LMFDB - Embedded newform 507.4.b.i.337.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(337,507)]

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, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("507.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	507.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$29.9139683729$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 70x^{8} + 1645x^{6} + 14700x^{4} + 44100x^{2} + 27648 $ x^10 + 70x^8 + 1645x^6 + 14700x^4 + 44100x^2 + 27648
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.7
Root			$2.04224i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.i.337.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.04224i q^{2} +3.00000 q^{3} +3.82924 q^{4} +12.0825i q^{5} +6.12673i q^{6} -29.7373i q^{7} +24.1582i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.04224i q^{2} +3.00000 q^{3} +3.82924 q^{4} +12.0825i q^{5} +6.12673i q^{6} -29.7373i q^{7} +24.1582i q^{8} +9.00000 q^{9} -24.6753 q^{10} -28.0636i q^{11} +11.4877 q^{12} +60.7308 q^{14} +36.2474i q^{15} -18.7029 q^{16} +50.6556 q^{17} +18.3802i q^{18} +105.148i q^{19} +46.2667i q^{20} -89.2119i q^{21} +57.3126 q^{22} +160.592 q^{23} +72.4746i q^{24} -20.9857 q^{25} +27.0000 q^{27} -113.871i q^{28} +140.105 q^{29} -74.0259 q^{30} +223.593i q^{31} +155.070i q^{32} -84.1907i q^{33} +103.451i q^{34} +359.300 q^{35} +34.4632 q^{36} +228.352i q^{37} -214.739 q^{38} -291.890 q^{40} -295.902i q^{41} +182.192 q^{42} -192.103 q^{43} -107.462i q^{44} +108.742i q^{45} +327.968i q^{46} -36.9300i q^{47} -56.1088 q^{48} -541.307 q^{49} -42.8579i q^{50} +151.967 q^{51} +149.102 q^{53} +55.1406i q^{54} +339.077 q^{55} +718.399 q^{56} +315.445i q^{57} +286.128i q^{58} -438.867i q^{59} +138.800i q^{60} +286.146 q^{61} -456.631 q^{62} -267.636i q^{63} -466.313 q^{64} +171.938 q^{66} -537.128i q^{67} +193.973 q^{68} +481.776 q^{69} +733.777i q^{70} +102.729i q^{71} +217.424i q^{72} +75.5209i q^{73} -466.350 q^{74} -62.9571 q^{75} +402.639i q^{76} -834.535 q^{77} +17.5526 q^{79} -225.977i q^{80} +81.0000 q^{81} +604.304 q^{82} +1463.08i q^{83} -341.614i q^{84} +612.044i q^{85} -392.322i q^{86} +420.315 q^{87} +677.965 q^{88} +334.905i q^{89} -222.078 q^{90} +614.946 q^{92} +670.779i q^{93} +75.4200 q^{94} -1270.45 q^{95} +465.209i q^{96} -748.756i q^{97} -1105.48i q^{98} -252.572i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 30 q^{3} - 60 q^{4} + 90 q^{9}+O(q^{10}) $ 10 * q + 30 * q^3 - 60 * q^4 + 90 * q^9	$ 10 q + 30 q^{3} - 60 q^{4} + 90 q^{9} - 80 q^{10} - 180 q^{12} - 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} - 960 q^{25} + 270 q^{27} + 990 q^{29} - 240 q^{30} - 120 q^{35} - 540 q^{36} - 1380 q^{38} + 2000 q^{40} - 180 q^{42} + 740 q^{43} + 1500 q^{48} - 1550 q^{49} - 630 q^{51} + 330 q^{53} + 520 q^{55} + 5340 q^{56} + 2750 q^{61} + 1560 q^{62} - 3140 q^{64} + 1740 q^{66} + 1200 q^{68} + 360 q^{69} - 4380 q^{74} - 2880 q^{75} - 4320 q^{77} + 1100 q^{79} + 810 q^{81} + 4780 q^{82} + 2970 q^{87} - 6340 q^{88} - 720 q^{90} - 1740 q^{92} + 6460 q^{94} + 2760 q^{95}+O(q^{100}) $ 10 * q + 30 * q^3 - 60 * q^4 + 90 * q^9 - 80 * q^10 - 180 * q^12 - 60 * q^14 + 500 * q^16 - 210 * q^17 + 580 * q^22 + 120 * q^23 - 960 * q^25 + 270 * q^27 + 990 * q^29 - 240 * q^30 - 120 * q^35 - 540 * q^36 - 1380 * q^38 + 2000 * q^40 - 180 * q^42 + 740 * q^43 + 1500 * q^48 - 1550 * q^49 - 630 * q^51 + 330 * q^53 + 520 * q^55 + 5340 * q^56 + 2750 * q^61 + 1560 * q^62 - 3140 * q^64 + 1740 * q^66 + 1200 * q^68 + 360 * q^69 - 4380 * q^74 - 2880 * q^75 - 4320 * q^77 + 1100 * q^79 + 810 * q^81 + 4780 * q^82 + 2970 * q^87 - 6340 * q^88 - 720 * q^90 - 1740 * q^92 + 6460 * q^94 + 2760 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.04224i	0.722042i	0.932558	+	0.361021i	$0.117572\pi$
$2$	2.04224i	0.722042i	−0.932558	+	0.361021i	$0.882428\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	3.82924	0.478656
$5$	12.0825i	1.08069i	0.841444	+	0.540344i	$0.181706\pi$
$5$	12.0825i	1.08069i	−0.841444	+	0.540344i	$0.818294\pi$
$6$	6.12673i	0.416871i
$7$	− 29.7373i	− 1.60566i	−0.596206	−	0.802832i	$-0.703326\pi$
$7$	− 29.7373i	− 1.60566i	0.596206	−	0.802832i	$-0.296674\pi$
$8$	24.1582i	1.06765i
$9$	9.00000	0.333333
$10$	−24.6753	−0.780302
$11$	− 28.0636i	− 0.769226i	−0.923078	−	0.384613i	$-0.874335\pi$
$11$	− 28.0636i	− 0.769226i	0.923078	−	0.384613i	$-0.125665\pi$
$12$	11.4877	0.276352
$13$	0	0
$13$	0	0
$14$	60.7308	1.15936
$15$	36.2474i	0.623935i
$16$	−18.7029	−0.292233
$17$	50.6556	0.722693	0.361347	−	0.932432i	$-0.382317\pi$
$17$	50.6556	0.722693	0.361347	+	0.932432i	$0.382317\pi$
$18$	18.3802i	0.240681i
$19$	105.148i	1.26962i	0.772670	+	0.634808i	$0.218921\pi$
$19$	105.148i	1.26962i	−0.772670	+	0.634808i	$0.781079\pi$
$20$	46.2667i	0.517277i
$21$	− 89.2119i	− 0.927030i
$22$	57.3126	0.555413
$23$	160.592	1.45590	0.727951	−	0.685629i	$-0.240473\pi$
$23$	160.592	1.45590	0.727951	+	0.685629i	$0.240473\pi$
$24$	72.4746i	0.616409i
$25$	−20.9857	−0.167886
$26$	0	0
$27$	27.0000	0.192450
$28$	− 113.871i	− 0.768560i
$29$	140.105	0.897132	0.448566	−	0.893750i	$-0.351935\pi$
$29$	140.105	0.897132	0.448566	+	0.893750i	$0.351935\pi$
$30$	−74.0259	−0.450507
$31$	223.593i	1.29544i	0.761880	+	0.647718i	$0.224276\pi$
$31$	223.593i	1.29544i	−0.761880	+	0.647718i	$0.775724\pi$
$32$	155.070i	0.856647i
$33$	− 84.1907i	− 0.444113i
$34$	103.451i	0.521815i
$35$	359.300	1.73522
$36$	34.4632	0.159552
$37$	228.352i	1.01462i	0.861765	+	0.507308i	$0.169359\pi$
$37$	228.352i	1.01462i	−0.861765	+	0.507308i	$0.830641\pi$
$38$	−214.739	−0.916716
$39$	0	0
$40$	−291.890	−1.15380
$41$	− 295.902i	− 1.12713i	−0.826073	−	0.563563i	$-0.809430\pi$
$41$	− 295.902i	− 1.12713i	0.826073	−	0.563563i	$-0.190570\pi$
$42$	182.192	0.669354
$43$	−192.103	−0.681291	−0.340645	−	0.940192i	$-0.610646\pi$
$43$	−192.103	−0.681291	−0.340645	+	0.940192i	$0.610646\pi$
$44$	− 107.462i	− 0.368194i
$45$	108.742i	0.360229i
$46$	327.968i	1.05122i
$47$	− 36.9300i	− 0.114613i	−0.998357	−	0.0573063i	$-0.981749\pi$
$47$	− 36.9300i	− 0.114613i	0.998357	−	0.0573063i	$-0.0182512\pi$
$48$	−56.1088	−0.168721
$49$	−541.307	−1.57815
$50$	− 42.8579i	− 0.121220i
$51$	151.967	0.417247
$52$	0	0
$53$	149.102	0.386429	0.193214	−	0.981157i	$-0.438109\pi$
$53$	149.102	0.386429	0.193214	+	0.981157i	$0.438109\pi$
$54$	55.1406i	0.138957i
$55$	339.077	0.831293
$56$	718.399	1.71429
$57$	315.445i	0.733013i
$58$	286.128i	0.647767i
$59$	− 438.867i	− 0.968400i	−0.874957	−	0.484200i	$-0.839111\pi$
$59$	− 438.867i	− 0.968400i	0.874957	−	0.484200i	$-0.160889\pi$
$60$	138.800i	0.298650i
$61$	286.146	0.600610	0.300305	−	0.953843i	$-0.402912\pi$
$61$	286.146	0.600610	0.300305	+	0.953843i	$0.402912\pi$
$62$	−456.631	−0.935358
$63$	− 267.636i	− 0.535221i
$64$	−466.313	−0.910768
$65$	0	0
$66$	171.938	0.320668
$67$	− 537.128i	− 0.979412i	−0.871888	−	0.489706i	$-0.837104\pi$
$67$	− 537.128i	− 0.979412i	0.871888	−	0.489706i	$-0.162896\pi$
$68$	193.973	0.345921
$69$	481.776	0.840566
$70$	733.777i	1.25290i
$71$	102.729i	0.171713i	0.996307	+	0.0858567i	$0.0273627\pi$
$71$	102.729i	0.171713i	−0.996307	+	0.0858567i	$0.972637\pi$
$72$	217.424i	0.355884i
$73$	75.5209i	0.121083i	0.998166	+	0.0605414i	$0.0192827\pi$
$73$	75.5209i	0.121083i	−0.998166	+	0.0605414i	$0.980717\pi$
$74$	−466.350	−0.732596
$75$	−62.9571	−0.0969288
$76$	402.639i	0.607709i
$77$	−834.535	−1.23512
$78$	0	0
$79$	17.5526	0.0249978	0.0124989	−	0.999922i	$-0.496021\pi$
$79$	17.5526	0.0249978	0.0124989	+	0.999922i	$0.496021\pi$
$80$	− 225.977i	− 0.315813i
$81$	81.0000	0.111111
$82$	604.304	0.813831
$83$	1463.08i	1.93487i	0.253122	+	0.967434i	$0.418543\pi$
$83$	1463.08i	1.93487i	−0.253122	+	0.967434i	$0.581457\pi$
$84$	− 341.614i	− 0.443728i
$85$	612.044i	0.781005i
$86$	− 392.322i	− 0.491920i
$87$	420.315	0.517960
$88$	677.965	0.821265
$89$	334.905i	0.398875i	0.979911	+	0.199438i	$0.0639115\pi$
$89$	334.905i	0.398875i	−0.979911	+	0.199438i	$0.936089\pi$
$90$	−222.078	−0.260101
$91$	0	0
$92$	614.946	0.696876
$93$	670.779i	0.747920i
$94$	75.4200	0.0827551
$95$	−1270.45	−1.37206
$96$	465.209i	0.494585i
$97$	− 748.756i	− 0.783760i	−0.920016	−	0.391880i	$-0.871825\pi$
$97$	− 748.756i	− 0.783760i	0.920016	−	0.391880i	$-0.128175\pi$
$98$	− 1105.48i	− 1.13949i
$99$	− 252.572i	− 0.256409i
$100$	−80.3594	−0.0803594
$101$	784.002	0.772387	0.386194	−	0.922418i	$-0.373790\pi$
$101$	784.002	0.772387	0.386194	+	0.922418i	$0.373790\pi$
$102$	310.353i	0.301270i
$103$	−396.040	−0.378864	−0.189432	−	0.981894i	$-0.560665\pi$
$103$	−396.040	−0.378864	−0.189432	+	0.981894i	$0.560665\pi$
$104$	0	0
$105$	1077.90	1.00183
$106$	304.502i	0.279018i
$107$	−1436.59	−1.29795	−0.648974	−	0.760810i	$-0.724802\pi$
$107$	−1436.59	−1.29795	−0.648974	+	0.760810i	$0.724802\pi$
$108$	103.390	0.0921173
$109$	− 1977.92i	− 1.73807i	−0.494746	−	0.869037i	$-0.664739\pi$
$109$	− 1977.92i	− 1.73807i	0.494746	−	0.869037i	$-0.335261\pi$
$110$	692.477i	0.600228i
$111$	685.056i	0.585789i
$112$	556.175i	0.469228i
$113$	122.405	0.101902	0.0509509	−	0.998701i	$-0.483775\pi$
$113$	122.405	0.101902	0.0509509	+	0.998701i	$0.483775\pi$
$114$	−644.216	−0.529266
$115$	1940.35i	1.57338i
$116$	536.496	0.429417
$117$	0	0
$118$	896.273	0.699225
$119$	− 1506.36i	− 1.16040i
$120$	−875.671	−0.666145
$121$	543.436	0.408291
$122$	584.379i	0.433665i
$123$	− 887.706i	− 0.650746i
$124$	856.192i	0.620067i
$125$	1256.75i	0.899256i
$126$	546.577	0.386452
$127$	−2309.61	−1.61374	−0.806868	−	0.590731i	$-0.798839\pi$
$127$	−2309.61	−1.61374	−0.806868	+	0.590731i	$0.798839\pi$
$128$	288.232i	0.199034i
$129$	−576.310	−0.393343
$130$	0	0
$131$	−1444.26	−0.963250	−0.481625	−	0.876377i	$-0.659953\pi$
$131$	−1444.26	−0.963250	−0.481625	+	0.876377i	$0.659953\pi$
$132$	− 322.387i	− 0.212577i
$133$	3126.83	2.03858
$134$	1096.94	0.707176
$135$	326.226i	0.207978i
$136$	1223.75i	0.771584i
$137$	735.918i	0.458932i	0.973317	+	0.229466i	$0.0736980\pi$
$137$	735.918i	0.458932i	−0.973317	+	0.229466i	$0.926302\pi$
$138$	983.904i	0.606924i
$139$	−1505.14	−0.918449	−0.459225	−	0.888320i	$-0.651873\pi$
$139$	−1505.14	−0.918449	−0.459225	+	0.888320i	$0.651873\pi$
$140$	1375.85	0.830573
$141$	− 110.790i	− 0.0661716i
$142$	−209.797	−0.123984
$143$	0	0
$144$	−168.326	−0.0974111
$145$	1692.81i	0.969520i
$146$	−154.232	−0.0874269
$147$	−1623.92	−0.911148
$148$	874.415i	0.485652i
$149$	427.843i	0.235237i	0.993059	+	0.117618i	$0.0375259\pi$
$149$	427.843i	0.235237i	−0.993059	+	0.117618i	$0.962474\pi$
$150$	− 128.574i	− 0.0699867i
$151$	1601.83i	0.863278i	0.902046	+	0.431639i	$0.142065\pi$
$151$	1601.83i	0.863278i	−0.902046	+	0.431639i	$0.857935\pi$
$152$	−2540.20	−1.35551
$153$	455.900	0.240898
$154$	− 1704.32i	− 0.891807i
$155$	−2701.55	−1.39996
$156$	0	0
$157$	−730.346	−0.371261	−0.185631	−	0.982620i	$-0.559433\pi$
$157$	−730.346	−0.371261	−0.185631	+	0.982620i	$0.559433\pi$
$158$	35.8467i	0.0180495i
$159$	447.306	0.223105
$160$	−1873.62	−0.925767
$161$	− 4775.58i	− 2.33769i
$162$	165.422i	0.0802269i
$163$	− 1898.36i	− 0.912215i	−0.889925	−	0.456107i	$-0.849243\pi$
$163$	− 1898.36i	− 0.912215i	0.889925	−	0.456107i	$-0.150757\pi$
$164$	− 1133.08i	− 0.539505i
$165$	1017.23	0.479947
$166$	−2987.97	−1.39706
$167$	− 1427.50i	− 0.661457i	−0.943726	−	0.330729i	$-0.892706\pi$
$167$	− 1427.50i	− 0.661457i	0.943726	−	0.330729i	$-0.107294\pi$
$168$	2155.20	0.989745
$169$	0	0
$170$	−1249.94	−0.563919
$171$	946.336i	0.423205i
$172$	−735.611	−0.326104
$173$	2044.40	0.898454	0.449227	−	0.893418i	$-0.351699\pi$
$173$	2044.40	0.898454	0.449227	+	0.893418i	$0.351699\pi$
$174$	858.385i	0.373988i
$175$	624.058i	0.269568i
$176$	524.871i	0.224793i
$177$	− 1316.60i	− 0.559106i
$178$	−683.958	−0.288004
$179$	3889.72	1.62420	0.812098	−	0.583521i	$-0.198325\pi$
$179$	3889.72	1.62420	0.812098	+	0.583521i	$0.198325\pi$
$180$	416.400i	0.172426i
$181$	2477.02	1.01721	0.508606	−	0.861000i	$-0.330161\pi$
$181$	2477.02	1.01721	0.508606	+	0.861000i	$0.330161\pi$
$182$	0	0
$183$	858.437	0.346762
$184$	3879.61i	1.55440i
$185$	−2759.05	−1.09648
$186$	−1369.89	−0.540029
$187$	− 1421.58i	− 0.555914i
$188$	− 141.414i	− 0.0548600i
$189$	− 802.907i	− 0.309010i
$190$	− 2594.57i	− 0.990683i
$191$	2276.81	0.862535	0.431267	−	0.902224i	$-0.358067\pi$
$191$	2276.81	0.862535	0.431267	+	0.902224i	$0.358067\pi$
$192$	−1398.94	−0.525832
$193$	− 3922.42i	− 1.46291i	−0.681888	−	0.731456i	$-0.738841\pi$
$193$	− 3922.42i	− 1.46291i	0.681888	−	0.731456i	$-0.261159\pi$
$194$	1529.14	0.565907
$195$	0	0
$196$	−2072.80	−0.755392
$197$	− 5063.23i	− 1.83117i	−0.402128	−	0.915584i	$-0.631729\pi$
$197$	− 5063.23i	− 1.83117i	0.402128	−	0.915584i	$-0.368271\pi$
$198$	515.814	0.185138
$199$	−3270.06	−1.16487	−0.582433	−	0.812879i	$-0.697899\pi$
$199$	−3270.06	−1.16487	−0.582433	+	0.812879i	$0.697899\pi$
$200$	− 506.977i	− 0.179243i
$201$	− 1611.38i	− 0.565464i
$202$	1601.12i	0.557696i
$203$	− 4166.34i	− 1.44049i
$204$	581.918	0.199718
$205$	3575.22	1.21807
$206$	− 808.810i	− 0.273555i
$207$	1445.33	0.485301
$208$	0	0
$209$	2950.84	0.976622
$210$	2201.33i	0.723363i
$211$	−2812.18	−0.917527	−0.458764	−	0.888558i	$-0.651708\pi$
$211$	−2812.18	−0.917527	−0.458764	+	0.888558i	$0.651708\pi$
$212$	570.948	0.184966
$213$	308.186i	0.0991388i
$214$	− 2933.87i	− 0.937173i
$215$	− 2321.08i	− 0.736262i
$216$	652.271i	0.205470i
$217$	6649.05	2.08003
$218$	4039.39	1.25496
$219$	226.563i	0.0699072i
$220$	1298.41	0.397903
$221$	0	0
$222$	−1399.05	−0.422964
$223$	− 917.736i	− 0.275588i	−0.990461	−	0.137794i	$-0.955999\pi$
$223$	− 917.736i	− 0.275588i	0.990461	−	0.137794i	$-0.0440012\pi$
$224$	4611.35	1.37549
$225$	−188.871	−0.0559619
$226$	249.981i	0.0735774i
$227$	− 1336.39i	− 0.390746i	−0.980729	−	0.195373i	$-0.937408\pi$
$227$	− 1336.39i	− 0.390746i	0.980729	−	0.195373i	$-0.0625917\pi$
$228$	1207.92i	0.350861i
$229$	− 164.820i	− 0.0475617i	−0.999717	−	0.0237808i	$-0.992430\pi$
$229$	− 164.820i	− 0.0475617i	0.999717	−	0.0237808i	$-0.00757039\pi$
$230$	−3962.66	−1.13604
$231$	−2503.60	−0.713096
$232$	3384.68i	0.957824i
$233$	4243.42	1.19312	0.596558	−	0.802570i	$-0.296535\pi$
$233$	4243.42	1.19312	0.596558	+	0.802570i	$0.296535\pi$
$234$	0	0
$235$	446.205	0.123860
$236$	− 1680.53i	− 0.463530i
$237$	52.6579	0.0144325
$238$	3076.35	0.837859
$239$	− 2491.07i	− 0.674200i	−0.941469	−	0.337100i	$-0.890554\pi$
$239$	− 2491.07i	− 0.674200i	0.941469	−	0.337100i	$-0.109446\pi$
$240$	− 677.932i	− 0.182335i
$241$	− 2917.40i	− 0.779776i	−0.920862	−	0.389888i	$-0.872514\pi$
$241$	− 2917.40i	− 0.779776i	0.920862	−	0.389888i	$-0.127486\pi$
$242$	1109.83i	0.294803i
$243$	243.000	0.0641500
$244$	1095.72	0.287485
$245$	− 6540.32i	− 1.70549i
$246$	1812.91	0.469866
$247$	0	0
$248$	−5401.60	−1.38307
$249$	4389.25i	1.11710i
$250$	−2566.58	−0.649300
$251$	1313.88	0.330403	0.165202	−	0.986260i	$-0.447173\pi$
$251$	1313.88	0.330403	0.165202	+	0.986260i	$0.447173\pi$
$252$	− 1024.84i	− 0.256187i
$253$	− 4506.79i	− 1.11992i
$254$	− 4716.78i	− 1.16519i
$255$	1836.13i	0.450914i
$256$	−4319.15	−1.05448
$257$	987.582	0.239703	0.119851	−	0.992792i	$-0.461758\pi$
$257$	987.582	0.239703	0.119851	+	0.992792i	$0.461758\pi$
$258$	− 1176.97i	− 0.284010i
$259$	6790.57	1.62913
$260$	0	0
$261$	1260.94	0.299044
$262$	− 2949.53i	− 0.695507i
$263$	−6986.45	−1.63803	−0.819017	−	0.573769i	$-0.805481\pi$
$263$	−6986.45	−1.63803	−0.819017	+	0.573769i	$0.805481\pi$
$264$	2033.90	0.474158
$265$	1801.52i	0.417609i
$266$	6385.74i	1.47194i
$267$	1004.72i	0.230291i
$268$	− 2056.79i	− 0.468801i
$269$	−5904.34	−1.33827	−0.669134	−	0.743142i	$-0.733335\pi$
$269$	−5904.34	−1.33827	−0.669134	+	0.743142i	$0.733335\pi$
$270$	−666.233	−0.150169
$271$	2131.54i	0.477793i	0.971045	+	0.238897i	$0.0767857\pi$
$271$	2131.54i	0.477793i	−0.971045	+	0.238897i	$0.923214\pi$
$272$	−947.408	−0.211195
$273$	0	0
$274$	−1502.92	−0.331368
$275$	588.934i	0.129142i
$276$	1844.84	0.402342
$277$	−4032.41	−0.874673	−0.437336	−	0.899298i	$-0.644078\pi$
$277$	−4032.41	−0.874673	−0.437336	+	0.899298i	$0.644078\pi$
$278$	− 3073.86i	− 0.663159i
$279$	2012.34i	0.431812i
$280$	8680.03i	1.85261i
$281$	2298.29i	0.487916i	0.969786	+	0.243958i	$0.0784459\pi$
$281$	2298.29i	0.487916i	−0.969786	+	0.243958i	$0.921554\pi$
$282$	226.260	0.0477787
$283$	−6656.80	−1.39825	−0.699127	−	0.714998i	$-0.746428\pi$
$283$	−6656.80	−1.39825	−0.699127	+	0.714998i	$0.746428\pi$
$284$	393.373i	0.0821916i
$285$	−3811.35	−0.792158
$286$	0	0
$287$	−8799.33	−1.80978
$288$	1395.63i	0.285549i
$289$	−2347.01	−0.477715
$290$	−3457.13	−0.700034
$291$	− 2246.27i	− 0.452504i
$292$	289.188i	0.0579570i
$293$	− 7466.99i	− 1.48883i	−0.667719	−	0.744413i	$-0.732729\pi$
$293$	− 7466.99i	− 1.48883i	0.667719	−	0.744413i	$-0.267271\pi$
$294$	− 3316.44i	− 0.657887i
$295$	5302.59	1.04654
$296$	−5516.57	−1.08326
$297$	− 757.717i	− 0.148038i
$298$	−873.759	−0.169851
$299$	0	0
$300$	−241.078	−0.0463955
$301$	5712.64i	1.09392i
$302$	−3271.32	−0.623323
$303$	2352.01	0.445938
$304$	− 1966.58i	− 0.371024i
$305$	3457.34i	0.649071i
$306$	931.059i	0.173938i
$307$	3965.99i	0.737299i	0.929568	+	0.368650i	$0.120180\pi$
$307$	3965.99i	0.737299i	−0.929568	+	0.368650i	$0.879820\pi$
$308$	−3195.64	−0.591196
$309$	−1188.12	−0.218737
$310$	− 5517.23i	− 1.01083i
$311$	−7372.29	−1.34419	−0.672097	−	0.740463i	$-0.734606\pi$
$311$	−7372.29	−1.34419	−0.672097	+	0.740463i	$0.734606\pi$
$312$	0	0
$313$	8249.55	1.48975	0.744875	−	0.667204i	$-0.232509\pi$
$313$	8249.55	1.48975	0.744875	+	0.667204i	$0.232509\pi$
$314$	− 1491.54i	− 0.268066i
$315$	3233.70	0.578407
$316$	67.2133	0.0119653
$317$	− 5575.26i	− 0.987817i	−0.869514	−	0.493909i	$-0.835568\pi$
$317$	− 5575.26i	− 0.987817i	0.869514	−	0.493909i	$-0.164432\pi$
$318$	913.507i	0.161091i
$319$	− 3931.85i	− 0.690098i
$320$	− 5634.21i	− 0.984256i
$321$	−4309.77	−0.749371
$322$	9752.88	1.68791
$323$	5326.35i	0.917543i
$324$	310.169	0.0531840
$325$	0	0
$326$	3876.91	0.658657
$327$	− 5933.75i	− 1.00348i
$328$	7148.46	1.20338
$329$	−1098.20	−0.184029
$330$	2077.43i	0.346542i
$331$	− 4157.36i	− 0.690361i	−0.938536	−	0.345180i	$-0.887818\pi$
$331$	− 4157.36i	− 0.690361i	0.938536	−	0.345180i	$-0.112182\pi$
$332$	5602.50i	0.926136i
$333$	2055.17i	0.338206i
$334$	2915.30	0.477600
$335$	6489.82	1.05844
$336$	1668.52i	0.270909i
$337$	−3225.18	−0.521326	−0.260663	−	0.965430i	$-0.583941\pi$
$337$	−3225.18	−0.521326	−0.260663	+	0.965430i	$0.583941\pi$
$338$	0	0
$339$	367.215	0.0588330
$340$	2343.67i	0.373833i
$341$	6274.82	0.996483
$342$	−1932.65	−0.305572
$343$	5897.11i	0.928321i
$344$	− 4640.87i	− 0.727381i
$345$	5821.04i	0.908389i
$346$	4175.15i	0.648721i
$347$	3290.49	0.509057	0.254529	−	0.967065i	$-0.418080\pi$
$347$	3290.49	0.509057	0.254529	+	0.967065i	$0.418080\pi$
$348$	1609.49	0.247924
$349$	4491.52i	0.688899i	0.938805	+	0.344449i	$0.111934\pi$
$349$	4491.52i	0.688899i	−0.938805	+	0.344449i	$0.888066\pi$
$350$	−1274.48	−0.194639
$351$	0	0
$352$	4351.81	0.658955
$353$	5897.88i	0.889270i	0.895712	+	0.444635i	$0.146667\pi$
$353$	5897.88i	0.889270i	−0.895712	+	0.444635i	$0.853333\pi$
$354$	2688.82	0.403698
$355$	−1241.21	−0.185569
$356$	1282.43i	0.190924i
$357$	− 4519.08i	− 0.669958i
$358$	7943.75i	1.17274i
$359$	9277.20i	1.36388i	0.731410	+	0.681938i	$0.238863\pi$
$359$	9277.20i	1.36388i	−0.731410	+	0.681938i	$0.761137\pi$
$360$	−2627.01	−0.384599
$361$	−4197.19	−0.611924
$362$	5058.67i	0.734469i
$363$	1630.31	0.235727
$364$	0	0
$365$	−912.477	−0.130853
$366$	1753.14i	0.250377i
$367$	−6574.36	−0.935092	−0.467546	−	0.883969i	$-0.654862\pi$
$367$	−6574.36	−0.935092	−0.467546	+	0.883969i	$0.654862\pi$
$368$	−3003.54	−0.425463
$369$	− 2663.12i	− 0.375708i
$370$	− 5634.65i	− 0.791707i
$371$	− 4433.89i	− 0.620474i
$372$	2568.58i	0.357996i
$373$	−5345.55	−0.742043	−0.371021	−	0.928624i	$-0.620992\pi$
$373$	−5345.55	−0.742043	−0.371021	+	0.928624i	$0.620992\pi$
$374$	2903.20	0.401393
$375$	3770.24i	0.519186i
$376$	892.162	0.122366
$377$	0	0
$378$	1639.73	0.223118
$379$	− 1038.51i	− 0.140751i	−0.997521	−	0.0703757i	$-0.977580\pi$
$379$	− 1038.51i	− 0.140751i	0.997521	−	0.0703757i	$-0.0224198\pi$
$380$	−4864.87	−0.656743
$381$	−6928.82	−0.931691
$382$	4649.80i	0.622786i
$383$	6749.19i	0.900437i	0.892918	+	0.450219i	$0.148654\pi$
$383$	6749.19i	0.900437i	−0.892918	+	0.450219i	$0.851346\pi$
$384$	864.696i	0.114912i
$385$	− 10083.2i	− 1.33478i
$386$	8010.54	1.05628
$387$	−1728.93	−0.227097
$388$	− 2867.17i	− 0.375151i
$389$	1246.11	0.162417	0.0812083	−	0.996697i	$-0.474122\pi$
$389$	1246.11	0.162417	0.0812083	+	0.996697i	$0.474122\pi$
$390$	0	0
$391$	8134.89	1.05217
$392$	− 13077.0i	− 1.68492i
$393$	−4332.78	−0.556133
$394$	10340.3	1.32218
$395$	212.079i	0.0270148i
$396$	− 967.161i	− 0.122731i
$397$	− 8355.69i	− 1.05632i	−0.849144	−	0.528161i	$-0.822882\pi$
$397$	− 8355.69i	− 1.05632i	0.849144	−	0.528161i	$-0.177118\pi$
$398$	− 6678.25i	− 0.841082i
$399$	9380.49	1.17697
$400$	392.494	0.0490618
$401$	− 3283.66i	− 0.408923i	−0.978875	−	0.204461i	$-0.934456\pi$
$401$	− 3283.66i	− 0.408923i	0.978875	−	0.204461i	$-0.0655443\pi$
$402$	3290.83	0.408288
$403$	0	0
$404$	3002.14	0.369708
$405$	978.679i	0.120076i
$406$	8508.68	1.04010
$407$	6408.37	0.780470
$408$	3671.24i	0.445474i
$409$	10928.9i	1.32127i	0.750706	+	0.660636i	$0.229713\pi$
$409$	10928.9i	1.32127i	−0.750706	+	0.660636i	$0.770287\pi$
$410$	7301.47i	0.879497i
$411$	2207.75i	0.264965i
$412$	−1516.53	−0.181345
$413$	−13050.7	−1.55492
$414$	2951.71i	0.350408i
$415$	−17677.6	−2.09099
$416$	0	0
$417$	−4515.43	−0.530267
$418$	6026.33i	0.705162i
$419$	7302.94	0.851485	0.425742	−	0.904844i	$-0.360013\pi$
$419$	7302.94	0.851485	0.425742	+	0.904844i	$0.360013\pi$
$420$	4127.54	0.479531
$421$	7580.99i	0.877612i	0.898582	+	0.438806i	$0.144599\pi$
$421$	7580.99i	0.877612i	−0.898582	+	0.438806i	$0.855401\pi$
$422$	− 5743.15i	− 0.662493i
$423$	− 332.370i	− 0.0382042i
$424$	3602.03i	0.412571i
$425$	−1063.04	−0.121330
$426$	−629.391	−0.0715823
$427$	− 8509.19i	− 0.964377i
$428$	−5501.06	−0.621270
$429$	0	0
$430$	4740.21	0.531612
$431$	− 10056.7i	− 1.12394i	−0.827159	−	0.561968i	$-0.810044\pi$
$431$	− 10056.7i	− 1.12394i	0.827159	−	0.561968i	$-0.189956\pi$
$432$	−504.979	−0.0562403
$433$	−2733.38	−0.303367	−0.151683	−	0.988429i	$-0.548469\pi$
$433$	−2733.38	−0.303367	−0.151683	+	0.988429i	$0.548469\pi$
$434$	13579.0i	1.50187i
$435$	5078.44i	0.559752i
$436$	− 7573.93i	− 0.831939i
$437$	16886.0i	1.84844i
$438$	−462.696	−0.0504759
$439$	−6744.23	−0.733223	−0.366611	−	0.930374i	$-0.619482\pi$
$439$	−6744.23	−0.733223	−0.366611	+	0.930374i	$0.619482\pi$
$440$	8191.48i	0.887531i
$441$	−4871.76	−0.526051
$442$	0	0
$443$	8655.69	0.928317	0.464158	−	0.885752i	$-0.346357\pi$
$443$	8655.69	0.928317	0.464158	+	0.885752i	$0.346357\pi$
$444$	2623.25i	0.280391i
$445$	−4046.48	−0.431059
$446$	1874.24	0.198986
$447$	1283.53i	0.135814i
$448$	13866.9i	1.46239i
$449$	6522.46i	0.685555i	0.939417	+	0.342777i	$0.111368\pi$
$449$	6522.46i	0.685555i	−0.939417	+	0.342777i	$0.888632\pi$
$450$	− 385.721i	− 0.0404068i
$451$	−8304.07	−0.867014
$452$	468.719	0.0487759
$453$	4805.49i	0.498414i
$454$	2729.23	0.282135
$455$	0	0
$456$	−7620.59	−0.782602
$457$	1551.23i	0.158782i	0.996844	+	0.0793909i	$0.0252975\pi$
$457$	1551.23i	0.158782i	−0.996844	+	0.0793909i	$0.974702\pi$
$458$	336.603	0.0343415
$459$	1367.70	0.139082
$460$	7430.06i	0.753105i
$461$	− 7766.25i	− 0.784621i	−0.919833	−	0.392310i	$-0.871676\pi$
$461$	− 7766.25i	− 0.784621i	0.919833	−	0.392310i	$-0.128324\pi$
$462$	− 5112.97i	− 0.514885i
$463$	− 2004.52i	− 0.201205i	−0.994927	−	0.100603i	$-0.967923\pi$
$463$	− 2004.52i	− 0.201205i	0.994927	−	0.100603i	$-0.0320771\pi$
$464$	−2620.37	−0.262172
$465$	−8104.66	−0.808268
$466$	8666.10i	0.861479i
$467$	−18674.3	−1.85042	−0.925209	−	0.379458i	$-0.876111\pi$
$467$	−18674.3	−1.85042	−0.925209	+	0.379458i	$0.876111\pi$
$468$	0	0
$469$	−15972.7	−1.57261
$470$	911.259i	0.0894324i
$471$	−2191.04	−0.214348
$472$	10602.2	1.03391
$473$	5391.11i	0.524067i
$474$	107.540i	0.0104209i
$475$	− 2206.61i	− 0.213150i
$476$	− 5768.22i	− 0.555433i
$477$	1341.92	0.128810
$478$	5087.36	0.486800
$479$	− 9313.02i	− 0.888357i	−0.895938	−	0.444178i	$-0.853496\pi$
$479$	− 9313.02i	− 0.888357i	0.895938	−	0.444178i	$-0.146504\pi$
$480$	−5620.86	−0.534492
$481$	0	0
$482$	5958.03	0.563031
$483$	− 14326.7i	− 1.34967i
$484$	2080.95	0.195431
$485$	9046.81	0.846999
$486$	496.265i	0.0463190i
$487$	3536.80i	0.329092i	0.986369	+	0.164546i	$0.0526158\pi$
$487$	3536.80i	0.329092i	−0.986369	+	0.164546i	$0.947384\pi$
$488$	6912.76i	0.641241i
$489$	− 5695.08i	− 0.526667i
$490$	13356.9	1.23144
$491$	3361.78	0.308992	0.154496	−	0.987993i	$-0.450625\pi$
$491$	3361.78	0.308992	0.154496	+	0.987993i	$0.450625\pi$
$492$	− 3399.24i	− 0.311483i
$493$	7097.10	0.648351
$494$	0	0
$495$	3051.69	0.277098
$496$	− 4181.84i	− 0.378569i
$497$	3054.87	0.275714
$498$	−8963.91	−0.806591
$499$	− 4027.43i	− 0.361308i	−0.983547	−	0.180654i	$-0.942179\pi$
$499$	− 4027.43i	− 0.361308i	0.983547	−	0.180654i	$-0.0578214\pi$
$500$	4812.40i	0.430434i
$501$	− 4282.50i	− 0.381892i
$502$	2683.26i	0.238565i
$503$	1766.67	0.156604	0.0783022	−	0.996930i	$-0.475050\pi$
$503$	1766.67	0.156604	0.0783022	+	0.996930i	$0.475050\pi$
$504$	6465.59	0.571429
$505$	9472.67i	0.834709i
$506$	9203.96	0.808628
$507$	0	0
$508$	−8844.05	−0.772424
$509$	6816.27i	0.593567i	0.954945	+	0.296784i	$0.0959140\pi$
$509$	6816.27i	0.593567i	−0.954945	+	0.296784i	$0.904086\pi$
$510$	−3749.83	−0.325579
$511$	2245.79	0.194418
$512$	− 6514.89i	− 0.562344i
$513$	2839.01i	0.244338i
$514$	2016.88i	0.173076i
$515$	− 4785.13i	− 0.409433i
$516$	−2206.83	−0.188276
$517$	−1036.39	−0.0881630
$518$	13868.0i	1.17630i
$519$	6133.19	0.518723
$520$	0	0
$521$	−5442.27	−0.457640	−0.228820	−	0.973469i	$-0.573487\pi$
$521$	−5442.27	−0.457640	−0.228820	+	0.973469i	$0.573487\pi$
$522$	2575.15i	0.215922i
$523$	20728.5	1.73307	0.866535	−	0.499117i	$-0.166342\pi$
$523$	20728.5	1.73307	0.866535	+	0.499117i	$0.166342\pi$
$524$	−5530.43	−0.461065
$525$	1872.17i	0.155635i
$526$	− 14268.0i	− 1.18273i
$527$	11326.2i	0.936202i
$528$	1574.61i	0.129785i
$529$	13622.8	1.11965
$530$	−3679.13	−0.301531
$531$	− 3949.80i	− 0.322800i
$532$	11973.4	0.975775
$533$	0	0
$534$	−2051.87	−0.166279
$535$	− 17357.5i	− 1.40268i
$536$	12976.0	1.04567
$537$	11669.2	0.937730
$538$	− 12058.1i	− 0.966285i
$539$	15191.0i	1.21396i
$540$	1249.20i	0.0995500i
$541$	− 8577.44i	− 0.681651i	−0.940127	−	0.340825i	$-0.889294\pi$
$541$	− 8577.44i	− 0.681651i	0.940127	−	0.340825i	$-0.110706\pi$
$542$	−4353.13	−0.344987
$543$	7431.05	0.587287
$544$	7855.14i	0.619093i
$545$	23898.1	1.87832
$546$	0	0
$547$	8723.99	0.681921	0.340961	−	0.940078i	$-0.389248\pi$
$547$	8723.99	0.681921	0.340961	+	0.940078i	$0.389248\pi$
$548$	2818.01i	0.219671i
$549$	2575.31	0.200203
$550$	−1202.75	−0.0932459
$551$	14731.8i	1.13901i
$552$	11638.8i	0.897431i
$553$	− 521.968i	− 0.0401380i
$554$	− 8235.17i	− 0.631550i
$555$	−8277.15	−0.633055
$556$	−5763.56	−0.439621
$557$	965.006i	0.0734087i	0.999326	+	0.0367043i	$0.0116860\pi$
$557$	965.006i	0.0734087i	−0.999326	+	0.0367043i	$0.988314\pi$
$558$	−4109.68	−0.311786
$559$	0	0
$560$	−6719.95	−0.507089
$561$	− 4264.73i	− 0.320957i
$562$	−4693.66	−0.352296
$563$	14605.2	1.09331	0.546657	−	0.837357i	$-0.315900\pi$
$563$	14605.2	1.09331	0.546657	+	0.837357i	$0.315900\pi$
$564$	− 424.242i	− 0.0316734i
$565$	1478.95i	0.110124i
$566$	− 13594.8i	− 1.00960i
$567$	− 2408.72i	− 0.178407i
$568$	−2481.74	−0.183330
$569$	−7802.48	−0.574863	−0.287432	−	0.957801i	$-0.592801\pi$
$569$	−7802.48	−0.574863	−0.287432	+	0.957801i	$0.592801\pi$
$570$	− 7783.71i	− 0.571971i
$571$	−11988.2	−0.878618	−0.439309	−	0.898336i	$-0.644777\pi$
$571$	−11988.2	−0.878618	−0.439309	+	0.898336i	$0.644777\pi$
$572$	0	0
$573$	6830.43	0.497985
$574$	− 17970.4i	− 1.30674i
$575$	−3370.14	−0.244425
$576$	−4196.82	−0.303589
$577$	5576.90i	0.402374i	0.979553	+	0.201187i	$0.0644798\pi$
$577$	5576.90i	0.402374i	−0.979553	+	0.201187i	$0.935520\pi$
$578$	− 4793.17i	− 0.344930i
$579$	− 11767.3i	− 0.844613i
$580$	6482.19i	0.464066i
$581$	43508.1	3.10675
$582$	4587.43	0.326727
$583$	− 4184.33i	− 0.297251i
$584$	−1824.45	−0.129274
$585$	0	0
$586$	15249.4	1.07499
$587$	26754.0i	1.88119i	0.339535	+	0.940593i	$0.389730\pi$
$587$	26754.0i	1.88119i	−0.339535	+	0.940593i	$0.610270\pi$
$588$	−6218.39	−0.436126
$589$	−23510.5	−1.64471
$590$	10829.2i	0.755644i
$591$	− 15189.7i	− 1.05722i
$592$	− 4270.85i	− 0.296505i
$593$	3589.40i	0.248565i	0.992247	+	0.124283i	$0.0396629\pi$
$593$	3589.40i	0.248565i	−0.992247	+	0.124283i	$0.960337\pi$
$594$	1547.44	0.106889
$595$	18200.5	1.25403
$596$	1638.32i	0.112597i
$597$	−9810.18	−0.672536
$598$	0	0
$599$	−7462.78	−0.509050	−0.254525	−	0.967066i	$-0.581919\pi$
$599$	−7462.78	−0.509050	−0.254525	+	0.967066i	$0.581919\pi$
$600$	− 1520.93i	− 0.103486i
$601$	−16511.0	−1.12063	−0.560316	−	0.828279i	$-0.689320\pi$
$601$	−16511.0	−1.12063	−0.560316	+	0.828279i	$0.689320\pi$
$602$	−11666.6	−0.789858
$603$	− 4834.15i	− 0.326471i
$604$	6133.80i	0.413213i
$605$	6566.04i	0.441235i
$606$	4803.37i	0.321986i
$607$	−11953.6	−0.799309	−0.399654	−	0.916666i	$-0.630870\pi$
$607$	−11953.6	−0.799309	−0.399654	+	0.916666i	$0.630870\pi$
$608$	−16305.3	−1.08761
$609$	− 12499.0i	− 0.831669i
$610$	−7060.73	−0.468657
$611$	0	0
$612$	1745.75	0.115307
$613$	− 4575.65i	− 0.301482i	−0.988573	−	0.150741i	$-0.951834\pi$
$613$	− 4575.65i	− 0.301482i	0.988573	−	0.150741i	$-0.0481660\pi$
$614$	−8099.51	−0.532361
$615$	10725.7	0.703253
$616$	− 20160.9i	− 1.31868i
$617$	− 19231.0i	− 1.25480i	−0.778697	−	0.627400i	$-0.784119\pi$
$617$	− 19231.0i	− 1.25480i	0.778697	−	0.627400i	$-0.215881\pi$
$618$	− 2426.43i	− 0.157937i
$619$	11715.6i	0.760727i	0.924837	+	0.380363i	$0.124201\pi$
$619$	11715.6i	0.760727i	−0.924837	+	0.380363i	$0.875799\pi$
$620$	−10344.9	−0.670099
$621$	4335.99	0.280189
$622$	− 15056.0i	− 0.970564i
$623$	9959.17	0.640459
$624$	0	0
$625$	−17807.8	−1.13970
$626$	16847.6i	1.07566i
$627$	8852.52	0.563853
$628$	−2796.68	−0.177706
$629$	11567.3i	0.733256i
$630$	6603.99i	0.417634i
$631$	8780.09i	0.553930i	0.960880	+	0.276965i	$0.0893286\pi$
$631$	8780.09i	0.553930i	−0.960880	+	0.276965i	$0.910671\pi$
$632$	424.040i	0.0266889i
$633$	−8436.53	−0.529735
$634$	11386.0	0.713245
$635$	− 27905.7i	− 1.74395i
$636$	1712.84	0.106790
$637$	0	0
$638$	8029.78	0.498279
$639$	924.558i	0.0572378i
$640$	−3482.55	−0.215094
$641$	−24991.7	−1.53996	−0.769980	−	0.638068i	$-0.779734\pi$
$641$	−24991.7	−1.53996	−0.769980	+	0.638068i	$0.779734\pi$
$642$	− 8801.60i	− 0.541077i
$643$	− 2353.86i	− 0.144365i	−0.997391	−	0.0721827i	$-0.977004\pi$
$643$	− 2353.86i	− 0.144365i	0.997391	−	0.0721827i	$-0.0229965\pi$
$644$	− 18286.8i	− 1.11895i
$645$	− 6963.24i	− 0.425081i
$646$	−10877.7	−0.662504
$647$	5910.80	0.359162	0.179581	−	0.983743i	$-0.442526\pi$
$647$	5910.80	0.359162	0.179581	+	0.983743i	$0.442526\pi$
$648$	1956.81i	0.118628i
$649$	−12316.2	−0.744918
$650$	0	0
$651$	19947.2	1.20091
$652$	− 7269.28i	− 0.436637i
$653$	5924.34	0.355034	0.177517	−	0.984118i	$-0.443194\pi$
$653$	5924.34	0.355034	0.177517	+	0.984118i	$0.443194\pi$
$654$	12118.2	0.724553
$655$	− 17450.2i	− 1.04097i
$656$	5534.23i	0.329383i
$657$	679.688i	0.0403610i
$658$	− 2242.79i	− 0.132877i
$659$	−12839.5	−0.758964	−0.379482	−	0.925199i	$-0.623898\pi$
$659$	−12839.5	−0.758964	−0.379482	+	0.925199i	$0.623898\pi$
$660$	3895.22	0.229729
$661$	− 10265.5i	− 0.604057i	−0.953299	−	0.302028i	$-0.902336\pi$
$661$	− 10265.5i	− 0.604057i	0.953299	−	0.302028i	$-0.0976638\pi$
$662$	8490.35	0.498469
$663$	0	0
$664$	−35345.4	−2.06577
$665$	37779.8i	2.20306i
$666$	−4197.15	−0.244199
$667$	22499.7	1.30614
$668$	− 5466.25i	− 0.316610i
$669$	− 2753.21i	− 0.159111i
$670$	13253.8i	0.764236i
$671$	− 8030.27i	− 0.462004i
$672$	13834.1	0.794137
$673$	−9862.82	−0.564909	−0.282454	−	0.959281i	$-0.591149\pi$
$673$	−9862.82	−0.564909	−0.282454	+	0.959281i	$0.591149\pi$
$674$	− 6586.61i	− 0.376419i
$675$	−566.614	−0.0323096
$676$	0	0
$677$	32615.5	1.85158	0.925788	−	0.378043i	$-0.123403\pi$
$677$	32615.5	1.85158	0.925788	+	0.378043i	$0.123403\pi$
$678$	749.943i	0.0424799i
$679$	−22266.0	−1.25845
$680$	−14785.9	−0.833841
$681$	− 4009.17i	− 0.225597i
$682$	12814.7i	0.719502i
$683$	21627.4i	1.21164i	0.795602	+	0.605820i	$0.207155\pi$
$683$	21627.4i	1.21164i	−0.795602	+	0.605820i	$0.792845\pi$
$684$	3623.75i	0.202570i
$685$	−8891.70	−0.495962
$686$	−12043.3	−0.670286
$687$	− 494.461i	− 0.0274598i
$688$	3592.90	0.199096
$689$	0	0
$690$	−11888.0	−0.655895
$691$	14233.1i	0.783580i	0.920055	+	0.391790i	$0.128144\pi$
$691$	14233.1i	0.783580i	−0.920055	+	0.391790i	$0.871856\pi$
$692$	7828.49	0.430050
$693$	−7510.81	−0.411706
$694$	6719.98i	0.367561i
$695$	− 18185.8i	− 0.992557i
$696$	10154.0i	0.553000i
$697$	− 14989.1i	− 0.814566i
$698$	−9172.78	−0.497414
$699$	12730.3	0.688845
$700$	2389.67i	0.129030i
$701$	28747.0	1.54887	0.774437	−	0.632651i	$-0.218033\pi$
$701$	28747.0	1.54887	0.774437	+	0.632651i	$0.218033\pi$
$702$	0	0
$703$	−24010.8	−1.28817
$704$	13086.4i	0.700586i
$705$	1338.61	0.0715109
$706$	−12044.9	−0.642090
$707$	− 23314.1i	− 1.24019i
$708$	− 5041.59i	− 0.267619i
$709$	1818.65i	0.0963339i	0.998839	+	0.0481670i	$0.0153380\pi$
$709$	1818.65i	0.0963339i	−0.998839	+	0.0481670i	$0.984662\pi$
$710$	− 2534.86i	− 0.133988i
$711$	157.974	0.00833260
$712$	−8090.70	−0.425859
$713$	35907.3i	1.88603i
$714$	9229.06	0.483738
$715$	0	0
$716$	14894.7	0.777431
$717$	− 7473.20i	− 0.389249i
$718$	−18946.3	−0.984776
$719$	−26141.8	−1.35595	−0.677973	−	0.735087i	$-0.737141\pi$
$719$	−26141.8	−1.35595	−0.677973	+	0.735087i	$0.737141\pi$
$720$	− 2033.80i	− 0.105271i
$721$	11777.2i	0.608328i
$722$	− 8571.68i	− 0.441835i
$723$	− 8752.19i	− 0.450204i
$724$	9485.10	0.486894
$725$	−2940.20	−0.150616
$726$	3329.48i	0.170205i
$727$	1340.10	0.0683652	0.0341826	−	0.999416i	$-0.489117\pi$
$727$	1340.10	0.0683652	0.0341826	+	0.999416i	$0.489117\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	− 1863.50i	− 0.0944811i
$731$	−9731.11	−0.492364
$732$	3287.16	0.165980
$733$	− 32517.1i	− 1.63854i	−0.573409	−	0.819269i	$-0.694380\pi$
$733$	− 32517.1i	− 1.63854i	0.573409	−	0.819269i	$-0.305620\pi$
$734$	− 13426.4i	− 0.675176i
$735$	− 19620.9i	− 0.984666i
$736$	24903.0i	1.24719i
$737$	−15073.7	−0.753389
$738$	5438.73	0.271277
$739$	22170.5i	1.10359i	0.833978	+	0.551797i	$0.186058\pi$
$739$	22170.5i	1.10359i	−0.833978	+	0.551797i	$0.813942\pi$
$740$	−10565.1	−0.524838
$741$	0	0
$742$	9055.07	0.448008
$743$	− 30387.8i	− 1.50043i	−0.661193	−	0.750216i	$-0.729949\pi$
$743$	− 30387.8i	− 1.50043i	0.661193	−	0.750216i	$-0.270051\pi$
$744$	−16204.8	−0.798518
$745$	−5169.39	−0.254217
$746$	− 10916.9i	− 0.535786i
$747$	13167.7i	0.644956i
$748$	− 5443.57i	− 0.266092i
$749$	42720.3i	2.08407i
$750$	−7699.75	−0.374874
$751$	16898.3	0.821077	0.410539	−	0.911843i	$-0.365341\pi$
$751$	16898.3	0.821077	0.410539	+	0.911843i	$0.365341\pi$
$752$	690.699i	0.0334936i
$753$	3941.63	0.190758
$754$	0	0
$755$	−19354.0	−0.932934
$756$	− 3074.53i	− 0.147909i
$757$	32925.8	1.58086	0.790428	−	0.612555i	$-0.209858\pi$
$757$	32925.8	1.58086	0.790428	+	0.612555i	$0.209858\pi$
$758$	2120.89	0.101628
$759$	− 13520.4i	− 0.646585i
$760$	− 30691.8i	− 1.46488i
$761$	16792.0i	0.799882i	0.916541	+	0.399941i	$0.130969\pi$
$761$	16792.0i	0.799882i	−0.916541	+	0.399941i	$0.869031\pi$
$762$	− 14150.3i	− 0.672720i
$763$	−58817.9	−2.79076
$764$	8718.46	0.412857
$765$	5508.39i	0.260335i
$766$	−13783.5	−0.650153
$767$	0	0
$768$	−12957.4	−0.608804
$769$	− 1541.57i	− 0.0722891i	−0.999347	−	0.0361445i	$-0.988492\pi$
$769$	− 1541.57i	− 0.0722891i	0.999347	−	0.0361445i	$-0.0115077\pi$
$770$	20592.4	0.963765
$771$	2962.75	0.138393
$772$	− 15019.9i	− 0.700231i
$773$	− 38071.2i	− 1.77144i	−0.464218	−	0.885721i	$-0.653665\pi$
$773$	− 38071.2i	− 1.77144i	0.464218	−	0.885721i	$-0.346335\pi$
$774$	− 3530.90i	− 0.163973i
$775$	− 4692.26i	− 0.217485i
$776$	18088.6	0.836782
$777$	20371.7	0.940580
$778$	2544.85i	0.117272i
$779$	31113.6	1.43102
$780$	0	0
$781$	2882.93	0.132086
$782$	16613.4i	0.759712i
$783$	3782.83	0.172653
$784$	10124.0	0.461189
$785$	− 8824.38i	− 0.401217i
$786$	− 8848.60i	− 0.401551i
$787$	− 20049.8i	− 0.908129i	−0.890969	−	0.454065i	$-0.849973\pi$
$787$	− 20049.8i	− 0.908129i	0.890969	−	0.454065i	$-0.150027\pi$
$788$	− 19388.3i	− 0.876498i
$789$	−20959.4	−0.945720
$790$	−433.117	−0.0195058
$791$	− 3640.00i	− 0.163620i
$792$	6101.69	0.273755
$793$	0	0
$794$	17064.4	0.762709
$795$	5404.55i	0.241106i
$796$	−12521.9	−0.557570
$797$	22401.9	0.995627	0.497813	−	0.867284i	$-0.334136\pi$
$797$	22401.9	0.995627	0.497813	+	0.867284i	$0.334136\pi$
$798$	19157.2i	0.849823i
$799$	− 1870.71i	− 0.0828298i
$800$	− 3254.24i	− 0.143819i
$801$	3014.15i	0.132958i
$802$	6706.02	0.295259
$803$	2119.39	0.0931401
$804$	− 6170.38i	− 0.270662i
$805$	57700.7	2.52631
$806$	0	0
$807$	−17713.0	−0.772649
$808$	18940.1i	0.824640i
$809$	−41966.4	−1.82381	−0.911903	−	0.410406i	$-0.865387\pi$
$809$	−41966.4	−1.82381	−0.911903	+	0.410406i	$0.865387\pi$
$810$	−1998.70	−0.0867002
$811$	− 13029.1i	− 0.564133i	−0.959395	−	0.282067i	$-0.908980\pi$
$811$	− 13029.1i	− 0.564133i	0.959395	−	0.282067i	$-0.0910199\pi$
$812$	− 15953.9i	− 0.689500i
$813$	6394.63i	0.275854i
$814$	13087.4i	0.563532i
$815$	22936.8	0.985819
$816$	−2842.22	−0.121933
$817$	− 20199.4i	− 0.864977i
$818$	−22319.5	−0.954014
$819$	0	0
$820$	13690.4	0.583036
$821$	8898.14i	0.378255i	0.981953	+	0.189127i	$0.0605659\pi$
$821$	8898.14i	0.378255i	−0.981953	+	0.189127i	$0.939434\pi$
$822$	−4508.77	−0.191316
$823$	−22723.6	−0.962448	−0.481224	−	0.876598i	$-0.659808\pi$
$823$	−22723.6	−0.962448	−0.481224	+	0.876598i	$0.659808\pi$
$824$	− 9567.61i	− 0.404494i
$825$	1766.80i	0.0745602i
$826$	− 26652.7i	− 1.12272i
$827$	− 19073.3i	− 0.801989i	−0.916081	−	0.400994i	$-0.868665\pi$
$827$	− 19073.3i	− 0.801989i	0.916081	−	0.400994i	$-0.131335\pi$
$828$	5534.52	0.232292
$829$	−42503.8	−1.78072	−0.890361	−	0.455255i	$-0.849548\pi$
$829$	−42503.8	−1.78072	−0.890361	+	0.455255i	$0.849548\pi$
$830$	− 36102.0i	− 1.50978i
$831$	−12097.2	−0.504992
$832$	0	0
$833$	−27420.2	−1.14052
$834$	− 9221.59i	− 0.382875i
$835$	17247.7	0.714829
$836$	11299.5	0.467465
$837$	6037.01i	0.249307i
$838$	14914.4i	0.614808i
$839$	19427.2i	0.799408i	0.916644	+	0.399704i	$0.130887\pi$
$839$	19427.2i	0.799408i	−0.916644	+	0.399704i	$0.869113\pi$
$840$	26040.1i	1.06960i
$841$	−4759.60	−0.195154
$842$	−15482.2	−0.633673
$843$	6894.86i	0.281698i
$844$	−10768.5	−0.439180
$845$	0	0
$846$	678.780	0.0275850
$847$	− 16160.3i	− 0.655578i
$848$	−2788.64	−0.112927
$849$	−19970.4	−0.807282
$850$	− 2170.99i	− 0.0876052i
$851$	36671.5i	1.47718i
$852$	1180.12i	0.0474533i
$853$	− 26851.8i	− 1.07783i	−0.842361	−	0.538914i	$-0.818835\pi$
$853$	− 26851.8i	− 1.07783i	0.842361	−	0.538914i	$-0.181165\pi$
$854$	17377.8	0.696320
$855$	−11434.1	−0.457353
$856$	− 34705.4i	− 1.38576i
$857$	−41539.4	−1.65573	−0.827864	−	0.560929i	$-0.810444\pi$
$857$	−41539.4	−1.65573	−0.827864	+	0.560929i	$0.810444\pi$
$858$	0	0
$859$	−11936.2	−0.474107	−0.237054	−	0.971497i	$-0.576182\pi$
$859$	−11936.2	−0.474107	−0.237054	+	0.971497i	$0.576182\pi$
$860$	− 8887.99i	− 0.352416i
$861$	−26398.0	−1.04488
$862$	20538.3	0.811529
$863$	41128.6i	1.62229i	0.584848	+	0.811143i	$0.301154\pi$
$863$	41128.6i	1.62229i	−0.584848	+	0.811143i	$0.698846\pi$
$864$	4186.88i	0.164862i
$865$	24701.3i	0.970948i
$866$	− 5582.22i	− 0.219043i
$867$	−7041.04	−0.275809
$868$	25460.8	0.995619
$869$	− 492.590i	− 0.0192290i
$870$	−10371.4	−0.404165
$871$	0	0
$872$	47782.9	1.85566
$873$	− 6738.81i	− 0.261253i
$874$	−34485.3	−1.33465
$875$	37372.3	1.44390
$876$	867.564i	0.0334615i
$877$	− 6406.86i	− 0.246687i	−0.992364	−	0.123343i	$-0.960638\pi$
$877$	− 6406.86i	− 0.246687i	0.992364	−	0.123343i	$-0.0393617\pi$
$878$	− 13773.4i	− 0.529417i
$879$	− 22401.0i	− 0.859574i
$880$	−6341.73	−0.242931
$881$	2938.07	0.112357	0.0561783	−	0.998421i	$-0.482108\pi$
$881$	2938.07	0.112357	0.0561783	+	0.998421i	$0.482108\pi$
$882$	− 9949.32i	− 0.379831i
$883$	−3022.06	−0.115176	−0.0575881	−	0.998340i	$-0.518341\pi$
$883$	−3022.06	−0.115176	−0.0575881	+	0.998340i	$0.518341\pi$
$884$	0	0
$885$	15907.8	0.604219
$886$	17677.0i	0.670284i
$887$	−10060.4	−0.380830	−0.190415	−	0.981704i	$-0.560983\pi$
$887$	−10060.4	−0.380830	−0.190415	+	0.981704i	$0.560983\pi$
$888$	−16549.7	−0.625419
$889$	68681.5i	2.59112i
$890$	− 8263.89i	− 0.311243i
$891$	− 2273.15i	− 0.0854696i
$892$	− 3514.24i	− 0.131912i
$893$	3883.13	0.145514
$894$	−2621.28	−0.0980634
$895$	46997.3i	1.75525i
$896$	8571.24	0.319582
$897$	0	0
$898$	−13320.5	−0.494999
$899$	31326.5i	1.16218i
$900$	−723.234	−0.0267865
$901$	7552.84	0.279269
$902$	− 16958.9i	− 0.626020i
$903$	17137.9i	0.631577i
$904$	2957.09i	0.108796i
$905$	29928.4i	1.09929i
$906$	−9813.97	−0.359876
$907$	43158.6	1.58000	0.789999	−	0.613108i	$-0.210081\pi$
$907$	43158.6	1.58000	0.789999	+	0.613108i	$0.210081\pi$
$908$	− 5117.36i	− 0.187033i
$909$	7056.02	0.257462
$910$	0	0
$911$	−32665.9	−1.18800	−0.594001	−	0.804464i	$-0.702453\pi$
$911$	−32665.9	−1.18800	−0.594001	+	0.804464i	$0.702453\pi$
$912$	− 5899.75i	− 0.214211i
$913$	41059.3	1.48835
$914$	−3167.98	−0.114647
$915$	10372.0i	0.374741i
$916$	− 631.137i	− 0.0227657i
$917$	42948.4i	1.54665i
$918$	2793.18i	0.100423i
$919$	18989.9	0.681633	0.340816	−	0.940130i	$-0.389297\pi$
$919$	18989.9	0.681633	0.340816	+	0.940130i	$0.389297\pi$
$920$	−46875.3	−1.67982
$921$	11898.0i	0.425680i
$922$	15860.6	0.566529
$923$	0	0
$924$	−9586.91	−0.341327
$925$	− 4792.12i	− 0.170340i
$926$	4093.72	0.145279
$927$	−3564.36	−0.126288
$928$	21726.0i	0.768525i
$929$	− 5596.81i	− 0.197659i	−0.995104	−	0.0988295i	$-0.968490\pi$
$929$	− 5596.81i	− 0.197659i	0.995104	−	0.0988295i	$-0.0315099\pi$
$930$	− 16551.7i	− 0.583603i
$931$	− 56917.6i	− 2.00365i
$932$	16249.1	0.571091
$933$	−22116.9	−0.776070
$934$	− 38137.5i	− 1.33608i
$935$	17176.1	0.600770
$936$	0	0
$937$	40294.4	1.40487	0.702433	−	0.711750i	$-0.252097\pi$
$937$	40294.4	1.40487	0.702433	+	0.711750i	$0.252097\pi$
$938$	− 32620.2i	− 1.13549i
$939$	24748.6	0.860108
$940$	1708.63	0.0592865
$941$	43648.8i	1.51213i	0.654499	+	0.756063i	$0.272880\pi$
$941$	43648.8i	1.51213i	−0.654499	+	0.756063i	$0.727120\pi$
$942$	− 4474.63i	− 0.154768i
$943$	− 47519.5i	− 1.64098i
$944$	8208.10i	0.282999i
$945$	9701.09	0.333943
$946$	−11010.0	−0.378398
$947$	10486.6i	0.359839i	0.983681	+	0.179919i	$0.0575837\pi$
$947$	10486.6i	0.359839i	−0.983681	+	0.179919i	$0.942416\pi$
$948$	201.640	0.00690819
$949$	0	0
$950$	4506.44	0.153903
$951$	− 16725.8i	− 0.570317i
$952$	36390.9	1.23890
$953$	−33058.8	−1.12369	−0.561846	−	0.827242i	$-0.689909\pi$
$953$	−33058.8	−1.12369	−0.561846	+	0.827242i	$0.689909\pi$
$954$	2740.52i	0.0930059i
$955$	27509.5i	0.932131i
$956$	− 9538.90i	− 0.322709i
$957$	− 11795.5i	− 0.398428i
$958$	19019.5	0.641431
$959$	21884.2	0.736891
$960$	− 16902.6i	− 0.568260i
$961$	−20202.8	−0.678152
$962$	0	0
$963$	−12929.3	−0.432650
$964$	− 11171.4i	− 0.373244i
$965$	47392.5	1.58095
$966$	29258.7	0.974515
$967$	− 53634.9i	− 1.78364i	−0.452389	−	0.891821i	$-0.649428\pi$
$967$	− 53634.9i	− 1.78364i	0.452389	−	0.891821i	$-0.350572\pi$
$968$	13128.4i	0.435913i
$969$	15979.1i	0.529743i
$970$	18475.8i	0.611569i
$971$	−4086.80	−0.135069	−0.0675344	−	0.997717i	$-0.521513\pi$
$971$	−4086.80	−0.135069	−0.0675344	+	0.997717i	$0.521513\pi$
$972$	930.506	0.0307058
$973$	44758.8i	1.47472i
$974$	−7223.00	−0.237618
$975$	0	0
$976$	−5351.76	−0.175518
$977$	14381.0i	0.470919i	0.971884	+	0.235459i	$0.0756594\pi$
$977$	14381.0i	0.470919i	−0.971884	+	0.235459i	$0.924341\pi$
$978$	11630.7	0.380276
$979$	9398.64	0.306825
$980$	− 25044.5i	− 0.816343i
$981$	− 17801.3i	− 0.579358i
$982$	6865.57i	0.223105i
$983$	12916.5i	0.419099i	0.977798	+	0.209549i	$0.0671997\pi$
$983$	12916.5i	0.419099i	−0.977798	+	0.209549i	$0.932800\pi$
$984$	21445.4	0.694770
$985$	61176.2	1.97892
$986$	14494.0i	0.468137i
$987$	−3294.59	−0.106249
$988$	0	0
$989$	−30850.3	−0.991893
$990$	6232.30i	0.200076i
$991$	−5838.98	−0.187166	−0.0935829	−	0.995611i	$-0.529832\pi$
$991$	−5838.98	−0.187166	−0.0935829	+	0.995611i	$0.529832\pi$
$992$	−34672.5	−1.10973
$993$	− 12472.1i	− 0.398580i
$994$	6238.79i	0.199077i
$995$	− 39510.3i	− 1.25886i
$996$	16807.5i	0.534705i
$997$	44290.1	1.40690	0.703452	−	0.710743i	$-0.251641\pi$
$997$	44290.1	1.40690	0.703452	+	0.710743i	$0.251641\pi$
$998$	8224.99	0.260879
$999$	6165.50i	0.195263i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.i.337.7		10
13.3	even	3		39.4.j.c.4.2	✓	10
13.4	even	6		39.4.j.c.10.2	yes	10
13.5	odd	4		507.4.a.r.1.7		10
13.8	odd	4		507.4.a.r.1.4		10
13.12	even	2	inner	507.4.b.i.337.4		10
39.5	even	4		1521.4.a.bk.1.4		10
39.8	even	4		1521.4.a.bk.1.7		10
39.17	odd	6		117.4.q.e.10.4		10
39.29	odd	6		117.4.q.e.82.4		10
52.3	odd	6		624.4.bv.h.433.4		10
52.43	odd	6		624.4.bv.h.49.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.j.c.4.2	✓	10	13.3	even	3
39.4.j.c.10.2	yes	10	13.4	even	6
117.4.q.e.10.4		10	39.17	odd	6
117.4.q.e.82.4		10	39.29	odd	6
507.4.a.r.1.4		10	13.8	odd	4
507.4.a.r.1.7		10	13.5	odd	4
507.4.b.i.337.4		10	13.12	even	2	inner
507.4.b.i.337.7		10	1.1	even	1	trivial
624.4.bv.h.49.2		10	52.43	odd	6
624.4.bv.h.433.4		10	52.3	odd	6
1521.4.a.bk.1.4		10	39.5	even	4
1521.4.a.bk.1.7		10	39.8	even	4

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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.04224i q^{2} +3.00000 q^{3} +3.82924 q^{4} +12.0825i q^{5} +6.12673i q^{6} -29.7373i q^{7} +24.1582i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.04224i q^{2} +3.00000 q^{3} +3.82924 q^{4} +12.0825i q^{5} +6.12673i q^{6} -29.7373i q^{7} +24.1582i q^{8} +9.00000 q^{9} -24.6753 q^{10} -28.0636i q^{11} +11.4877 q^{12} +60.7308 q^{14} +36.2474i q^{15} -18.7029 q^{16} +50.6556 q^{17} +18.3802i q^{18} +105.148i q^{19} +46.2667i q^{20} -89.2119i q^{21} +57.3126 q^{22} +160.592 q^{23} +72.4746i q^{24} -20.9857 q^{25} +27.0000 q^{27} -113.871i q^{28} +140.105 q^{29} -74.0259 q^{30} +223.593i q^{31} +155.070i q^{32} -84.1907i q^{33} +103.451i q^{34} +359.300 q^{35} +34.4632 q^{36} +228.352i q^{37} -214.739 q^{38} -291.890 q^{40} -295.902i q^{41} +182.192 q^{42} -192.103 q^{43} -107.462i q^{44} +108.742i q^{45} +327.968i q^{46} -36.9300i q^{47} -56.1088 q^{48} -541.307 q^{49} -42.8579i q^{50} +151.967 q^{51} +149.102 q^{53} +55.1406i q^{54} +339.077 q^{55} +718.399 q^{56} +315.445i q^{57} +286.128i q^{58} -438.867i q^{59} +138.800i q^{60} +286.146 q^{61} -456.631 q^{62} -267.636i q^{63} -466.313 q^{64} +171.938 q^{66} -537.128i q^{67} +193.973 q^{68} +481.776 q^{69} +733.777i q^{70} +102.729i q^{71} +217.424i q^{72} +75.5209i q^{73} -466.350 q^{74} -62.9571 q^{75} +402.639i q^{76} -834.535 q^{77} +17.5526 q^{79} -225.977i q^{80} +81.0000 q^{81} +604.304 q^{82} +1463.08i q^{83} -341.614i q^{84} +612.044i q^{85} -392.322i q^{86} +420.315 q^{87} +677.965 q^{88} +334.905i q^{89} -222.078 q^{90} +614.946 q^{92} +670.779i q^{93} +75.4200 q^{94} -1270.45 q^{95} +465.209i q^{96} -748.756i q^{97} -1105.48i q^{98} -252.572i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 30 q^{3} - 60 q^{4} + 90 q^{9}+O(q^{10}) \) 10 * q + 30 * q^3 - 60 * q^4 + 90 * q^9	\( 10 q + 30 q^{3} - 60 q^{4} + 90 q^{9} - 80 q^{10} - 180 q^{12} - 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} - 960 q^{25} + 270 q^{27} + 990 q^{29} - 240 q^{30} - 120 q^{35} - 540 q^{36} - 1380 q^{38} + 2000 q^{40} - 180 q^{42} + 740 q^{43} + 1500 q^{48} - 1550 q^{49} - 630 q^{51} + 330 q^{53} + 520 q^{55} + 5340 q^{56} + 2750 q^{61} + 1560 q^{62} - 3140 q^{64} + 1740 q^{66} + 1200 q^{68} + 360 q^{69} - 4380 q^{74} - 2880 q^{75} - 4320 q^{77} + 1100 q^{79} + 810 q^{81} + 4780 q^{82} + 2970 q^{87} - 6340 q^{88} - 720 q^{90} - 1740 q^{92} + 6460 q^{94} + 2760 q^{95}+O(q^{100}) \) 10 * q + 30 * q^3 - 60 * q^4 + 90 * q^9 - 80 * q^10 - 180 * q^12 - 60 * q^14 + 500 * q^16 - 210 * q^17 + 580 * q^22 + 120 * q^23 - 960 * q^25 + 270 * q^27 + 990 * q^29 - 240 * q^30 - 120 * q^35 - 540 * q^36 - 1380 * q^38 + 2000 * q^40 - 180 * q^42 + 740 * q^43 + 1500 * q^48 - 1550 * q^49 - 630 * q^51 + 330 * q^53 + 520 * q^55 + 5340 * q^56 + 2750 * q^61 + 1560 * q^62 - 3140 * q^64 + 1740 * q^66 + 1200 * q^68 + 360 * q^69 - 4380 * q^74 - 2880 * q^75 - 4320 * q^77 + 1100 * q^79 + 810 * q^81 + 4780 * q^82 + 2970 * q^87 - 6340 * q^88 - 720 * q^90 - 1740 * q^92 + 6460 * q^94 + 2760 * q^95

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