LMFDB - Embedded newform 507.4.a.p.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(1,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, 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("507.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	507.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:	$29.9139683729$
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:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.100291$ of defining polynomial
Character	$\chi$	$=$	507.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.34727 q^{2} +3.00000 q^{3} -2.49032 q^{4} +15.3991 q^{5} +7.04181 q^{6} -10.1317 q^{7} -24.6236 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.34727 q^{2} +3.00000 q^{3} -2.49032 q^{4} +15.3991 q^{5} +7.04181 q^{6} -10.1317 q^{7} -24.6236 q^{8} +9.00000 q^{9} +36.1458 q^{10} +15.0669 q^{11} -7.47096 q^{12} -23.7819 q^{14} +46.1972 q^{15} -37.8757 q^{16} +90.8352 q^{17} +21.1254 q^{18} +114.640 q^{19} -38.3486 q^{20} -30.3952 q^{21} +35.3661 q^{22} +75.7635 q^{23} -73.8709 q^{24} +112.132 q^{25} +27.0000 q^{27} +25.2313 q^{28} +214.817 q^{29} +108.437 q^{30} -284.476 q^{31} +108.084 q^{32} +45.2007 q^{33} +213.215 q^{34} -156.019 q^{35} -22.4129 q^{36} +358.878 q^{37} +269.091 q^{38} -379.181 q^{40} +313.154 q^{41} -71.3458 q^{42} -296.702 q^{43} -37.5214 q^{44} +138.592 q^{45} +177.837 q^{46} -316.691 q^{47} -113.627 q^{48} -240.348 q^{49} +263.203 q^{50} +272.506 q^{51} +163.911 q^{53} +63.3763 q^{54} +232.016 q^{55} +249.480 q^{56} +343.920 q^{57} +504.233 q^{58} +254.149 q^{59} -115.046 q^{60} -935.247 q^{61} -667.742 q^{62} -91.1856 q^{63} +556.709 q^{64} +106.098 q^{66} -240.494 q^{67} -226.209 q^{68} +227.291 q^{69} -366.220 q^{70} +947.455 q^{71} -221.613 q^{72} -430.712 q^{73} +842.384 q^{74} +336.395 q^{75} -285.490 q^{76} -152.654 q^{77} -496.620 q^{79} -583.251 q^{80} +81.0000 q^{81} +735.058 q^{82} +392.527 q^{83} +75.6938 q^{84} +1398.78 q^{85} -696.439 q^{86} +644.451 q^{87} -371.002 q^{88} +979.895 q^{89} +325.312 q^{90} -188.675 q^{92} -853.428 q^{93} -743.360 q^{94} +1765.35 q^{95} +324.253 q^{96} -553.356 q^{97} -564.162 q^{98} +135.602 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 6 q^{2} + 27 q^{3} + 44 q^{4} + 33 q^{5} + 18 q^{6} + 83 q^{7} + 87 q^{8} + 81 q^{9}+O(q^{10}) $ 9 * q + 6 * q^2 + 27 * q^3 + 44 * q^4 + 33 * q^5 + 18 * q^6 + 83 * q^7 + 87 * q^8 + 81 * q^9	\( 9 q + 6 q^{2} + 27 q^{3} + 44 q^{4} + 33 q^{5} + 18 q^{6} + 83 q^{7} + 87 q^{8} + 81 q^{9} - 54 q^{10} + 85 q^{11} + 132 q^{12} + 158 q^{14} + 99 q^{15} + 216 q^{16} + 178 q^{17} + 54 q^{18} + 352 q^{19} + 402 q^{20} + 249 q^{21} - 630 q^{22} + 150 q^{23} + 261 q^{24} - 20 q^{25} + 243 q^{27} + 940 q^{28} - 97 q^{29} - 162 q^{30} + 717 q^{31} + 707 q^{32} + 255 q^{33} + 632 q^{34} - 418 q^{35} + 396 q^{36} + 1108 q^{37} - 660 q^{38} - 1506 q^{40} + 334 q^{41} + 474 q^{42} + 242 q^{43} - 307 q^{44} + 297 q^{45} + 979 q^{46} - 184 q^{47} + 648 q^{48} - 38 q^{49} - 2031 q^{50} + 534 q^{51} - 151 q^{53} + 162 q^{54} + 2064 q^{55} + 2276 q^{56} + 1056 q^{57} + 1161 q^{58} + 537 q^{59} + 1206 q^{60} - 1340 q^{61} + 347 q^{62} + 747 q^{63} + 893 q^{64} - 1890 q^{66} + 2308 q^{67} + 2785 q^{68} + 450 q^{69} - 1420 q^{70} + 96 q^{71} + 783 q^{72} + 2505 q^{73} - 1191 q^{74} - 60 q^{75} + 2409 q^{76} - 2142 q^{77} - 1591 q^{79} - 2671 q^{80} + 729 q^{81} + 1517 q^{82} + 1539 q^{83} + 2820 q^{84} + 4296 q^{85} - 3763 q^{86} - 291 q^{87} - 3716 q^{88} - 592 q^{89} - 486 q^{90} + 515 q^{92} + 2151 q^{93} - 692 q^{94} + 4158 q^{95} + 2121 q^{96} + 1445 q^{97} + 1457 q^{98} + 765 q^{99}+O(q^{100}) \) 9 * q + 6 * q^2 + 27 * q^3 + 44 * q^4 + 33 * q^5 + 18 * q^6 + 83 * q^7 + 87 * q^8 + 81 * q^9 - 54 * q^10 + 85 * q^11 + 132 * q^12 + 158 * q^14 + 99 * q^15 + 216 * q^16 + 178 * q^17 + 54 * q^18 + 352 * q^19 + 402 * q^20 + 249 * q^21 - 630 * q^22 + 150 * q^23 + 261 * q^24 - 20 * q^25 + 243 * q^27 + 940 * q^28 - 97 * q^29 - 162 * q^30 + 717 * q^31 + 707 * q^32 + 255 * q^33 + 632 * q^34 - 418 * q^35 + 396 * q^36 + 1108 * q^37 - 660 * q^38 - 1506 * q^40 + 334 * q^41 + 474 * q^42 + 242 * q^43 - 307 * q^44 + 297 * q^45 + 979 * q^46 - 184 * q^47 + 648 * q^48 - 38 * q^49 - 2031 * q^50 + 534 * q^51 - 151 * q^53 + 162 * q^54 + 2064 * q^55 + 2276 * q^56 + 1056 * q^57 + 1161 * q^58 + 537 * q^59 + 1206 * q^60 - 1340 * q^61 + 347 * q^62 + 747 * q^63 + 893 * q^64 - 1890 * q^66 + 2308 * q^67 + 2785 * q^68 + 450 * q^69 - 1420 * q^70 + 96 * q^71 + 783 * q^72 + 2505 * q^73 - 1191 * q^74 - 60 * q^75 + 2409 * q^76 - 2142 * q^77 - 1591 * q^79 - 2671 * q^80 + 729 * q^81 + 1517 * q^82 + 1539 * q^83 + 2820 * q^84 + 4296 * q^85 - 3763 * q^86 - 291 * q^87 - 3716 * q^88 - 592 * q^89 - 486 * q^90 + 515 * q^92 + 2151 * q^93 - 692 * q^94 + 4158 * q^95 + 2121 * q^96 + 1445 * q^97 + 1457 * q^98 + 765 * q^99

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 $

$2$	2.34727	0.829886	0.414943	−	0.909848i	$-0.363802\pi$
$2$	2.34727	0.829886	0.414943	+	0.909848i	$0.363802\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−2.49032	−0.311290
$5$	15.3991	1.37734	0.688668	−	0.725077i	$-0.258196\pi$
$5$	15.3991	1.37734	0.688668	+	0.725077i	$0.258196\pi$
$6$	7.04181	0.479135
$7$	−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
$9$	9.00000	0.333333
$10$	36.1458	1.14303
$11$	15.0669	0.412985	0.206493	−	0.978448i	$-0.433795\pi$
$11$	15.0669	0.412985	0.206493	+	0.978448i	$0.433795\pi$
$12$	−7.47096	−0.179723
$13$	0	0
$13$	0	0
$14$	−23.7819	−0.453999
$15$	46.1972	0.795205
$16$	−37.8757	−0.591809
$17$	90.8352	1.29593	0.647964	−	0.761671i	$-0.275621\pi$
$17$	90.8352	1.29593	0.647964	+	0.761671i	$0.275621\pi$
$18$	21.1254	0.276629
$19$	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
$21$	−30.3952	−0.315847
$22$	35.3661	0.342731
$23$	75.7635	0.686860	0.343430	−	0.939178i	$-0.388411\pi$
$23$	75.7635	0.686860	0.343430	+	0.939178i	$0.388411\pi$
$24$	−73.8709	−0.628284
$25$	112.132	0.897052
$26$	0	0
$27$	27.0000	0.192450
$28$	25.2313	0.170295
$29$	214.817	1.37553	0.687767	−	0.725931i	$-0.258591\pi$
$29$	214.817	1.37553	0.687767	+	0.725931i	$0.258591\pi$
$30$	108.437	0.659929
$31$	−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
$33$	45.2007	0.238437
$34$	213.215	1.07547
$35$	−156.019	−0.753488
$36$	−22.4129	−0.103763
$37$	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	$-0.296589\pi$
$41$	313.154	1.19284	0.596421	+	0.802672i	$0.296589\pi$
$42$	−71.3458	−0.262116
$43$	−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
$45$	138.592	0.459112
$46$	177.837	0.570015
$47$	−316.691	−0.982854	−0.491427	−	0.870919i	$-0.663525\pi$
$47$	−316.691	−0.982854	−0.491427	+	0.870919i	$0.663525\pi$
$48$	−113.627	−0.341681
$49$	−240.348	−0.700723
$50$	263.203	0.744451
$51$	272.506	0.748204
$52$	0	0
$53$	163.911	0.424810	0.212405	−	0.977182i	$-0.431870\pi$
$53$	163.911	0.424810	0.212405	+	0.977182i	$0.431870\pi$
$54$	63.3763	0.159712
$55$	232.016	0.568819
$56$	249.480	0.595324
$57$	343.920	0.799180
$58$	504.233	1.14154
$59$	254.149	0.560803	0.280401	−	0.959883i	$-0.409532\pi$
$59$	254.149	0.560803	0.280401	+	0.959883i	$0.409532\pi$
$60$	−115.046	−0.247539
$61$	−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
$63$	−91.1856	−0.182354
$64$	556.709	1.08732
$65$	0	0
$66$	106.098	0.197876
$67$	−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
$69$	227.291	0.396559
$70$	−366.220	−0.625309
$71$	947.455	1.58369	0.791847	−	0.610720i	$-0.209120\pi$
$71$	947.455	1.58369	0.791847	+	0.610720i	$0.209120\pi$
$72$	−221.613	−0.362740
$73$	−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
$75$	336.395	0.517913
$76$	−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
$81$	81.0000	0.111111
$82$	735.058	0.989922
$83$	392.527	0.519102	0.259551	−	0.965729i	$-0.416425\pi$
$83$	392.527	0.519102	0.259551	+	0.965729i	$0.416425\pi$
$84$	75.6938	0.0983199
$85$	1398.78	1.78493
$86$	−696.439	−0.873244
$87$	644.451	0.794165
$88$	−371.002	−0.449419
$89$	979.895	1.16706	0.583532	−	0.812090i	$-0.301670\pi$
$89$	979.895	1.16706	0.583532	+	0.812090i	$0.301670\pi$
$90$	325.312	0.381010
$91$	0	0
$92$	−188.675	−0.213813
$93$	−853.428	−0.951574
$94$	−743.360	−0.815656
$95$	1765.35	1.90654
$96$	324.253	0.344729
$97$	−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
$99$	135.602	0.137662
$100$	−279.243	−0.279243
$101$	−763.951	−0.752633	−0.376317	−	0.926491i	$-0.622810\pi$
$101$	−763.951	−0.752633	−0.376317	+	0.926491i	$0.622810\pi$
$102$	639.645	0.620924
$103$	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
$105$	−468.058	−0.435027
$106$	384.744	0.352544
$107$	183.699	0.165971	0.0829855	−	0.996551i	$-0.473554\pi$
$107$	183.699	0.165971	0.0829855	+	0.996551i	$0.473554\pi$
$108$	−67.2386	−0.0599078
$109$	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
$111$	1076.63	0.920627
$112$	383.747	0.323756
$113$	417.288	0.347391	0.173695	−	0.984799i	$-0.444429\pi$
$113$	417.288	0.347391	0.173695	+	0.984799i	$0.444429\pi$
$114$	807.273	0.663228
$115$	1166.69	0.946037
$116$	−534.963	−0.428190
$117$	0	0
$118$	596.556	0.465402
$119$	−920.318	−0.708953
$120$	−1137.54	−0.865358
$121$	−1103.99	−0.829443
$122$	−2195.28	−1.62911
$123$	939.463	0.688687
$124$	708.436	0.513060
$125$	−198.162	−0.141793
$126$	−214.037	−0.151333
$127$	−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
$129$	−890.105	−0.607514
$130$	0	0
$131$	−1475.61	−0.984159	−0.492080	−	0.870550i	$-0.663763\pi$
$131$	−1475.61	−0.984159	−0.492080	+	0.870550i	$0.663763\pi$
$132$	−112.564	−0.0742231
$133$	−1161.50	−0.757255
$134$	−564.503	−0.363923
$135$	415.775	0.265068
$136$	−2236.69	−1.41026
$137$	−1900.71	−1.18532	−0.592660	−	0.805453i	$-0.701922\pi$
$137$	−1900.71	−1.18532	−0.592660	+	0.805453i	$0.701922\pi$
$138$	533.512	0.329099
$139$	−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
$141$	−950.073	−0.567451
$142$	2223.93	1.31428
$143$	0	0
$144$	−340.882	−0.197270
$145$	3307.98	1.89457
$146$	−1011.00	−0.573088
$147$	−721.044	−0.404563
$148$	−893.721	−0.496374
$149$	−1370.92	−0.753761	−0.376881	−	0.926262i	$-0.623003\pi$
$149$	−1370.92	−0.753761	−0.376881	+	0.926262i	$0.623003\pi$
$150$	789.609	0.429809
$151$	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
$153$	817.517	0.431976
$154$	−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
$159$	491.734	0.245264
$160$	1664.40	0.822389
$161$	−767.616	−0.375755
$162$	190.129	0.0922095
$163$	−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
$165$	696.049	0.328408
$166$	921.368	0.430795
$167$	−490.724	−0.227385	−0.113693	−	0.993516i	$-0.536268\pi$
$167$	−490.724	−0.227385	−0.113693	+	0.993516i	$0.536268\pi$
$168$	748.440	0.343711
$169$	0	0
$170$	3283.31	1.48129
$171$	1031.76	0.461407
$172$	738.882	0.327554
$173$	2008.13	0.882518	0.441259	−	0.897380i	$-0.354532\pi$
$173$	2008.13	0.882518	0.441259	+	0.897380i	$0.354532\pi$
$174$	1512.70	0.659066
$175$	−1136.09	−0.490743
$176$	−570.670	−0.244408
$177$	762.446	0.323779
$178$	2300.08	0.968529
$179$	−2152.60	−0.898843	−0.449421	−	0.893320i	$-0.648370\pi$
$179$	−2152.60	−0.898843	−0.449421	+	0.893320i	$0.648370\pi$
$180$	−345.138	−0.142917
$181$	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
$183$	−2805.74	−1.13337
$184$	−1865.57	−0.747455
$185$	5526.39	2.19626
$186$	−2003.23	−0.789697
$187$	1368.60	0.535200
$188$	788.662	0.305953
$189$	−273.557	−0.105282
$190$	4143.75	1.58221
$191$	4464.71	1.69139	0.845695	−	0.533667i	$-0.179186\pi$
$191$	4464.71	1.69139	0.845695	+	0.533667i	$0.179186\pi$
$192$	1670.13	0.627766
$193$	−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	$-0.319288\pi$
$197$	2973.59	1.07543	0.537714	+	0.843127i	$0.319288\pi$
$198$	318.295	0.114244
$199$	−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
$201$	−721.481	−0.253181
$202$	−1793.20	−0.624600
$203$	−2176.47	−0.752503
$204$	−678.626	−0.232909
$205$	4822.29	1.64294
$206$	428.419	0.144900
$207$	681.872	0.228953
$208$	0	0
$209$	1727.27	0.571663
$210$	−1098.66	−0.361022
$211$	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
$213$	2842.37	0.914346
$214$	431.192	0.137737
$215$	−4568.93	−1.44930
$216$	−664.838	−0.209428
$217$	2882.24	0.901654
$218$	4165.05	1.29400
$219$	−1292.14	−0.398696
$220$	−577.795	−0.177068
$221$	0	0
$222$	2527.15	0.764015
$223$	−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
$225$	1009.18	0.299017
$226$	979.487	0.288294
$227$	287.067	0.0839354	0.0419677	−	0.999119i	$-0.486637\pi$
$227$	287.067	0.0839354	0.0419677	+	0.999119i	$0.486637\pi$
$228$	−856.470	−0.248777
$229$	−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
$231$	−457.961	−0.130440
$232$	−5289.57	−1.49688
$233$	−970.620	−0.272908	−0.136454	−	0.990646i	$-0.543571\pi$
$233$	−970.620	−0.272908	−0.136454	+	0.990646i	$0.543571\pi$
$234$	0	0
$235$	−4876.75	−1.35372
$236$	−632.912	−0.174572
$237$	−1489.86	−0.408341
$238$	−2160.24	−0.588350
$239$	−5007.40	−1.35524	−0.677619	−	0.735413i	$-0.736988\pi$
$239$	−5007.40	−1.35524	−0.677619	+	0.735413i	$0.736988\pi$
$240$	−1749.75	−0.470609
$241$	−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
$243$	243.000	0.0641500
$244$	2329.07	0.611078
$245$	−3701.14	−0.965130
$246$	2205.17	0.571531
$247$	0	0
$248$	7004.83	1.79358
$249$	1177.58	0.299704
$250$	−465.141	−0.117672
$251$	−6087.80	−1.53091	−0.765455	−	0.643489i	$-0.777486\pi$
$251$	−6087.80	−1.53091	−0.765455	+	0.643489i	$0.777486\pi$
$252$	227.081	0.0567650
$253$	1141.52	0.283663
$254$	−4581.14	−1.13168
$255$	4196.34	1.03053
$256$	−3416.01	−0.833987
$257$	−5096.34	−1.23697	−0.618484	−	0.785797i	$-0.712253\pi$
$257$	−5096.34	−1.23697	−0.618484	+	0.785797i	$0.712253\pi$
$258$	−2089.32	−0.504167
$259$	−3636.06	−0.872330
$260$	0	0
$261$	1933.35	0.458511
$262$	−3463.66	−0.816739
$263$	3405.50	0.798450	0.399225	−	0.916853i	$-0.369279\pi$
$263$	3405.50	0.798450	0.399225	+	0.916853i	$0.369279\pi$
$264$	−1113.00	−0.259472
$265$	2524.08	0.585106
$266$	−2726.36	−0.628435
$267$	2939.68	0.673804
$268$	598.906	0.136507
$269$	−2720.44	−0.616611	−0.308306	−	0.951287i	$-0.599762\pi$
$269$	−2720.44	−0.616611	−0.308306	+	0.951287i	$0.599762\pi$
$270$	975.937	0.219976
$271$	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
$276$	−566.026	−0.123445
$277$	−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
$279$	−2560.28	−0.549392
$280$	3841.76	0.819961
$281$	−652.800	−0.138586	−0.0692932	−	0.997596i	$-0.522074\pi$
$281$	−652.800	−0.138586	−0.0692932	+	0.997596i	$0.522074\pi$
$282$	−2230.08	−0.470919
$283$	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
$285$	5296.05	1.10074
$286$	0	0
$287$	−3172.80	−0.652558
$288$	972.759	0.199029
$289$	3338.04	0.679430
$290$	7764.73	1.57228
$291$	−1660.07	−0.334416
$292$	1072.61	0.214965
$293$	−1912.72	−0.381372	−0.190686	−	0.981651i	$-0.561071\pi$
$293$	−1912.72	−0.381372	−0.190686	+	0.981651i	$0.561071\pi$
$294$	−1692.49	−0.335741
$295$	3913.66	0.772413
$296$	−8836.87	−1.73525
$297$	406.806	0.0794791
$298$	−3217.93	−0.625535
$299$	0	0
$300$	−837.730	−0.161221
$301$	3006.10	0.575644
$302$	2764.06	0.526669
$303$	−2291.85	−0.434533
$304$	−4342.07	−0.819194
$305$	−14401.9	−2.70378
$306$	1918.93	0.358491
$307$	−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
$309$	547.553	0.100807
$310$	−10282.6	−1.88391
$311$	1893.76	0.345291	0.172645	−	0.984984i	$-0.444769\pi$
$311$	1893.76	0.345291	0.172645	+	0.984984i	$0.444769\pi$
$312$	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
$315$	−1404.17	−0.251163
$316$	1236.74	0.220165
$317$	7594.53	1.34559	0.672794	−	0.739830i	$-0.265094\pi$
$317$	7594.53	1.34559	0.672794	+	0.739830i	$0.265094\pi$
$318$	1154.23	0.203541
$319$	3236.62	0.568076
$320$	8572.81	1.49761
$321$	551.098	0.0958234
$322$	−1801.80	−0.311834
$323$	10413.3	1.79385
$324$	−201.716	−0.0345878
$325$	0	0
$326$	−312.492	−0.0530901
$327$	5323.27	0.900236
$328$	−7710.99	−1.29807
$329$	3208.63	0.537682
$330$	1633.82	0.272541
$331$	−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
$333$	3229.90	0.531524
$334$	−1151.86	−0.188704
$335$	−3703.38	−0.603992
$336$	1151.24	0.186921
$337$	−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
$339$	1251.86	0.200566
$340$	−3483.41	−0.555630
$341$	−4286.17	−0.680672
$342$	2421.82	0.382915
$343$	5910.33	0.930401
$344$	7305.87	1.14508
$345$	3500.06	0.546195
$346$	4713.63	0.732389
$347$	−2506.81	−0.387817	−0.193908	−	0.981020i	$-0.562116\pi$
$347$	−2506.81	−0.387817	−0.193908	+	0.981020i	$0.562116\pi$
$348$	−1604.89	−0.247216
$349$	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	$-0.737138\pi$
$353$	−8992.88	−1.35593	−0.677964	+	0.735095i	$0.737138\pi$
$354$	1789.67	0.268700
$355$	14589.9	2.18128
$356$	−2440.25	−0.363295
$357$	−2760.96	−0.409314
$358$	−5052.74	−0.745937
$359$	−4566.64	−0.671359	−0.335679	−	0.941976i	$-0.608966\pi$
$359$	−4566.64	−0.671359	−0.335679	+	0.941976i	$0.608966\pi$
$360$	−3412.63	−0.499615
$361$	6283.31	0.916068
$362$	1959.24	0.284463
$363$	−3311.97	−0.478879
$364$	0	0
$365$	−6632.57	−0.951136
$366$	−6585.84	−0.940566
$367$	−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
$369$	2818.39	0.397614
$370$	12971.9	1.82264
$371$	−1660.71	−0.232398
$372$	2125.31	0.296215
$373$	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
$375$	−594.487	−0.0818645
$376$	7798.08	1.06956
$377$	0	0
$378$	−642.112	−0.0873722
$379$	−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
$381$	−5855.06	−0.787307
$382$	10479.9	1.40366
$383$	10258.0	1.36856	0.684282	−	0.729217i	$-0.260116\pi$
$383$	10258.0	1.36856	0.684282	+	0.729217i	$0.260116\pi$
$384$	1326.22	0.176245
$385$	−2350.73	−0.311180
$386$	−7745.63	−1.02135
$387$	−2670.31	−0.350749
$388$	1378.03	0.180307
$389$	−4771.57	−0.621924	−0.310962	−	0.950422i	$-0.600651\pi$
$389$	−4771.57	−0.621924	−0.310962	+	0.950422i	$0.600651\pi$
$390$	0	0
$391$	6882.00	0.890122
$392$	5918.24	0.762541
$393$	−4426.84	−0.568205
$394$	6979.82	0.892483
$395$	−7647.50	−0.974145
$396$	−337.693	−0.0428528
$397$	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
$399$	−3484.50	−0.437201
$400$	−4247.07	−0.530883
$401$	7534.63	0.938308	0.469154	−	0.883116i	$-0.344559\pi$
$401$	7534.63	0.938308	0.469154	+	0.883116i	$0.344559\pi$
$402$	−1693.51	−0.210111
$403$	0	0
$404$	1902.48	0.234287
$405$	1247.33	0.153037
$406$	−5108.76	−0.624491
$407$	5407.18	0.658535
$408$	−6710.08	−0.814211
$409$	−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
$411$	−5702.14	−0.684345
$412$	−454.528	−0.0543519
$413$	−2574.97	−0.306794
$414$	1600.54	0.190005
$415$	6044.56	0.714978
$416$	0	0
$417$	−6980.28	−0.819726
$418$	4054.36	0.474415
$419$	8080.97	0.942199	0.471100	−	0.882080i	$-0.343857\pi$
$419$	8080.97	0.942199	0.471100	+	0.882080i	$0.343857\pi$
$420$	1165.61	0.135419
$421$	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
$423$	−2850.22	−0.327618
$424$	−4036.09	−0.462287
$425$	10185.5	1.16252
$426$	6671.80	0.758803
$427$	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	$-0.460027\pi$
$431$	2241.39	0.250496	0.125248	+	0.992125i	$0.460027\pi$
$432$	−1022.65	−0.113894
$433$	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
$435$	9923.94	1.09383
$436$	−4418.88	−0.485380
$437$	8685.52	0.950766
$438$	−3033.00	−0.330872
$439$	−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
$441$	−2163.13	−0.233574
$442$	0	0
$443$	−2537.44	−0.272138	−0.136069	−	0.990699i	$-0.543447\pi$
$443$	−2537.44	−0.272138	−0.136069	+	0.990699i	$0.543447\pi$
$444$	−2681.16	−0.286582
$445$	15089.5	1.60744
$446$	−1608.48	−0.170771
$447$	−4112.77	−0.435184
$448$	−5640.43	−0.594833
$449$	7790.61	0.818846	0.409423	−	0.912345i	$-0.365730\pi$
$449$	7790.61	0.818846	0.409423	+	0.912345i	$0.365730\pi$
$450$	2368.83	0.248150
$451$	4718.26	0.492626
$452$	−1039.18	−0.108139
$453$	3532.70	0.366403
$454$	673.825	0.0696568
$455$	0	0
$456$	−8468.55	−0.869685
$457$	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
$459$	2452.55	0.249401
$460$	−2905.43	−0.294492
$461$	4482.57	0.452872	0.226436	−	0.974026i	$-0.427293\pi$
$461$	4482.57	0.452872	0.226436	+	0.974026i	$0.427293\pi$
$462$	−1074.96	−0.108250
$463$	−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
$465$	−13142.0	−1.31064
$466$	−2278.31	−0.226482
$467$	14778.8	1.46441	0.732207	−	0.681082i	$-0.238490\pi$
$467$	14778.8	1.46441	0.732207	+	0.681082i	$0.238490\pi$
$468$	0	0
$469$	2436.62	0.239899
$470$	−11447.0	−1.12343
$471$	4864.71	0.475911
$472$	−6258.06	−0.610277
$473$	−4470.37	−0.434562
$474$	−3497.11	−0.338877
$475$	12854.8	1.24172
$476$	2291.89	0.220690
$477$	1475.20	0.141603
$478$	−11753.7	−1.12469
$479$	11171.2	1.06560	0.532801	−	0.846240i	$-0.321139\pi$
$479$	11171.2	1.06560	0.532801	+	0.846240i	$0.321139\pi$
$480$	4993.20	0.474807
$481$	0	0
$482$	−1267.74	−0.119801
$483$	−2302.85	−0.216942
$484$	2749.29	0.258197
$485$	−8521.18	−0.797787
$486$	570.387	0.0532372
$487$	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
$489$	−399.390	−0.0369347
$490$	−8687.57	−0.800948
$491$	−1035.04	−0.0951338	−0.0475669	−	0.998868i	$-0.515147\pi$
$491$	−1035.04	−0.0951338	−0.0475669	+	0.998868i	$0.515147\pi$
$492$	−2339.56	−0.214381
$493$	19512.9	1.78259
$494$	0	0
$495$	2088.15	0.189606
$496$	10774.7	0.975404
$497$	−9599.37	−0.866379
$498$	2764.10	0.248720
$499$	−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
$501$	−1472.17	−0.131281
$502$	−14289.7	−1.27048
$503$	−13552.0	−1.20130	−0.600651	−	0.799511i	$-0.705092\pi$
$503$	−13552.0	−1.20130	−0.600651	+	0.799511i	$0.705092\pi$
$504$	2245.32	0.198441
$505$	−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	$-0.429058\pi$
$509$	5076.46	0.442064	0.221032	+	0.975267i	$0.429058\pi$
$510$	9849.94	0.855221
$511$	4363.86	0.377781
$512$	−11554.9	−0.997380
$513$	3095.28	0.266393
$514$	−11962.5	−1.02654
$515$	2810.60	0.240486
$516$	2216.65	0.189113
$517$	−4771.55	−0.405904
$518$	−8534.81	−0.723934
$519$	6024.40	0.509522
$520$	0	0
$521$	−8493.73	−0.714236	−0.357118	−	0.934059i	$-0.616241\pi$
$521$	−8493.73	−0.714236	−0.357118	+	0.934059i	$0.616241\pi$
$522$	4538.10	0.380512
$523$	−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
$525$	−3408.26	−0.283331
$526$	7993.64	0.662622
$527$	−25840.4	−2.13592
$528$	−1712.01	−0.141109
$529$	−6426.89	−0.528223
$530$	5924.71	0.485571
$531$	2287.34	0.186934
$532$	2892.51	0.235726
$533$	0	0
$534$	6900.24	0.559181
$535$	2828.80	0.228598
$536$	5921.82	0.477208
$537$	−6457.80	−0.518947
$538$	−6385.62	−0.511717
$539$	−3621.30	−0.289388
$540$	−1035.41	−0.0825131
$541$	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
$543$	2504.07	0.197900
$544$	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
$549$	−8417.23	−0.654351
$550$	3965.65	0.307447
$551$	24626.6	1.90404
$552$	−5596.72	−0.431544
$553$	5031.63	0.386920
$554$	−15407.4	−1.18158
$555$	16579.2	1.26801
$556$	5794.38	0.441972
$557$	−8550.51	−0.650443	−0.325221	−	0.945638i	$-0.605439\pi$
$557$	−8550.51	−0.650443	−0.325221	+	0.945638i	$0.605439\pi$
$558$	−6009.68	−0.455932
$559$	0	0
$560$	5909.35	0.445921
$561$	4105.81	0.308998
$562$	−1532.30	−0.115011
$563$	−6569.35	−0.491768	−0.245884	−	0.969299i	$-0.579078\pi$
$563$	−6569.35	−0.491768	−0.245884	+	0.969299i	$0.579078\pi$
$564$	2365.99	0.176642
$565$	6425.85	0.478473
$566$	14108.5	1.04775
$567$	−820.671	−0.0607847
$568$	−23329.8	−1.72341
$569$	23766.1	1.75102	0.875508	−	0.483204i	$-0.160527\pi$
$569$	23766.1	1.75102	0.875508	+	0.483204i	$0.160527\pi$
$570$	12431.3	0.913488
$571$	−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
$573$	13394.1	0.976524
$574$	−7447.41	−0.541549
$575$	8495.48	0.616150
$576$	5010.38	0.362441
$577$	−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
$579$	−9899.53	−0.710553
$580$	−8237.93	−0.589761
$581$	−3976.98	−0.283981
$582$	−3896.63	−0.277527
$583$	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	$-0.575928\pi$
$587$	−6720.67	−0.472558	−0.236279	+	0.971685i	$0.575928\pi$
$588$	1795.63	0.125936
$589$	−32612.3	−2.28144
$590$	9186.41	0.641014
$591$	8920.77	0.620899
$592$	−13592.8	−0.943681
$593$	26349.8	1.82471	0.912357	−	0.409395i	$-0.134260\pi$
$593$	26349.8	1.82471	0.912357	+	0.409395i	$0.134260\pi$
$594$	954.884	0.0659586
$595$	−14172.1	−0.976466
$596$	3414.04	0.234638
$597$	−16292.4	−1.11693
$598$	0	0
$599$	−6136.81	−0.418603	−0.209302	−	0.977851i	$-0.567119\pi$
$599$	−6136.81	−0.418603	−0.209302	+	0.977851i	$0.567119\pi$
$600$	−8283.25	−0.563604
$601$	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
$603$	−2164.44	−0.146174
$604$	−2932.51	−0.197553
$605$	−17000.4	−1.14242
$606$	−5379.60	−0.360613
$607$	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
$609$	−6529.40	−0.434458
$610$	−33805.3	−2.24383
$611$	0	0
$612$	−2035.88	−0.134470
$613$	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
$615$	14466.9	0.948553
$616$	3758.89	0.245860
$617$	16549.5	1.07983	0.539917	−	0.841718i	$-0.318456\pi$
$617$	16549.5	1.07983	0.539917	+	0.841718i	$0.318456\pi$
$618$	1285.26	0.0836579
$619$	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
$621$	2045.62	0.132186
$622$	4445.17	0.286552
$623$	−9928.03	−0.638456
$624$	0	0
$625$	−17068.0	−1.09235
$626$	10736.9	0.685513
$627$	5181.80	0.330050
$628$	−4038.23	−0.256597
$629$	32598.8	2.06645
$630$	−3295.98	−0.208436
$631$	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
$633$	3684.76	0.231368
$634$	17826.4	1.11668
$635$	−30054.2	−1.87821
$636$	−1224.58	−0.0763484
$637$	0	0
$638$	7597.23	0.471438
$639$	8527.10	0.527898
$640$	6807.51	0.420454
$641$	11067.7	0.681979	0.340989	−	0.940067i	$-0.389238\pi$
$641$	11067.7	0.681979	0.340989	+	0.940067i	$0.389238\pi$
$642$	1293.58	0.0795224
$643$	−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
$645$	−13706.8	−0.836751
$646$	24442.9	1.48869
$647$	−4447.03	−0.270217	−0.135109	−	0.990831i	$-0.543138\pi$
$647$	−4447.03	−0.270217	−0.135109	+	0.990831i	$0.543138\pi$
$648$	−1994.51	−0.120913
$649$	3829.23	0.231603
$650$	0	0
$651$	8646.71	0.520570
$652$	331.537	0.0199141
$653$	−19426.2	−1.16418	−0.582088	−	0.813126i	$-0.697764\pi$
$653$	−19426.2	−1.16418	−0.582088	+	0.813126i	$0.697764\pi$
$654$	12495.1	0.747093
$655$	−22723.1	−1.35552
$656$	−11861.0	−0.705934
$657$	−3876.41	−0.230187
$658$	7531.52	0.446215
$659$	−14099.4	−0.833438	−0.416719	−	0.909035i	$-0.636820\pi$
$659$	−14099.4	−0.833438	−0.416719	+	0.909035i	$0.636820\pi$
$660$	−1733.38	−0.102230
$661$	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
$666$	7581.45	0.441104
$667$	16275.3	0.944800
$668$	1222.06	0.0707828
$669$	−2055.77	−0.118805
$670$	−8692.83	−0.501244
$671$	−14091.3	−0.810712
$672$	−3285.25	−0.188588
$673$	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
$675$	3027.55	0.172638
$676$	0	0
$677$	−10255.2	−0.582186	−0.291093	−	0.956695i	$-0.594019\pi$
$677$	−10255.2	−0.582186	−0.291093	+	0.956695i	$0.594019\pi$
$678$	2938.46	0.166447
$679$	5606.46	0.316872
$680$	−34443.0	−1.94239
$681$	861.202	0.0484601
$682$	−10060.8	−0.564880
$683$	10605.7	0.594169	0.297085	−	0.954851i	$-0.403986\pi$
$683$	10605.7	0.594169	0.297085	+	0.954851i	$0.403986\pi$
$684$	−2569.41	−0.143631
$685$	−29269.2	−1.63258
$686$	13873.1	0.772127
$687$	−6906.69	−0.383561
$688$	11237.8	0.622728
$689$	0	0
$690$	8215.60	0.453279
$691$	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
$693$	−1373.88	−0.0753096
$694$	−5884.15	−0.321844
$695$	−35829.9	−1.95555
$696$	−15868.7	−0.864227
$697$	28445.4	1.54584
$698$	17690.2	0.959287
$699$	−2911.86	−0.157563
$700$	2829.22	0.152764
$701$	−24016.9	−1.29402	−0.647008	−	0.762483i	$-0.723980\pi$
$701$	−24016.9	−1.29402	−0.647008	+	0.762483i	$0.723980\pi$
$702$	0	0
$703$	41141.7	2.20724
$704$	8387.88	0.449048
$705$	−14630.2	−0.781570
$706$	−21108.7	−1.12527
$707$	7740.15	0.411737
$708$	−1898.74	−0.100789
$709$	−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
$711$	−4469.58	−0.235756
$712$	−24128.6	−1.27002
$713$	−21552.9	−1.13207
$714$	−6480.71	−0.339684
$715$	0	0
$716$	5360.66	0.279801
$717$	−15022.2	−0.782447
$718$	−10719.1	−0.557151
$719$	10951.7	0.568050	0.284025	−	0.958817i	$-0.408330\pi$
$719$	10951.7	0.568050	0.284025	+	0.958817i	$0.408330\pi$
$720$	−5249.26	−0.271706
$721$	−1849.22	−0.0955182
$722$	14748.6	0.760231
$723$	−1620.28	−0.0833454
$724$	−2078.65	−0.106702
$725$	24087.7	1.23393
$726$	−7774.08	−0.397415
$727$	−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
$729$	729.000	0.0370370
$730$	−15568.4	−0.789334
$731$	−26951.0	−1.36364
$732$	6987.20	0.352806
$733$	−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
$735$	−11103.4	−0.557218
$736$	8188.85	0.410116
$737$	−3623.49	−0.181103
$738$	6615.52	0.329974
$739$	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	$-0.392095\pi$
$743$	13469.7	0.665079	0.332539	+	0.943089i	$0.392095\pi$
$744$	21014.5	1.03552
$745$	−21111.0	−1.03818
$746$	17954.0	0.881158
$747$	3532.75	0.173034
$748$	−3408.26	−0.166602
$749$	−1861.19	−0.0907965
$750$	−1395.42	−0.0679381
$751$	−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
$753$	−18263.4	−0.883871
$754$	0	0
$755$	18133.4	0.874096
$756$	681.244	0.0327733
$757$	−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
$759$	3424.56	0.163773
$760$	−43469.3	−2.07473
$761$	18238.2	0.868772	0.434386	−	0.900727i	$-0.356965\pi$
$761$	18238.2	0.868772	0.434386	+	0.900727i	$0.356965\pi$
$762$	−13743.4	−0.653375
$763$	−17978.0	−0.853009
$764$	−11118.6	−0.526513
$765$	12589.0	0.594976
$766$	24078.4	1.13575
$767$	0	0
$768$	−10248.0	−0.481502
$769$	−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
$771$	−15289.0	−0.714164
$772$	8217.67	0.383109
$773$	−13795.6	−0.641904	−0.320952	−	0.947095i	$-0.604003\pi$
$773$	−13795.6	−0.641904	−0.320952	+	0.947095i	$0.604003\pi$
$774$	−6267.95	−0.291081
$775$	−31898.7	−1.47850
$776$	13625.6	0.630325
$777$	−10908.2	−0.503640
$778$	−11200.2	−0.516126
$779$	35900.0	1.65116
$780$	0	0
$781$	14275.2	0.654043
$782$	16153.9	0.738699
$783$	5800.05	0.264722
$784$	9103.36	0.414694
$785$	24970.7	1.13534
$786$	−10391.0	−0.471545
$787$	−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
$789$	10216.5	0.460985
$790$	−17950.7	−0.808429
$791$	−4227.85	−0.190044
$792$	−3339.01	−0.149806
$793$	0	0
$794$	5378.11	0.240380
$795$	7572.25	0.337811
$796$	13524.5	0.602214
$797$	−31576.4	−1.40338	−0.701690	−	0.712482i	$-0.747571\pi$
$797$	−31576.4	−1.40338	−0.701690	+	0.712482i	$0.747571\pi$
$798$	−8179.07	−0.362827
$799$	−28766.7	−1.27371
$800$	12119.7	0.535619
$801$	8819.05	0.389021
$802$	17685.8	0.778688
$803$	−6489.50	−0.285192
$804$	1796.72	0.0788126
$805$	−11820.6	−0.517541
$806$	0	0
$807$	−8161.33	−0.356001
$808$	18811.2	0.819031
$809$	31130.1	1.35287	0.676437	−	0.736501i	$-0.263523\pi$
$809$	31130.1	1.35287	0.676437	+	0.736501i	$0.263523\pi$
$810$	2927.81	0.127003
$811$	−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
$813$	20862.8	0.899988
$814$	12692.1	0.546509
$815$	−2050.08	−0.0881119
$816$	−10321.4	−0.442794
$817$	−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	$-0.787633\pi$
$821$	−36960.1	−1.57115	−0.785576	+	0.618765i	$0.787633\pi$
$822$	−13384.5	−0.567928
$823$	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
$825$	5068.42	0.213891
$826$	−6044.15	−0.254604
$827$	−11533.4	−0.484952	−0.242476	−	0.970157i	$-0.577960\pi$
$827$	−11533.4	−0.484952	−0.242476	+	0.970157i	$0.577960\pi$
$828$	−1698.08	−0.0712709
$829$	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
$831$	−19691.9	−0.822026
$832$	0	0
$833$	−21832.1	−0.908087
$834$	−16384.6	−0.680279
$835$	−7556.69	−0.313186
$836$	−4301.45	−0.177953
$837$	−7680.85	−0.317191
$838$	18968.2	0.781917
$839$	−1855.42	−0.0763484	−0.0381742	−	0.999271i	$-0.512154\pi$
$839$	−1855.42	−0.0763484	−0.0381742	+	0.999271i	$0.512154\pi$
$840$	11525.3	0.473405
$841$	21757.3	0.892094
$842$	18949.8	0.775599
$843$	−1958.40	−0.0800129
$844$	−3058.75	−0.124747
$845$	0	0
$846$	−6690.24	−0.271885
$847$	11185.3	0.453757
$848$	−6208.26	−0.251406
$849$	18031.9	0.728918
$850$	23908.1	0.964755
$851$	27189.9	1.09525
$852$	−7078.40	−0.284627
$853$	−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
$855$	15888.1	0.635512
$856$	−4523.35	−0.180613
$857$	449.310	0.0179091	0.00895457	−	0.999960i	$-0.497150\pi$
$857$	449.310	0.0179091	0.00895457	+	0.999960i	$0.497150\pi$
$858$	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
$861$	−9518.39	−0.376755
$862$	5261.14	0.207883
$863$	−25097.5	−0.989953	−0.494976	−	0.868906i	$-0.664823\pi$
$863$	−25097.5	−0.989953	−0.494976	+	0.868906i	$0.664823\pi$
$864$	2918.28	0.114910
$865$	30923.4	1.21552
$866$	24031.0	0.942965
$867$	10014.1	0.392269
$868$	−7177.69	−0.280676
$869$	−7482.53	−0.292091
$870$	23294.2	0.907755
$871$	0	0
$872$	−43692.7	−1.69681
$873$	−4980.21	−0.193075
$874$	20387.3	0.789027
$875$	2007.73	0.0775698
$876$	3217.83	0.124110
$877$	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
$879$	−5738.15	−0.220185
$880$	−8787.79	−0.336632
$881$	48638.7	1.86002	0.930010	−	0.367534i	$-0.119798\pi$
$881$	48638.7	1.86002	0.930010	+	0.367534i	$0.119798\pi$
$882$	−5077.46	−0.193840
$883$	−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
$885$	11741.0	0.445953
$886$	−5956.05	−0.225843
$887$	42359.7	1.60350	0.801748	−	0.597663i	$-0.203904\pi$
$887$	42359.7	1.60350	0.801748	+	0.597663i	$0.203904\pi$
$888$	−26510.6	−1.00184
$889$	19774.0	0.746005
$890$	35419.1	1.33399
$891$	1220.42	0.0458873
$892$	1706.51	0.0640561
$893$	−36305.4	−1.36049
$894$	−9653.78	−0.361153
$895$	−33148.1	−1.23801
$896$	−4478.96	−0.167000
$897$	0	0
$898$	18286.7	0.679548
$899$	−61110.3	−2.26712
$900$	−2513.19	−0.0930811
$901$	14888.9	0.550524
$902$	11075.0	0.408823
$903$	9018.31	0.332348
$904$	−10275.1	−0.378038
$905$	12853.5	0.472114
$906$	8292.19	0.304072
$907$	−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
$909$	−6875.56	−0.250878
$910$	0	0
$911$	2471.71	0.0898919	0.0449459	−	0.998989i	$-0.485688\pi$
$911$	2471.71	0.0898919	0.0449459	+	0.998989i	$0.485688\pi$
$912$	−13026.2	−0.472962
$913$	5914.17	0.214382
$914$	15491.3	0.560620
$915$	−43205.8	−1.56103
$916$	5733.29	0.206805
$917$	14950.5	0.538396
$918$	5756.80	0.206975
$919$	−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
$921$	−5949.22	−0.212849
$922$	10521.8	0.375832
$923$	0	0
$924$	1140.47	0.0406047
$925$	40241.5	1.43041
$926$	−8933.38	−0.317029
$927$	1642.66	0.0582007
$928$	23218.3	0.821314
$929$	−53249.1	−1.88057	−0.940283	−	0.340394i	$-0.889439\pi$
$929$	−53249.1	−1.88057	−0.940283	+	0.340394i	$0.889439\pi$
$930$	−30847.8	−1.08768
$931$	−27553.5	−0.969955
$932$	2417.16	0.0849534
$933$	5681.28	0.199354
$934$	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
$939$	13722.6	0.476911
$940$	12144.7	0.421399
$941$	−8005.71	−0.277342	−0.138671	−	0.990339i	$-0.544283\pi$
$941$	−8005.71	−0.277342	−0.138671	+	0.990339i	$0.544283\pi$
$942$	11418.8	0.394952
$943$	23725.7	0.819315
$944$	−9626.07	−0.331888
$945$	−4212.52	−0.145009
$946$	−10493.2	−0.360637
$947$	−17468.6	−0.599424	−0.299712	−	0.954030i	$-0.596891\pi$
$947$	−17468.6	−0.599424	−0.299712	+	0.954030i	$0.596891\pi$
$948$	3710.23	0.127113
$949$	0	0
$950$	30173.6	1.03048
$951$	22783.6	0.776875
$952$	22661.6	0.771498
$953$	30681.4	1.04288	0.521442	−	0.853287i	$-0.325394\pi$
$953$	30681.4	1.04288	0.521442	+	0.853287i	$0.325394\pi$
$954$	3462.70	0.117515
$955$	68752.5	2.32961
$956$	12470.0	0.421872
$957$	9709.87	0.327979
$958$	26221.8	0.884328
$959$	19257.5	0.648444
$960$	25718.4	0.864644
$961$	51135.6	1.71648
$962$	0	0
$963$	1653.29	0.0553237
$964$	1345.00	0.0449373
$965$	−50814.5	−1.69511
$966$	−5405.41	−0.180037
$967$	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
$969$	31240.0	1.03568
$970$	−20001.5	−0.662072
$971$	7185.98	0.237496	0.118748	−	0.992924i	$-0.462112\pi$
$971$	7185.98	0.237496	0.118748	+	0.992924i	$0.462112\pi$
$972$	−605.148	−0.0199693
$973$	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	$-0.317283\pi$
$977$	33165.3	1.08603	0.543015	+	0.839723i	$0.317283\pi$
$978$	−937.477	−0.0306516
$979$	14764.0	0.481980
$980$	9217.01	0.300435
$981$	15969.8	0.519752
$982$	−2429.52	−0.0789502
$983$	−11658.7	−0.378287	−0.189143	−	0.981949i	$-0.560571\pi$
$983$	−11658.7	−0.378287	−0.189143	+	0.981949i	$0.560571\pi$
$984$	−23133.0	−0.749444
$985$	45790.5	1.48123
$986$	45802.1	1.47935
$987$	9625.89	0.310431
$988$	0	0
$989$	−22479.2	−0.722746
$990$	4901.45	0.157352
$991$	−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
$993$	−5215.75	−0.166684
$994$	−22532.3	−0.718996
$995$	−83629.5	−2.66456
$996$	−2932.56	−0.0932948
$997$	−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
$999$	9689.70	0.306876

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.o.1.4	✓	9	13.12	even	2
507.4.a.p.1.6	yes	9	1.1	even	1	trivial
507.4.b.k.337.7		18	13.5	odd	4
507.4.b.k.337.12		18	13.8	odd	4
1521.4.a.bf.1.4		9	3.2	odd	2
1521.4.a.bi.1.6		9	39.38	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:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.a (trivial)

\(f(q)\)	\(=\)	\(q+2.34727 q^{2} +3.00000 q^{3} -2.49032 q^{4} +15.3991 q^{5} +7.04181 q^{6} -10.1317 q^{7} -24.6236 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.34727 q^{2} +3.00000 q^{3} -2.49032 q^{4} +15.3991 q^{5} +7.04181 q^{6} -10.1317 q^{7} -24.6236 q^{8} +9.00000 q^{9} +36.1458 q^{10} +15.0669 q^{11} -7.47096 q^{12} -23.7819 q^{14} +46.1972 q^{15} -37.8757 q^{16} +90.8352 q^{17} +21.1254 q^{18} +114.640 q^{19} -38.3486 q^{20} -30.3952 q^{21} +35.3661 q^{22} +75.7635 q^{23} -73.8709 q^{24} +112.132 q^{25} +27.0000 q^{27} +25.2313 q^{28} +214.817 q^{29} +108.437 q^{30} -284.476 q^{31} +108.084 q^{32} +45.2007 q^{33} +213.215 q^{34} -156.019 q^{35} -22.4129 q^{36} +358.878 q^{37} +269.091 q^{38} -379.181 q^{40} +313.154 q^{41} -71.3458 q^{42} -296.702 q^{43} -37.5214 q^{44} +138.592 q^{45} +177.837 q^{46} -316.691 q^{47} -113.627 q^{48} -240.348 q^{49} +263.203 q^{50} +272.506 q^{51} +163.911 q^{53} +63.3763 q^{54} +232.016 q^{55} +249.480 q^{56} +343.920 q^{57} +504.233 q^{58} +254.149 q^{59} -115.046 q^{60} -935.247 q^{61} -667.742 q^{62} -91.1856 q^{63} +556.709 q^{64} +106.098 q^{66} -240.494 q^{67} -226.209 q^{68} +227.291 q^{69} -366.220 q^{70} +947.455 q^{71} -221.613 q^{72} -430.712 q^{73} +842.384 q^{74} +336.395 q^{75} -285.490 q^{76} -152.654 q^{77} -496.620 q^{79} -583.251 q^{80} +81.0000 q^{81} +735.058 q^{82} +392.527 q^{83} +75.6938 q^{84} +1398.78 q^{85} -696.439 q^{86} +644.451 q^{87} -371.002 q^{88} +979.895 q^{89} +325.312 q^{90} -188.675 q^{92} -853.428 q^{93} -743.360 q^{94} +1765.35 q^{95} +324.253 q^{96} -553.356 q^{97} -564.162 q^{98} +135.602 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 6 q^{2} + 27 q^{3} + 44 q^{4} + 33 q^{5} + 18 q^{6} + 83 q^{7} + 87 q^{8} + 81 q^{9}+O(q^{10}) \) 9 * q + 6 * q^2 + 27 * q^3 + 44 * q^4 + 33 * q^5 + 18 * q^6 + 83 * q^7 + 87 * q^8 + 81 * q^9	\( 9 q + 6 q^{2} + 27 q^{3} + 44 q^{4} + 33 q^{5} + 18 q^{6} + 83 q^{7} + 87 q^{8} + 81 q^{9} - 54 q^{10} + 85 q^{11} + 132 q^{12} + 158 q^{14} + 99 q^{15} + 216 q^{16} + 178 q^{17} + 54 q^{18} + 352 q^{19} + 402 q^{20} + 249 q^{21} - 630 q^{22} + 150 q^{23} + 261 q^{24} - 20 q^{25} + 243 q^{27} + 940 q^{28} - 97 q^{29} - 162 q^{30} + 717 q^{31} + 707 q^{32} + 255 q^{33} + 632 q^{34} - 418 q^{35} + 396 q^{36} + 1108 q^{37} - 660 q^{38} - 1506 q^{40} + 334 q^{41} + 474 q^{42} + 242 q^{43} - 307 q^{44} + 297 q^{45} + 979 q^{46} - 184 q^{47} + 648 q^{48} - 38 q^{49} - 2031 q^{50} + 534 q^{51} - 151 q^{53} + 162 q^{54} + 2064 q^{55} + 2276 q^{56} + 1056 q^{57} + 1161 q^{58} + 537 q^{59} + 1206 q^{60} - 1340 q^{61} + 347 q^{62} + 747 q^{63} + 893 q^{64} - 1890 q^{66} + 2308 q^{67} + 2785 q^{68} + 450 q^{69} - 1420 q^{70} + 96 q^{71} + 783 q^{72} + 2505 q^{73} - 1191 q^{74} - 60 q^{75} + 2409 q^{76} - 2142 q^{77} - 1591 q^{79} - 2671 q^{80} + 729 q^{81} + 1517 q^{82} + 1539 q^{83} + 2820 q^{84} + 4296 q^{85} - 3763 q^{86} - 291 q^{87} - 3716 q^{88} - 592 q^{89} - 486 q^{90} + 515 q^{92} + 2151 q^{93} - 692 q^{94} + 4158 q^{95} + 2121 q^{96} + 1445 q^{97} + 1457 q^{98} + 765 q^{99}+O(q^{100}) \) 9 * q + 6 * q^2 + 27 * q^3 + 44 * q^4 + 33 * q^5 + 18 * q^6 + 83 * q^7 + 87 * q^8 + 81 * q^9 - 54 * q^10 + 85 * q^11 + 132 * q^12 + 158 * q^14 + 99 * q^15 + 216 * q^16 + 178 * q^17 + 54 * q^18 + 352 * q^19 + 402 * q^20 + 249 * q^21 - 630 * q^22 + 150 * q^23 + 261 * q^24 - 20 * q^25 + 243 * q^27 + 940 * q^28 - 97 * q^29 - 162 * q^30 + 717 * q^31 + 707 * q^32 + 255 * q^33 + 632 * q^34 - 418 * q^35 + 396 * q^36 + 1108 * q^37 - 660 * q^38 - 1506 * q^40 + 334 * q^41 + 474 * q^42 + 242 * q^43 - 307 * q^44 + 297 * q^45 + 979 * q^46 - 184 * q^47 + 648 * q^48 - 38 * q^49 - 2031 * q^50 + 534 * q^51 - 151 * q^53 + 162 * q^54 + 2064 * q^55 + 2276 * q^56 + 1056 * q^57 + 1161 * q^58 + 537 * q^59 + 1206 * q^60 - 1340 * q^61 + 347 * q^62 + 747 * q^63 + 893 * q^64 - 1890 * q^66 + 2308 * q^67 + 2785 * q^68 + 450 * q^69 - 1420 * q^70 + 96 * q^71 + 783 * q^72 + 2505 * q^73 - 1191 * q^74 - 60 * q^75 + 2409 * q^76 - 2142 * q^77 - 1591 * q^79 - 2671 * q^80 + 729 * q^81 + 1517 * q^82 + 1539 * q^83 + 2820 * q^84 + 4296 * q^85 - 3763 * q^86 - 291 * q^87 - 3716 * q^88 - 592 * q^89 - 486 * q^90 + 515 * q^92 + 2151 * q^93 - 692 * q^94 + 4158 * q^95 + 2121 * q^96 + 1445 * q^97 + 1457 * q^98 + 765 * q^99

Introduction

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