.18158
$555$	0	0
$556$	5794.38	0.441972
$557$	8550.51	0.650443	0.325221	−	0.945638i	$-0.394561\pi$
$557$	8550.51	0.650443	0.325221	+	0.945638i	$0.394561\pi$
$558$	0	0
$559$	0	0
$560$	−5909.35	−0.445921
$561$	0	0
$562$	−1532.30	−0.115011
$563$	6569.35	0.491768	0.245884	−	0.969299i	$-0.420922\pi$
$563$	6569.35	0.491768	0.245884	+	0.969299i	$0.420922\pi$
$564$	0	0
$565$	6425.85	0.478473
$566$	−14108.5	−1.04775
$567$	0	0
$568$	−23329.8	−1.72341
$569$	−23766.1	−1.75102	−0.875508	−	0.483204i	$-0.839473\pi$
$569$	−23766.1	−1.75102	−0.875508	+	0.483204i	$0.839473\pi$
$570$	0	0
$571$	−24971.7	−1.83018	−0.915091	−	0.403248i	$-0.867881\pi$
$571$	−24971.7	−1.83018	−0.915091	+	0.403248i	$0.867881\pi$
$572$	0	0
$573$	0	0
$574$	−7447.41	−0.541549
$575$	−8495.48	−0.616150
$576$	0	0
$577$	−16643.4	−1.20082	−0.600409	−	0.799693i	$-0.704996\pi$
$577$	−16643.4	−1.20082	−0.600409	+	0.799693i	$0.704996\pi$
$578$	−7835.28	−0.563849
$579$	0	0
$580$	−8237.93	−0.589761
$581$	3976.98	0.283981
$582$	0	0
$583$	2469.64	0.175441
$584$	−10605.7	−0.751484
$585$	0	0
$586$	−4489.66	−0.316495
$587$	6720.67	0.472558	0.236279	−	0.971685i	$-0.424072\pi$
$587$	6720.67	0.472558	0.236279	+	0.971685i	$0.424072\pi$
$588$	0	0
$589$	−32612.3	−2.28144
$590$	−9186.41	−0.641014
$591$	0	0
$592$	−13592.8	−0.943681
$593$	−26349.8	−1.82471	−0.912357	−	0.409395i	$-0.865740\pi$
$593$	−26349.8	−1.82471	−0.912357	+	0.409395i	$0.865740\pi$
$594$	0	0
$595$	−14172.1	−0.976466
$596$	−3414.04	−0.234638
$597$	0	0
$598$	0	0
$599$	6136.81	0.418603	0.209302	−	0.977851i	$-0.432881\pi$
$599$	6136.81	0.418603	0.209302	+	0.977851i	$0.432881\pi$
$600$	0	0
$601$	12493.7	0.847966	0.423983	−	0.905670i	$-0.360632\pi$
$601$	12493.7	0.847966	0.423983	+	0.905670i	$0.360632\pi$
$602$	−7056.14	−0.477719
$603$	0	0
$604$	−2932.51	−0.197553
$605$	17000.4	1.14242
$606$	0	0
$607$	18696.6	1.25020	0.625099	−	0.780545i	$-0.285059\pi$
$607$	18696.6	1.25020	0.625099	+	0.780545i	$0.285059\pi$
$608$	−12390.8	−0.826501
$609$	0	0
$610$	−33805.3	−2.24383
$611$	0	0
$612$	0	0
$613$	1238.64	0.0816123	0.0408062	−	0.999167i	$-0.487007\pi$
$613$	1238.64	0.0816123	0.0408062	+	0.999167i	$0.487007\pi$
$614$	4654.81	0.305949
$615$	0	0
$616$	3758.89	0.245860
$617$	−16549.5	−1.07983	−0.539917	−	0.841718i	$-0.681544\pi$
$617$	−16549.5	−1.07983	−0.539917	+	0.841718i	$0.681544\pi$
$618$	0	0
$619$	13945.4	0.905513	0.452756	−	0.891634i	$-0.350441\pi$
$619$	13945.4	0.905513	0.452756	+	0.891634i	$0.350441\pi$
$620$	−10909.3	−0.706656
$621$	0	0
$622$	4445.17	0.286552
$623$	9928.03	0.638456
$624$	0	0
$625$	−17068.0	−1.09235
$626$	−10736.9	−0.685513
$627$	0	0
$628$	−4038.23	−0.256597
$629$	−32598.8	−2.06645
$630$	0	0
$631$	15343.1	0.967987	0.483994	−	0.875072i	$-0.339186\pi$
$631$	15343.1	0.967987	0.483994	+	0.875072i	$0.339186\pi$
$632$	−12228.6	−0.769663
$633$	0	0
$634$	17826.4	1.11668
$635$	30054.2	1.87821
$636$	0	0
$637$	0	0
$638$	−7597.23	−0.471438
$639$	0	0
$640$	6807.51	0.420454
$641$	−11067.7	−0.681979	−0.340989	−	0.940067i	$-0.610762\pi$
$641$	−11067.7	−0.681979	−0.340989	+	0.940067i	$0.610762\pi$
$642$	0	0
$643$	−25118.7	−1.54057	−0.770284	−	0.637701i	$-0.779885\pi$
$643$	−25118.7	−1.54057	−0.770284	+	0.637701i	$0.779885\pi$
$644$	−1911.61	−0.116969
$645$	0	0
$646$	24442.9	1.48869
$647$	4447.03	0.270217	0.135109	−	0.990831i	$-0.456862\pi$
$647$	4447.03	0.270217	0.135109	+	0.990831i	$0.456862\pi$
$648$	0	0
$649$	3829.23	0.231603
$650$	0	0
$651$	0	0
$652$	331.537	0.0199141
$653$	19426.2	1.16418	0.582088	−	0.813126i	$-0.302236\pi$
$653$	19426.2	1.16418	0.582088	+	0.813126i	$0.302236\pi$
$654$	0	0
$655$	−22723.1	−1.35552
$656$	11861.0	0.705934
$657$	0	0
$658$	7531.52	0.446215
$659$	14099.4	0.833438	0.416719	−	0.909035i	$-0.363180\pi$
$659$	14099.4	0.833438	0.416719	+	0.909035i	$0.363180\pi$
$660$	0	0
$661$	2754.25	0.162070	0.0810348	−	0.996711i	$-0.474177\pi$
$661$	2754.25	0.162070	0.0810348	+	0.996711i	$0.474177\pi$
$662$	4080.93	0.239592
$663$	0	0
$664$	−9665.44	−0.564898
$665$	17886.0	1.04299
$666$	0	0
$667$	16275.3	0.944800
$668$	−1222.06	−0.0707828
$669$	0	0
$670$	−8692.83	−0.501244
$671$	14091.3	0.810712
$672$	0	0
$673$	11936.8	0.683700	0.341850	−	0.939754i	$-0.388946\pi$
$673$	11936.8	0.683700	0.341850	+	0.939754i	$0.388946\pi$
$674$	6362.53	0.363613
$675$	0	0
$676$	0	0
$677$	10255.2	0.582186	0.291093	−	0.956695i	$-0.405981\pi$
$677$	10255.2	0.582186	0.291093	+	0.956695i	$0.405981\pi$
$678$	0	0
$679$	5606.46	0.316872
$680$	34443.0	1.94239
$681$	0	0
$682$	−10060.8	−0.564880
$683$	−10605.7	−0.594169	−0.297085	−	0.954851i	$-0.596014\pi$
$683$	−10605.7	−0.594169	−0.297085	+	0.954851i	$0.596014\pi$
$684$	0	0
$685$	−29269.2	−1.63258
$686$	−13873.1	−0.772127
$687$	0	0
$688$	11237.8	0.622728
$689$	0	0
$690$	0	0
$691$	4660.14	0.256556	0.128278	−	0.991738i	$-0.459055\pi$
$691$	4660.14	0.256556	0.128278	+	0.991738i	$0.459055\pi$
$692$	5000.90	0.274719
$693$	0	0
$694$	−5884.15	−0.321844
$695$	35829.9	1.95555
$696$	0	0
$697$	28445.4	1.54584
$698$	−17690.2	−0.959287
$699$	0	0
$700$	2829.22	0.152764
$701$	24016.9	1.29402	0.647008	−	0.762483i	$-0.276020\pi$
$701$	24016.9	1.29402	0.647008	+	0.762483i	$0.276020\pi$
$702$	0	0
$703$	41141.7	2.20724
$704$	−8387.88	−0.449048
$705$	0	0
$706$	−21108.7	−1.12527
$707$	−7740.15	−0.411737
$708$	0	0
$709$	−7252.56	−0.384169	−0.192084	−	0.981378i	$-0.561525\pi$
$709$	−7252.56	−0.384169	−0.192084	+	0.981378i	$0.561525\pi$
$710$	−34246.5	−1.81021
$711$	0	0
$712$	−24128.6	−1.27002
$713$	21552.9	1.13207
$714$	0	0
$715$	0	0
$716$	−5360.66	−0.279801
$717$	0	0
$718$	−10719.1	−0.557151
$719$	−10951.7	−0.568050	−0.284025	−	0.958817i	$-0.591670\pi$
$719$	−10951.7	−0.568050	−0.284025	+	0.958817i	$0.591670\pi$
$720$	0	0
$721$	−1849.22	−0.0955182
$722$	−14748.6	−0.760231
$723$	0	0
$724$	−2078.65	−0.106702
$725$	−24087.7	−1.23393
$726$	0	0
$727$	−27856.0	−1.42108	−0.710538	−	0.703658i	$-0.751549\pi$
$727$	−27856.0	−1.42108	−0.710538	+	0.703658i	$0.751549\pi$
$728$	0	0
$729$	0	0
$730$	−15568.4	−0.789334
$731$	26951.0	1.36364
$732$	0	0
$733$	−31213.8	−1.57286	−0.786431	−	0.617678i	$-0.788073\pi$
$733$	−31213.8	−1.57286	−0.786431	+	0.617678i	$0.788073\pi$
$734$	15138.7	0.761281
$735$	0	0
$736$	8188.85	0.410116
$737$	3623.49	0.181103
$738$	0	0
$739$	14423.1	0.717946	0.358973	−	0.933348i	$-0.383127\pi$
$739$	14423.1	0.717946	0.358973	+	0.933348i	$0.383127\pi$
$740$	13762.5	0.683674
$741$	0	0
$742$	−3898.13	−0.192864
$743$	−13469.7	−0.665079	−0.332539	−	0.943089i	$-0.607905\pi$
$743$	−13469.7	−0.665079	−0.332539	+	0.943089i	$0.607905\pi$
$744$	0	0
$745$	−21111.0	−1.03818
$746$	−17954.0	−0.881158
$747$	0	0
$748$	−3408.26	−0.166602
$749$	1861.19	0.0907965
$750$	0	0
$751$	−32033.6	−1.55649	−0.778245	−	0.627961i	$-0.783890\pi$
$751$	−32033.6	−1.55649	−0.778245	+	0.627961i	$0.783890\pi$
$752$	−11994.9	−0.581661
$753$	0	0
$754$	0	0
$755$	−18133.4	−0.874096
$756$	0	0
$757$	−26097.2	−1.25300	−0.626498	−	0.779423i	$-0.715512\pi$
$757$	−26097.2	−1.25300	−0.626498	+	0.779423i	$0.715512\pi$
$758$	24171.6	1.15825
$759$	0	0
$760$	−43469.3	−2.07473
$761$	−18238.2	−0.868772	−0.434386	−	0.900727i	$-0.643035\pi$
$761$	−18238.2	−0.868772	−0.434386	+	0.900727i	$0.643035\pi$
$762$	0	0
$763$	−17978.0	−0.853009
$764$	11118.6	0.526513
$765$	0	0
$766$	24078.4	1.13575
$767$	0	0
$768$	0	0
$769$	−18817.0	−0.882390	−0.441195	−	0.897411i	$-0.645445\pi$
$769$	−18817.0	−0.882390	−0.441195	+	0.897411i	$0.645445\pi$
$770$	5517.79	0.258243
$771$	0	0
$772$	8217.67	0.383109
$773$	13795.6	0.641904	0.320952	−	0.947095i	$-0.395997\pi$
$773$	13795.6	0.641904	0.320952	+	0.947095i	$0.395997\pi$
$774$	0	0
$775$	−31898.7	−1.47850
$776$	−13625.6	−0.630325
$777$	0	0
$778$	−11200.2	−0.516126
$779$	−35900.0	−1.65116
$780$	0	0
$781$	14275.2	0.654043
$782$	−16153.9	−0.738699
$783$	0	0
$784$	9103.36	0.414694
$785$	−24970.7	−1.13534
$786$	0	0
$787$	−40545.4	−1.83645	−0.918227	−	0.396055i	$-0.870379\pi$
$787$	−40545.4	−1.83645	−0.918227	+	0.396055i	$0.870379\pi$
$788$	7405.19	0.334770
$789$	0	0
$790$	−17950.7	−0.808429
$791$	4227.85	0.190044
$792$	0	0
$793$	0	0
$794$	−5378.11	−0.240380
$795$	0	0
$796$	13524.5	0.602214
$797$	31576.4	1.40338	0.701690	−	0.712482i	$-0.252429\pi$
$797$	31576.4	1.40338	0.701690	+	0.712482i	$0.252429\pi$
$798$	0	0
$799$	−28766.7	−1.27371
$800$	−12119.7	−0.535619
$801$	0	0
$802$	17685.8	0.778688
$803$	6489.50	0.285192
$804$	0	0
$805$	−11820.6	−0.517541
$806$	0	0
$807$	0	0
$808$	18811.2	0.819031
$809$	−31130.1	−1.35287	−0.676437	−	0.736501i	$-0.736477\pi$
$809$	−31130.1	−1.35287	−0.676437	+	0.736501i	$0.736477\pi$
$810$	0	0
$811$	−8733.71	−0.378153	−0.189076	−	0.981962i	$-0.560549\pi$
$811$	−8733.71	−0.378153	−0.189076	+	0.981962i	$0.560549\pi$
$812$	−5420.10	−0.234247
$813$	0	0
$814$	12692.1	0.546509
$815$	2050.08	0.0881119
$816$	0	0
$817$	−34013.8	−1.45654
$818$	3563.08	0.152298
$819$	0	0
$820$	−12009.0	−0.511431
$821$	36960.1	1.57115	0.785576	−	0.618765i	$-0.212367\pi$
$821$	36960.1	1.57115	0.785576	+	0.618765i	$0.212367\pi$
$822$	0	0
$823$	20509.7	0.868679	0.434340	−	0.900749i	$-0.356982\pi$
$823$	20509.7	0.868679	0.434340	+	0.900749i	$0.356982\pi$
$824$	4494.25	0.190006
$825$	0	0
$826$	−6044.15	−0.254604
$827$	11533.4	0.484952	0.242476	−	0.970157i	$-0.422040\pi$
$827$	11533.4	0.484952	0.242476	+	0.970157i	$0.422040\pi$
$828$	0	0
$829$	34096.4	1.42849	0.714244	−	0.699897i	$-0.246771\pi$
$829$	34096.4	1.42849	0.714244	+	0.699897i	$0.246771\pi$
$830$	−14188.2	−0.593350
$831$	0	0
$832$	0	0
$833$	21832.1	0.908087
$834$	0	0
$835$	−7556.69	−0.313186
$836$	4301.45	0.177953
$837$	0	0
$838$	18968.2	0.781917
$839$	1855.42	0.0763484	0.0381742	−	0.999271i	$-0.487846\pi$
$839$	1855.42	0.0763484	0.0381742	+	0.999271i	$0.487846\pi$
$840$	0	0
$841$	21757.3	0.892094
$842$	−18949.8	−0.775599
$843$	0	0
$844$	−3058.75	−0.124747
$845$	0	0
$846$	0	0
$847$	11185.3	0.453757
$848$	6208.26	0.251406
$849$	0	0
$850$	23908.1	0.964755
$851$	−27189.9	−1.09525
$852$	0	0
$853$	−28668.7	−1.15076	−0.575380	−	0.817886i	$-0.695146\pi$
$853$	−28668.7	−1.15076	−0.575380	+	0.817886i	$0.695146\pi$
$854$	−22242.0	−0.891224
$855$	0	0
$856$	−4523.35	−0.180613
$857$	−449.310	−0.0179091	−0.00895457	−	0.999960i	$-0.502850\pi$
$857$	−449.310	−0.0179091	−0.00895457	+	0.999960i	$0.502850\pi$
$858$	0	0
$859$	33466.9	1.32931	0.664654	−	0.747151i	$-0.268579\pi$
$859$	33466.9	1.32931	0.664654	+	0.747151i	$0.268579\pi$
$860$	−11378.1	−0.451151
$861$	0	0
$862$	5261.14	0.207883
$863$	25097.5	0.989953	0.494976	−	0.868906i	$-0.335177\pi$
$863$	25097.5	0.989953	0.494976	+	0.868906i	$0.335177\pi$
$864$	0	0
$865$	30923.4	1.21552
$866$	−24031.0	−0.942965
$867$	0	0
$868$	−7177.69	−0.280676
$869$	7482.53	0.292091
$870$	0	0
$871$	0	0
$872$	43692.7	1.69681
$873$	0	0
$874$	20387.3	0.789027
$875$	−2007.73	−0.0775698
$876$	0	0
$877$	27015.2	1.04018	0.520090	−	0.854112i	$-0.325899\pi$
$877$	27015.2	1.04018	0.520090	+	0.854112i	$0.325899\pi$
$878$	12714.4	0.488715
$879$	0	0
$880$	−8787.79	−0.336632
$881$	−48638.7	−1.86002	−0.930010	−	0.367534i	$-0.880202\pi$
$881$	−48638.7	−1.86002	−0.930010	+	0.367534i	$0.880202\pi$
$882$	0	0
$883$	−25479.0	−0.971048	−0.485524	−	0.874223i	$-0.661371\pi$
$883$	−25479.0	−0.971048	−0.485524	+	0.874223i	$0.661371\pi$
$884$	0	0
$885$	0	0
$886$	−5956.05	−0.225843
$887$	−42359.7	−1.60350	−0.801748	−	0.597663i	$-0.796096\pi$
$887$	−42359.7	−1.60350	−0.801748	+	0.597663i	$0.796096\pi$
$888$	0	0
$889$	19774.0	0.746005
$890$	−35419.1	−1.33399
$891$	0	0
$892$	1706.51	0.0640561
$893$	36305.4	1.36049
$894$	0	0
$895$	−33148.1	−1.23801
$896$	4478.96	0.167000
$897$	0	0
$898$	18286.7	0.679548
$899$	61110.3	2.26712
$900$	0	0
$901$	14888.9	0.550524
$902$	−11075.0	−0.408823
$903$	0	0
$904$	−10275.1	−0.378038
$905$	−12853.5	−0.472114
$906$	0	0
$907$	−5314.91	−0.194574	−0.0972871	−	0.995256i	$-0.531016\pi$
$907$	−5314.91	−0.194574	−0.0972871	+	0.995256i	$0.531016\pi$
$908$	714.890	0.0261283
$909$	0	0
$910$	0	0
$911$	−2471.71	−0.0898919	−0.0449459	−	0.998989i	$-0.514312\pi$
$911$	−2471.71	−0.0898919	−0.0449459	+	0.998989i	$0.514312\pi$
$912$	0	0
$913$	5914.17	0.214382
$914$	−15491.3	−0.560620
$915$	0	0
$916$	5733.29	0.206805
$917$	−14950.5	−0.538396
$918$	0	0
$919$	−9636.13	−0.345883	−0.172942	−	0.984932i	$-0.555327\pi$
$919$	−9636.13	−0.345883	−0.172942	+	0.984932i	$0.555327\pi$
$920$	28728.1	1.02950
$921$	0	0
$922$	10521.8	0.375832
$923$	0	0
$924$	0	0
$925$	40241.5	1.43041
$926$	8933.38	0.317029
$927$	0	0
$928$	23218.3	0.821314
$929$	53249.1	1.88057	0.940283	−	0.340394i	$-0.110561\pi$
$929$	53249.1	1.88057	0.940283	+	0.340394i	$0.110561\pi$
$930$	0	0
$931$	−27553.5	−0.969955
$932$	−2417.16	−0.0849534
$933$	0	0
$934$	34689.8	1.21530
$935$	−21075.3	−0.737149
$936$	0	0
$937$	−41122.6	−1.43374	−0.716871	−	0.697205i	$-0.754427\pi$
$937$	−41122.6	−1.43374	−0.716871	+	0.697205i	$0.754427\pi$
$938$	−5719.40	−0.199088
$939$	0	0
$940$	12144.7	0.421399
$941$	8005.71	0.277342	0.138671	−	0.990339i	$-0.455717\pi$
$941$	8005.71	0.277342	0.138671	+	0.990339i	$0.455717\pi$
$942$	0	0
$943$	23725.7	0.819315
$944$	9626.07	0.331888
$945$	0	0
$946$	−10493.2	−0.360637
$947$	17468.6	0.599424	0.299712	−	0.954030i	$-0.403109\pi$
$947$	17468.6	0.599424	0.299712	+	0.954030i	$0.403109\pi$
$948$	0	0
$949$	0	0
$950$	−30173.6	−1.03048
$951$	0	0
$952$	22661.6	0.771498
$953$	−30681.4	−1.04288	−0.521442	−	0.853287i	$-0.674606\pi$
$953$	−30681.4	−1.04288	−0.521442	+	0.853287i	$0.674606\pi$
$954$	0	0
$955$	68752.5	2.32961
$956$	−12470.0	−0.421872
$957$	0	0
$958$	26221.8	0.884328
$959$	−19257.5	−0.648444
$960$	0	0
$961$	51135.6	1.71648
$962$	0	0
$963$	0	0
$964$	1345.00	0.0449373
$965$	50814.5	1.69511
$966$	0	0
$967$	15616.0	0.519313	0.259656	−	0.965701i	$-0.416391\pi$
$967$	15616.0	0.519313	0.259656	+	0.965701i	$0.416391\pi$
$968$	−27184.2	−0.902617
$969$	0	0
$970$	−20001.5	−0.662072
$971$	−7185.98	−0.237496	−0.118748	−	0.992924i	$-0.537888\pi$
$971$	−7185.98	−0.237496	−0.118748	+	0.992924i	$0.537888\pi$
$972$	0	0
$973$	23574.1	0.776723
$974$	−14192.9	−0.466911
$975$	0	0
$976$	35423.2	1.16175
$977$	−33165.3	−1.08603	−0.543015	−	0.839723i	$-0.682717\pi$
$977$	−33165.3	−1.08603	−0.543015	+	0.839723i	$0.682717\pi$
$978$	0	0
$979$	14764.0	0.481980
$980$	−9217.01	−0.300435
$981$	0	0
$982$	−2429.52	−0.0789502
$983$	11658.7	0.378287	0.189143	−	0.981949i	$-0.439429\pi$
$983$	11658.7	0.378287	0.189143	+	0.981949i	$0.439429\pi$
$984$	0	0
$985$	45790.5	1.48123
$986$	−45802.1	−1.47935
$987$	0	0
$988$	0	0
$989$	22479.2	0.722746
$990$	0	0
$991$	−1957.32	−0.0627409	−0.0313704	−	0.999508i	$-0.509987\pi$
$991$	−1957.32	−0.0627409	−0.0313704	+	0.999508i	$0.509987\pi$
$992$	30747.4	0.984104
$993$	0	0
$994$	−22532.3	−0.718996
$995$	83629.5	2.66456
$996$	0	0
$997$	−11434.8	−0.363233	−0.181617	−	0.983369i	$-0.558133\pi$
$997$	−11434.8	−0.363233	−0.181617	+	0.983369i	$0.558133\pi$
$998$	27459.0	0.870942
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bf.1.4		9
3.2	odd	2		507.4.a.p.1.6	yes	9
13.12	even	2		1521.4.a.bi.1.6		9
39.5	even	4		507.4.b.k.337.7		18
39.8	even	4		507.4.b.k.337.12		18
39.38	odd	2		507.4.a.o.1.4	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.o.1.4	✓	9	39.38	odd	2
507.4.a.p.1.6	yes	9	3.2	odd	2
507.4.b.k.337.7		18	39.5	even	4
507.4.b.k.337.12		18	39.8	even	4
1521.4.a.bf.1.4		9	1.1	even	1	trivial
1521.4.a.bi.1.6		9	13.12	even	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-2.34727 q^{2} -2.49032 q^{4} -15.3991 q^{5} -10.1317 q^{7} +24.6236 q^{8} +36.1458 q^{10} -15.0669 q^{11} +23.7819 q^{14} -37.8757 q^{16} -90.8352 q^{17} +114.640 q^{19} +38.3486 q^{20} +35.3661 q^{22} -75.7635 q^{23} +112.132 q^{25} +25.2313 q^{28} -214.817 q^{29} -284.476 q^{31} -108.084 q^{32} +213.215 q^{34} +156.019 q^{35} +358.878 q^{37} -269.091 q^{38} -379.181 q^{40} -313.154 q^{41} -296.702 q^{43} +37.5214 q^{44} +177.837 q^{46} +316.691 q^{47} -240.348 q^{49} -263.203 q^{50} -163.911 q^{53} +232.016 q^{55} -249.480 q^{56} +504.233 q^{58} -254.149 q^{59} -935.247 q^{61} +667.742 q^{62} +556.709 q^{64} -240.494 q^{67} +226.209 q^{68} -366.220 q^{70} -947.455 q^{71} -430.712 q^{73} -842.384 q^{74} -285.490 q^{76} +152.654 q^{77} -496.620 q^{79} +583.251 q^{80} +735.058 q^{82} -392.527 q^{83} +1398.78 q^{85} +696.439 q^{86} -371.002 q^{88} -979.895 q^{89} +188.675 q^{92} -743.360 q^{94} -1765.35 q^{95} -553.356 q^{97} +564.162 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 6 q^{2} + 44 q^{4} - 33 q^{5} + 83 q^{7} - 87 q^{8} - 54 q^{10} - 85 q^{11} - 158 q^{14} + 216 q^{16} - 178 q^{17} + 352 q^{19} - 402 q^{20} - 630 q^{22} - 150 q^{23} - 20 q^{25} + 940 q^{28} + 97 q^{29}+ \cdots - 1457 q^{98}+O(q^{100}) \) 9 * q - 6 * q^2 + 44 * q^4 - 33 * q^5 + 83 * q^7 - 87 * q^8 - 54 * q^10 - 85 * q^11 - 158 * q^14 + 216 * q^16 - 178 * q^17 + 352 * q^19 - 402 * q^20 - 630 * q^22 - 150 * q^23 - 20 * q^25 + 940 * q^28 + 97 * q^29 + 717 * q^31 - 707 * q^32 + 632 * q^34 + 418 * q^35 + 1108 * q^37 + 660 * q^38 - 1506 * q^40 - 334 * q^41 + 242 * q^43 + 307 * q^44 + 979 * q^46 + 184 * q^47 - 38 * q^49 + 2031 * q^50 + 151 * q^53 + 2064 * q^55 - 2276 * q^56 + 1161 * q^58 - 537 * q^59 - 1340 * q^61 - 347 * q^62 + 893 * q^64 + 2308 * q^67 - 2785 * q^68 - 1420 * q^70 - 96 * q^71 + 2505 * q^73 + 1191 * q^74 + 2409 * q^76 + 2142 * q^77 - 1591 * q^79 + 2671 * q^80 + 1517 * q^82 - 1539 * q^83 + 4296 * q^85 + 3763 * q^86 - 3716 * q^88 + 592 * q^89 - 515 * q^92 - 692 * q^94 - 4158 * q^95 + 1445 * q^97 - 1457 * q^98

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