LMFDB - Embedded newform 507.4.b.h.337.5

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 54x^{6} + 889x^{4} + 4584x^{2} + 5776 $ x^8 + 54x^6 + 889x^4 + 4584*x^2 + 5776
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 13^{2} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.5
Root			$2.46610i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.h.337.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.46610i q^{2} -3.00000 q^{3} +5.85055 q^{4} -9.85055i q^{5} -4.39830i q^{6} +29.9396i q^{7} +20.3063i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+1.46610i q^{2} -3.00000 q^{3} +5.85055 q^{4} -9.85055i q^{5} -4.39830i q^{6} +29.9396i q^{7} +20.3063i q^{8} +9.00000 q^{9} +14.4419 q^{10} +46.9257i q^{11} -17.5516 q^{12} -43.8945 q^{14} +29.5516i q^{15} +17.0333 q^{16} +48.2616 q^{17} +13.1949i q^{18} -120.300i q^{19} -57.6311i q^{20} -89.8188i q^{21} -68.7979 q^{22} -130.697 q^{23} -60.9189i q^{24} +27.9667 q^{25} -27.0000 q^{27} +175.163i q^{28} -194.946 q^{29} -43.3257 q^{30} +32.0123i q^{31} +187.423i q^{32} -140.777i q^{33} +70.7565i q^{34} +294.921 q^{35} +52.6549 q^{36} -32.4250i q^{37} +176.372 q^{38} +200.028 q^{40} +241.825i q^{41} +131.684 q^{42} -96.4087 q^{43} +274.541i q^{44} -88.6549i q^{45} -191.615i q^{46} +539.015i q^{47} -51.0998 q^{48} -553.380 q^{49} +41.0021i q^{50} -144.785 q^{51} -152.277 q^{53} -39.5847i q^{54} +462.244 q^{55} -607.963 q^{56} +360.901i q^{57} -285.810i q^{58} +327.792i q^{59} +172.893i q^{60} -98.4180 q^{61} -46.9332 q^{62} +269.456i q^{63} -138.515 q^{64} +206.394 q^{66} +441.151i q^{67} +282.357 q^{68} +392.091 q^{69} +432.385i q^{70} -345.049i q^{71} +182.757i q^{72} +773.839i q^{73} +47.5383 q^{74} -83.9002 q^{75} -703.822i q^{76} -1404.94 q^{77} -150.332 q^{79} -167.787i q^{80} +81.0000 q^{81} -354.540 q^{82} -337.966i q^{83} -525.489i q^{84} -475.403i q^{85} -141.345i q^{86} +584.837 q^{87} -952.889 q^{88} +169.913i q^{89} +129.977 q^{90} -764.649 q^{92} -96.0368i q^{93} -790.250 q^{94} -1185.02 q^{95} -562.269i q^{96} -214.201i q^{97} -811.311i q^{98} +422.332i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{3} - 44 q^{4} + 72 q^{9}+O(q^{10}) $ 8 * q - 24 * q^3 - 44 * q^4 + 72 * q^9	$ 8 q - 24 q^{3} - 44 q^{4} + 72 q^{9} + 124 q^{10} + 132 q^{12} + 80 q^{14} + 244 q^{16} - 196 q^{17} + 440 q^{22} - 208 q^{23} + 116 q^{25} - 216 q^{27} + 388 q^{29} - 372 q^{30} + 176 q^{35} - 396 q^{36} - 664 q^{38} - 1996 q^{40} - 240 q^{42} - 900 q^{43} - 732 q^{48} - 2140 q^{49} + 588 q^{51} + 524 q^{53} + 408 q^{55} - 4328 q^{56} - 1856 q^{61} - 5560 q^{62} - 2052 q^{64} - 1320 q^{66} + 3572 q^{68} + 624 q^{69} + 2316 q^{74} - 348 q^{75} - 5016 q^{77} - 1492 q^{79} + 648 q^{81} - 3468 q^{82} - 1164 q^{87} - 6120 q^{88} + 1116 q^{90} + 664 q^{92} - 1544 q^{94} - 4408 q^{95}+O(q^{100}) $ 8 * q - 24 * q^3 - 44 * q^4 + 72 * q^9 + 124 * q^10 + 132 * q^12 + 80 * q^14 + 244 * q^16 - 196 * q^17 + 440 * q^22 - 208 * q^23 + 116 * q^25 - 216 * q^27 + 388 * q^29 - 372 * q^30 + 176 * q^35 - 396 * q^36 - 664 * q^38 - 1996 * q^40 - 240 * q^42 - 900 * q^43 - 732 * q^48 - 2140 * q^49 + 588 * q^51 + 524 * q^53 + 408 * q^55 - 4328 * q^56 - 1856 * q^61 - 5560 * q^62 - 2052 * q^64 - 1320 * q^66 + 3572 * q^68 + 624 * q^69 + 2316 * q^74 - 348 * q^75 - 5016 * q^77 - 1492 * q^79 + 648 * q^81 - 3468 * q^82 - 1164 * q^87 - 6120 * q^88 + 1116 * q^90 + 664 * q^92 - 1544 * q^94 - 4408 * 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 $

$2$	1.46610i	0.518345i	0.965831	+	0.259173i	$0.0834498\pi$
$2$	1.46610i	0.518345i	−0.965831	+	0.259173i	$0.916550\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	5.85055	0.731318
$5$	− 9.85055i	− 0.881060i	−0.897738	−	0.440530i	$-0.854791\pi$
$5$	− 9.85055i	− 0.881060i	0.897738	−	0.440530i	$-0.145209\pi$
$6$	− 4.39830i	− 0.299267i
$7$	29.9396i	1.61659i	0.588780	+	0.808293i	$0.299608\pi$
$7$	29.9396i	1.61659i	−0.588780	+	0.808293i	$0.700392\pi$
$8$	20.3063i	0.897421i
$9$	9.00000	0.333333
$10$	14.4419	0.456693
$11$	46.9257i	1.28624i	0.765765	+	0.643120i	$0.222360\pi$
$11$	46.9257i	1.28624i	−0.765765	+	0.643120i	$0.777640\pi$
$12$	−17.5516	−0.422227
$13$	0	0
$13$	0	0
$14$	−43.8945	−0.837950
$15$	29.5516i	0.508680i
$16$	17.0333	0.266145
$17$	48.2616	0.688539	0.344270	−	0.938871i	$-0.388127\pi$
$17$	48.2616	0.688539	0.344270	+	0.938871i	$0.388127\pi$
$18$	13.1949i	0.172782i
$19$	− 120.300i	− 1.45257i	−0.687396	−	0.726283i	$-0.741246\pi$
$19$	− 120.300i	− 1.45257i	0.687396	−	0.726283i	$-0.258754\pi$
$20$	− 57.6311i	− 0.644335i
$21$	− 89.8188i	− 0.933337i
$22$	−68.7979	−0.666717
$23$	−130.697	−1.18488	−0.592440	−	0.805615i	$-0.701835\pi$
$23$	−130.697	−1.18488	−0.592440	+	0.805615i	$0.701835\pi$
$24$	− 60.9189i	− 0.518126i
$25$	27.9667	0.223734
$26$	0	0
$27$	−27.0000	−0.192450
$28$	175.163i	1.18224i
$29$	−194.946	−1.24829	−0.624147	−	0.781307i	$-0.714553\pi$
$29$	−194.946	−1.24829	−0.624147	+	0.781307i	$0.714553\pi$
$30$	−43.3257	−0.263672
$31$	32.0123i	0.185470i	0.995691	+	0.0927351i	$0.0295610\pi$
$31$	32.0123i	0.185470i	−0.995691	+	0.0927351i	$0.970439\pi$
$32$	187.423i	1.03538i
$33$	− 140.777i	− 0.742611i
$34$	70.7565i	0.356901i
$35$	294.921	1.42431
$36$	52.6549	0.243773
$37$	− 32.4250i	− 0.144071i	−0.997402	−	0.0720355i	$-0.977051\pi$
$37$	− 32.4250i	− 0.144071i	0.997402	−	0.0720355i	$-0.0229495\pi$
$38$	176.372	0.752931
$39$	0	0
$40$	200.028	0.790681
$41$	241.825i	0.921140i	0.887623	+	0.460570i	$0.152355\pi$
$41$	241.825i	0.921140i	−0.887623	+	0.460570i	$0.847645\pi$
$42$	131.684	0.483791
$43$	−96.4087	−0.341911	−0.170956	−	0.985279i	$-0.554686\pi$
$43$	−96.4087	−0.341911	−0.170956	+	0.985279i	$0.554686\pi$
$44$	274.541i	0.940651i
$45$	− 88.6549i	− 0.293687i
$46$	− 191.615i	− 0.614177i
$47$	539.015i	1.67284i	0.548090	+	0.836419i	$0.315355\pi$
$47$	539.015i	1.67284i	−0.548090	+	0.836419i	$0.684645\pi$
$48$	−51.0998	−0.153659
$49$	−553.380	−1.61335
$50$	41.0021i	0.115971i
$51$	−144.785	−0.397528
$52$	0	0
$53$	−152.277	−0.394657	−0.197328	−	0.980337i	$-0.563226\pi$
$53$	−152.277	−0.394657	−0.197328	+	0.980337i	$0.563226\pi$
$54$	− 39.5847i	− 0.0997556i
$55$	462.244	1.13325
$56$	−607.963	−1.45076
$57$	360.901i	0.838640i
$58$	− 285.810i	− 0.647047i
$59$	327.792i	0.723304i	0.932313	+	0.361652i	$0.117787\pi$
$59$	327.792i	0.723304i	−0.932313	+	0.361652i	$0.882213\pi$
$60$	172.893i	0.372007i
$61$	−98.4180	−0.206576	−0.103288	−	0.994651i	$-0.532936\pi$
$61$	−98.4180	−0.206576	−0.103288	+	0.994651i	$0.532936\pi$
$62$	−46.9332	−0.0961376
$63$	269.456i	0.538862i
$64$	−138.515	−0.270537
$65$	0	0
$66$	206.394	0.384929
$67$	441.151i	0.804405i	0.915551	+	0.402202i	$0.131755\pi$
$67$	441.151i	0.804405i	−0.915551	+	0.402202i	$0.868245\pi$
$68$	282.357	0.503541
$69$	392.091	0.684090
$70$	432.385i	0.738284i
$71$	− 345.049i	− 0.576757i	−0.957516	−	0.288379i	$-0.906884\pi$
$71$	− 345.049i	− 0.576757i	0.957516	−	0.288379i	$-0.0931161\pi$
$72$	182.757i	0.299140i
$73$	773.839i	1.24070i	0.784326	+	0.620349i	$0.213009\pi$
$73$	773.839i	1.24070i	−0.784326	+	0.620349i	$0.786991\pi$
$74$	47.5383	0.0746785
$75$	−83.9002	−0.129173
$76$	− 703.822i	− 1.06229i
$77$	−1404.94	−2.07932
$78$	0	0
$79$	−150.332	−0.214097	−0.107049	−	0.994254i	$-0.534140\pi$
$79$	−150.332	−0.214097	−0.107049	+	0.994254i	$0.534140\pi$
$80$	− 167.787i	− 0.234489i
$81$	81.0000	0.111111
$82$	−354.540	−0.477469
$83$	− 337.966i	− 0.446947i	−0.974710	−	0.223473i	$-0.928260\pi$
$83$	− 337.966i	− 0.446947i	0.974710	−	0.223473i	$-0.0717396\pi$
$84$	− 525.489i	− 0.682566i
$85$	− 475.403i	− 0.606644i
$86$	− 141.345i	− 0.177228i
$87$	584.837	0.720703
$88$	−952.889	−1.15430
$89$	169.913i	0.202368i	0.994868	+	0.101184i	$0.0322631\pi$
$89$	169.913i	0.202368i	−0.994868	+	0.101184i	$0.967737\pi$
$90$	129.977	0.152231
$91$	0	0
$92$	−764.649	−0.866524
$93$	− 96.0368i	− 0.107081i
$94$	−790.250	−0.867108
$95$	−1185.02	−1.27980
$96$	− 562.269i	− 0.597774i
$97$	− 214.201i	− 0.224215i	−0.993696	−	0.112107i	$-0.964240\pi$
$97$	− 214.201i	− 0.224215i	0.993696	−	0.112107i	$-0.0357601\pi$
$98$	− 811.311i	− 0.836273i
$99$	422.332i	0.428747i
$100$	163.621	0.163621
$101$	−1595.11	−1.57148	−0.785741	−	0.618556i	$-0.787718\pi$
$101$	−1595.11	−1.57148	−0.785741	+	0.618556i	$0.787718\pi$
$102$	− 212.269i	− 0.206057i
$103$	1570.30	1.50219	0.751096	−	0.660193i	$-0.229525\pi$
$103$	1570.30	1.50219	0.751096	+	0.660193i	$0.229525\pi$
$104$	0	0
$105$	−884.764	−0.822325
$106$	− 223.253i	− 0.204568i
$107$	−6.73302	−0.00608323	−0.00304161	−	0.999995i	$-0.500968\pi$
$107$	−6.73302	−0.00608323	−0.00304161	+	0.999995i	$0.500968\pi$
$108$	−157.965	−0.140742
$109$	542.422i	0.476648i	0.971186	+	0.238324i	$0.0765979\pi$
$109$	542.422i	0.476648i	−0.971186	+	0.238324i	$0.923402\pi$
$110$	677.697i	0.587417i
$111$	97.2749i	0.0831795i
$112$	509.969i	0.430246i
$113$	−1442.82	−1.20114	−0.600572	−	0.799571i	$-0.705060\pi$
$113$	−1442.82	−1.20114	−0.600572	+	0.799571i	$0.705060\pi$
$114$	−529.117	−0.434705
$115$	1287.44i	1.04395i
$116$	−1140.54	−0.912900
$117$	0	0
$118$	−480.577	−0.374921
$119$	1444.93i	1.11308i
$120$	−600.085	−0.456500
$121$	−871.025	−0.654414
$122$	− 144.291i	− 0.107078i
$123$	− 725.476i	− 0.531821i
$124$	187.289i	0.135638i
$125$	− 1506.81i	− 1.07818i
$126$	−395.051	−0.279317
$127$	2492.84	1.74176	0.870881	−	0.491494i	$-0.163549\pi$
$127$	2492.84	1.74176	0.870881	+	0.491494i	$0.163549\pi$
$128$	1296.31i	0.895144i
$129$	289.226	0.197403
$130$	0	0
$131$	744.561	0.496585	0.248292	−	0.968685i	$-0.420131\pi$
$131$	744.561	0.496585	0.248292	+	0.968685i	$0.420131\pi$
$132$	− 823.624i	− 0.543085i
$133$	3601.74	2.34820
$134$	−646.772	−0.416959
$135$	265.965i	0.169560i
$136$	980.016i	0.617909i
$137$	− 222.167i	− 0.138547i	−0.997598	−	0.0692736i	$-0.977932\pi$
$137$	− 222.167i	− 0.138547i	0.997598	−	0.0692736i	$-0.0220682\pi$
$138$	574.846i	0.354595i
$139$	−777.145	−0.474220	−0.237110	−	0.971483i	$-0.576200\pi$
$139$	−777.145	−0.474220	−0.237110	+	0.971483i	$0.576200\pi$
$140$	1725.45	1.04162
$141$	− 1617.04i	− 0.965814i
$142$	505.876	0.298959
$143$	0	0
$144$	153.299	0.0887149
$145$	1920.32i	1.09982i
$146$	−1134.53	−0.643110
$147$	1660.14	0.931469
$148$	− 189.704i	− 0.105362i
$149$	− 1778.29i	− 0.977742i	−0.872356	−	0.488871i	$-0.837409\pi$
$149$	− 1778.29i	− 0.977742i	0.872356	−	0.488871i	$-0.162591\pi$
$150$	− 123.006i	− 0.0669561i
$151$	1166.00i	0.628394i	0.949358	+	0.314197i	$0.101735\pi$
$151$	1166.00i	0.628394i	−0.949358	+	0.314197i	$0.898265\pi$
$152$	2442.85	1.30356
$153$	434.355	0.229513
$154$	− 2059.78i	− 1.07780i
$155$	315.338	0.163410
$156$	0	0
$157$	517.628	0.263129	0.131564	−	0.991308i	$-0.458000\pi$
$157$	517.628	0.263129	0.131564	+	0.991308i	$0.458000\pi$
$158$	− 220.402i	− 0.110976i
$159$	456.830	0.227855
$160$	1846.22	0.912227
$161$	− 3913.02i	− 1.91546i
$162$	118.754i	0.0575939i
$163$	610.188i	0.293212i	0.989195	+	0.146606i	$0.0468350\pi$
$163$	610.188i	0.293212i	−0.989195	+	0.146606i	$0.953165\pi$
$164$	1414.81i	0.673647i
$165$	−1386.73	−0.654285
$166$	495.493	0.231673
$167$	2983.01i	1.38223i	0.722745	+	0.691115i	$0.242880\pi$
$167$	2983.01i	1.38223i	−0.722745	+	0.691115i	$0.757120\pi$
$168$	1823.89	0.837596
$169$	0	0
$170$	696.990	0.314451
$171$	− 1082.70i	− 0.484189i
$172$	−564.044	−0.250046
$173$	−978.212	−0.429896	−0.214948	−	0.976625i	$-0.568958\pi$
$173$	−978.212	−0.429896	−0.214948	+	0.976625i	$0.568958\pi$
$174$	857.431i	0.373573i
$175$	837.313i	0.361685i
$176$	799.298i	0.342326i
$177$	− 983.377i	− 0.417599i
$178$	−249.110	−0.104897
$179$	−1852.94	−0.773717	−0.386858	−	0.922139i	$-0.626440\pi$
$179$	−1852.94	−0.773717	−0.386858	+	0.922139i	$0.626440\pi$
$180$	− 518.680i	− 0.214778i
$181$	−852.777	−0.350201	−0.175101	−	0.984551i	$-0.556025\pi$
$181$	−852.777	−0.350201	−0.175101	+	0.984551i	$0.556025\pi$
$182$	0	0
$183$	295.254	0.119267
$184$	− 2653.98i	− 1.06334i
$185$	−319.403	−0.126935
$186$	140.800	0.0555050
$187$	2264.71i	0.885627i
$188$	3153.53i	1.22338i
$189$	− 808.369i	− 0.311112i
$190$	− 1737.36i	− 0.663377i
$191$	4441.29	1.68252	0.841258	−	0.540633i	$-0.181815\pi$
$191$	4441.29	1.68252	0.841258	+	0.540633i	$0.181815\pi$
$192$	415.545	0.156195
$193$	− 2482.43i	− 0.925851i	−0.886397	−	0.462925i	$-0.846800\pi$
$193$	− 2482.43i	− 0.925851i	0.886397	−	0.462925i	$-0.153200\pi$
$194$	314.041	0.116221
$195$	0	0
$196$	−3237.57	−1.17987
$197$	− 1260.23i	− 0.455775i	−0.973687	−	0.227888i	$-0.926818\pi$
$197$	− 1260.23i	− 0.455775i	0.973687	−	0.227888i	$-0.0731819\pi$
$198$	−619.181	−0.222239
$199$	5520.96	1.96669	0.983344	−	0.181756i	$-0.0581782\pi$
$199$	5520.96	1.96669	0.983344	+	0.181756i	$0.0581782\pi$
$200$	567.901i	0.200783i
$201$	− 1323.45i	− 0.464423i
$202$	− 2338.60i	− 0.814570i
$203$	− 5836.60i	− 2.01798i
$204$	−847.071	−0.290720
$205$	2382.11	0.811580
$206$	2302.21i	0.778654i
$207$	−1176.27	−0.394960
$208$	0	0
$209$	5645.18	1.86835
$210$	− 1297.15i	− 0.426248i
$211$	−4527.79	−1.47728	−0.738640	−	0.674100i	$-0.764532\pi$
$211$	−4527.79	−1.47728	−0.738640	+	0.674100i	$0.764532\pi$
$212$	−890.901	−0.288620
$213$	1035.15i	0.332991i
$214$	− 9.87129i	− 0.00315321i
$215$	949.679i	0.301244i
$216$	− 548.270i	− 0.172709i
$217$	−958.435	−0.299829
$218$	−795.245	−0.247068
$219$	− 2321.52i	− 0.716318i
$220$	2704.38	0.828770
$221$	0	0
$222$	−142.615	−0.0431157
$223$	− 4481.17i	− 1.34566i	−0.739798	−	0.672829i	$-0.765079\pi$
$223$	− 4481.17i	− 1.34566i	0.739798	−	0.672829i	$-0.234921\pi$
$224$	−5611.37	−1.67377
$225$	251.701	0.0745780
$226$	− 2115.32i	− 0.622607i
$227$	5759.75i	1.68409i	0.539408	+	0.842044i	$0.318648\pi$
$227$	5759.75i	1.68409i	−0.539408	+	0.842044i	$0.681352\pi$
$228$	2111.47i	0.613312i
$229$	− 4635.08i	− 1.33753i	−0.743473	−	0.668766i	$-0.766823\pi$
$229$	− 4635.08i	− 1.33753i	0.743473	−	0.668766i	$-0.233177\pi$
$230$	−1887.51	−0.541126
$231$	4214.81	1.20050
$232$	− 3958.63i	− 1.12024i
$233$	−5886.33	−1.65505	−0.827524	−	0.561431i	$-0.810251\pi$
$233$	−5886.33	−1.65505	−0.827524	+	0.561431i	$0.810251\pi$
$234$	0	0
$235$	5309.59	1.47387
$236$	1917.76i	0.528965i
$237$	450.996	0.123609
$238$	−2118.42	−0.576961
$239$	− 2135.84i	− 0.578060i	−0.957320	−	0.289030i	$-0.906667\pi$
$239$	− 2135.84i	− 0.578060i	0.957320	−	0.289030i	$-0.0933327\pi$
$240$	503.361i	0.135382i
$241$	4692.92i	1.25435i	0.778880	+	0.627173i	$0.215788\pi$
$241$	4692.92i	1.25435i	−0.778880	+	0.627173i	$0.784212\pi$
$242$	− 1277.01i	− 0.339212i
$243$	−243.000	−0.0641500
$244$	−575.799	−0.151073
$245$	5451.09i	1.42146i
$246$	1063.62	0.275667
$247$	0	0
$248$	−650.051	−0.166445
$249$	1013.90i	0.258045i
$250$	2209.13	0.558871
$251$	3902.88	0.981464	0.490732	−	0.871310i	$-0.336729\pi$
$251$	3902.88	0.981464	0.490732	+	0.871310i	$0.336729\pi$
$252$	1576.47i	0.394080i
$253$	− 6133.06i	− 1.52404i
$254$	3654.76i	0.902834i
$255$	1426.21i	0.350246i
$256$	−3008.64	−0.734531
$257$	4130.83	1.00262	0.501312	−	0.865267i	$-0.332851\pi$
$257$	4130.83	1.00262	0.501312	+	0.865267i	$0.332851\pi$
$258$	424.035i	0.102323i
$259$	970.790	0.232903
$260$	0	0
$261$	−1754.51	−0.416098
$262$	1091.60i	0.257402i
$263$	6352.17	1.48932	0.744661	−	0.667443i	$-0.232611\pi$
$263$	6352.17	1.48932	0.744661	+	0.667443i	$0.232611\pi$
$264$	2858.67	0.666435
$265$	1500.01i	0.347716i
$266$	5280.52i	1.21718i
$267$	− 509.740i	− 0.116837i
$268$	2580.97i	0.588276i
$269$	181.524	0.0411439	0.0205719	−	0.999788i	$-0.493451\pi$
$269$	181.524	0.0411439	0.0205719	+	0.999788i	$0.493451\pi$
$270$	−389.931	−0.0878906
$271$	3460.35i	0.775651i	0.921733	+	0.387825i	$0.126774\pi$
$271$	3460.35i	0.775651i	−0.921733	+	0.387825i	$0.873226\pi$
$272$	822.053	0.183251
$273$	0	0
$274$	325.719	0.0718153
$275$	1312.36i	0.287776i
$276$	2293.95	0.500288
$277$	6437.94	1.39646	0.698228	−	0.715876i	$-0.253972\pi$
$277$	6437.94	1.39646	0.698228	+	0.715876i	$0.253972\pi$
$278$	− 1139.37i	− 0.245810i
$279$	288.110i	0.0618234i
$280$	5988.77i	1.27820i
$281$	− 2974.26i	− 0.631421i	−0.948855	−	0.315711i	$-0.897757\pi$
$281$	− 2974.26i	− 0.631421i	0.948855	−	0.315711i	$-0.102243\pi$
$282$	2370.75	0.500625
$283$	−3035.72	−0.637649	−0.318825	−	0.947814i	$-0.603288\pi$
$283$	−3035.72	−0.637649	−0.318825	+	0.947814i	$0.603288\pi$
$284$	− 2018.72i	− 0.421793i
$285$	3555.07	0.738891
$286$	0	0
$287$	−7240.15	−1.48910
$288$	1686.81i	0.345125i
$289$	−2583.81	−0.525914
$290$	−2815.39	−0.570087
$291$	642.604i	0.129451i
$292$	4527.38i	0.907346i
$293$	− 1955.74i	− 0.389952i	−0.980808	−	0.194976i	$-0.937537\pi$
$293$	− 1955.74i	− 0.389952i	0.980808	−	0.194976i	$-0.0624628\pi$
$294$	2433.93i	0.482823i
$295$	3228.93	0.637274
$296$	658.431	0.129292
$297$	− 1267.00i	− 0.247537i
$298$	2607.16	0.506808
$299$	0	0
$300$	−490.862	−0.0944665
$301$	− 2886.44i	− 0.552729i
$302$	−1709.47	−0.325725
$303$	4785.34	0.907295
$304$	− 2049.10i	− 0.386593i
$305$	969.471i	0.182006i
$306$	636.808i	0.118967i
$307$	− 1027.56i	− 0.191029i	−0.995428	−	0.0955147i	$-0.969550\pi$
$307$	− 1027.56i	− 0.191029i	0.995428	−	0.0955147i	$-0.0304497\pi$
$308$	−8219.66	−1.52064
$309$	−4710.89	−0.867291
$310$	462.318i	0.0847029i
$311$	−3405.61	−0.620947	−0.310474	−	0.950582i	$-0.600488\pi$
$311$	−3405.61	−0.620947	−0.310474	+	0.950582i	$0.600488\pi$
$312$	0	0
$313$	4813.20	0.869196	0.434598	−	0.900625i	$-0.356890\pi$
$313$	4813.20	0.869196	0.434598	+	0.900625i	$0.356890\pi$
$314$	758.895i	0.136391i
$315$	2654.29	0.474770
$316$	−879.525	−0.156573
$317$	− 1141.33i	− 0.202219i	−0.994875	−	0.101110i	$-0.967761\pi$
$317$	− 1141.33i	− 0.202219i	0.994875	−	0.101110i	$-0.0322393\pi$
$318$	669.759i	0.118108i
$319$	− 9147.98i	− 1.60561i
$320$	1364.45i	0.238359i
$321$	20.1991	0.00351215
$322$	5736.88	0.992870
$323$	− 5805.88i	− 1.00015i
$324$	473.894	0.0812576
$325$	0	0
$326$	−894.597	−0.151985
$327$	− 1627.27i	− 0.275193i
$328$	−4910.58	−0.826650
$329$	−16137.9	−2.70429
$330$	− 2033.09i	− 0.339145i
$331$	− 7652.57i	− 1.27076i	−0.772198	−	0.635382i	$-0.780843\pi$
$331$	− 7652.57i	− 1.27076i	0.772198	−	0.635382i	$-0.219157\pi$
$332$	− 1977.29i	− 0.326860i
$333$	− 291.825i	− 0.0480237i
$334$	−4373.40	−0.716472
$335$	4345.57	0.708729
$336$	− 1529.91i	− 0.248403i
$337$	2503.69	0.404702	0.202351	−	0.979313i	$-0.435142\pi$
$337$	2503.69	0.404702	0.202351	+	0.979313i	$0.435142\pi$
$338$	0	0
$339$	4328.47	0.693481
$340$	− 2781.37i	− 0.443650i
$341$	−1502.20	−0.238559
$342$	1587.35	0.250977
$343$	− 6298.69i	− 0.991537i
$344$	− 1957.71i	− 0.306838i
$345$	− 3862.31i	− 0.602724i
$346$	− 1434.16i	− 0.222835i
$347$	4496.12	0.695574	0.347787	−	0.937574i	$-0.386933\pi$
$347$	4496.12	0.695574	0.347787	+	0.937574i	$0.386933\pi$
$348$	3421.62	0.527063
$349$	3577.61i	0.548726i	0.961626	+	0.274363i	$0.0884670\pi$
$349$	3577.61i	0.548726i	−0.961626	+	0.274363i	$0.911533\pi$
$350$	−1227.59	−0.187478
$351$	0	0
$352$	−8794.96	−1.33174
$353$	8045.17i	1.21303i	0.795070	+	0.606517i	$0.207434\pi$
$353$	8045.17i	1.21303i	−0.795070	+	0.606517i	$0.792566\pi$
$354$	1441.73	0.216461
$355$	−3398.92	−0.508157
$356$	994.086i	0.147996i
$357$	− 4334.80i	− 0.642639i
$358$	− 2716.60i	− 0.401052i
$359$	− 2172.90i	− 0.319447i	−0.987162	−	0.159724i	$-0.948940\pi$
$359$	− 2172.90i	− 0.319447i	0.987162	−	0.159724i	$-0.0510603\pi$
$360$	1800.25	0.263560
$361$	−7613.14	−1.10995
$362$	− 1250.26i	− 0.181525i
$363$	2613.08	0.377826
$364$	0	0
$365$	7622.74	1.09313
$366$	432.872i	0.0618213i
$367$	7662.76	1.08990	0.544949	−	0.838469i	$-0.316549\pi$
$367$	7662.76	1.08990	0.544949	+	0.838469i	$0.316549\pi$
$368$	−2226.20	−0.315349
$369$	2176.43i	0.307047i
$370$	− 468.278i	− 0.0657962i
$371$	− 4559.10i	− 0.637996i
$372$	− 561.868i	− 0.0783105i
$373$	10542.5	1.46346	0.731732	−	0.681593i	$-0.238712\pi$
$373$	10542.5	1.46346	0.731732	+	0.681593i	$0.238712\pi$
$374$	−3320.30	−0.459060
$375$	4520.42i	0.622489i
$376$	−10945.4	−1.50124
$377$	0	0
$378$	1185.15	0.161264
$379$	− 5475.54i	− 0.742110i	−0.928611	−	0.371055i	$-0.878996\pi$
$379$	− 5475.54i	− 0.742110i	0.928611	−	0.371055i	$-0.121004\pi$
$380$	−6933.03	−0.935939
$381$	−7478.52	−1.00561
$382$	6511.39i	0.872124i
$383$	− 808.085i	− 0.107810i	−0.998546	−	0.0539050i	$-0.982833\pi$
$383$	− 808.085i	− 0.107810i	0.998546	−	0.0539050i	$-0.0171668\pi$
$384$	− 3888.92i	− 0.516811i
$385$	13839.4i	1.83200i
$386$	3639.49	0.479910
$387$	−867.679	−0.113970
$388$	− 1253.19i	− 0.163972i
$389$	7060.26	0.920230	0.460115	−	0.887859i	$-0.347808\pi$
$389$	7060.26	0.920230	0.460115	+	0.887859i	$0.347808\pi$
$390$	0	0
$391$	−6307.65	−0.815836
$392$	− 11237.1i	− 1.44786i
$393$	−2233.68	−0.286703
$394$	1847.63	0.236249
$395$	1480.85i	0.188633i
$396$	2470.87i	0.313550i
$397$	− 1419.89i	− 0.179502i	−0.995964	−	0.0897511i	$-0.971393\pi$
$397$	− 1419.89i	− 0.179502i	0.995964	−	0.0897511i	$-0.0286072\pi$
$398$	8094.29i	1.01942i
$399$	−10805.2	−1.35573
$400$	476.365	0.0595456
$401$	10670.7i	1.32886i	0.747353	+	0.664428i	$0.231325\pi$
$401$	10670.7i	1.32886i	−0.747353	+	0.664428i	$0.768675\pi$
$402$	1940.31	0.240732
$403$	0	0
$404$	−9332.28	−1.14925
$405$	− 797.894i	− 0.0978955i
$406$	8557.05	1.04601
$407$	1521.56	0.185310
$408$	− 2940.05i	− 0.356750i
$409$	− 6351.56i	− 0.767883i	−0.923357	−	0.383942i	$-0.874566\pi$
$409$	− 6351.56i	− 0.767883i	0.923357	−	0.383942i	$-0.125434\pi$
$410$	3492.42i	0.420678i
$411$	666.500i	0.0799903i
$412$	9187.09	1.09858
$413$	−9813.97	−1.16928
$414$	− 1724.54i	− 0.204726i
$415$	−3329.15	−0.393787
$416$	0	0
$417$	2331.44	0.273791
$418$	8276.40i	0.968450i
$419$	5617.87	0.655015	0.327507	−	0.944849i	$-0.393791\pi$
$419$	5617.87	0.655015	0.327507	+	0.944849i	$0.393791\pi$
$420$	−5176.35	−0.601382
$421$	1518.29i	0.175765i	0.996131	+	0.0878825i	$0.0280100\pi$
$421$	1518.29i	0.175765i	−0.996131	+	0.0878825i	$0.971990\pi$
$422$	− 6638.20i	− 0.765741i
$423$	4851.13i	0.557613i
$424$	− 3092.17i	− 0.354173i
$425$	1349.72	0.154050
$426$	−1517.63	−0.172604
$427$	− 2946.60i	− 0.333948i
$428$	−39.3918	−0.00444878
$429$	0	0
$430$	−1392.33	−0.156149
$431$	4970.13i	0.555459i	0.960659	+	0.277730i	$0.0895819\pi$
$431$	4970.13i	0.555459i	−0.960659	+	0.277730i	$0.910418\pi$
$432$	−459.898	−0.0512196
$433$	3298.71	0.366111	0.183055	−	0.983103i	$-0.441401\pi$
$433$	3298.71	0.366111	0.183055	+	0.983103i	$0.441401\pi$
$434$	− 1405.16i	− 0.155415i
$435$	− 5760.97i	− 0.634982i
$436$	3173.46i	0.348581i
$437$	15722.9i	1.72112i
$438$	3403.58	0.371300
$439$	6048.47	0.657581	0.328790	−	0.944403i	$-0.393359\pi$
$439$	6048.47	0.657581	0.328790	+	0.944403i	$0.393359\pi$
$440$	9386.47i	1.01701i
$441$	−4980.42	−0.537784
$442$	0	0
$443$	6822.62	0.731722	0.365861	−	0.930670i	$-0.380775\pi$
$443$	6822.62	0.731722	0.365861	+	0.930670i	$0.380775\pi$
$444$	569.111i	0.0608307i
$445$	1673.74	0.178299
$446$	6569.86	0.697515
$447$	5334.88i	0.564500i
$448$	− 4147.09i	− 0.437347i
$449$	410.410i	0.0431369i	0.999767	+	0.0215684i	$0.00686598\pi$
$449$	410.410i	0.0431369i	−0.999767	+	0.0215684i	$0.993134\pi$
$450$	369.019i	0.0386571i
$451$	−11347.8	−1.18481
$452$	−8441.30	−0.878419
$453$	− 3497.99i	− 0.362804i
$454$	−8444.38	−0.872939
$455$	0	0
$456$	−7328.56	−0.752612
$457$	16642.8i	1.70354i	0.523915	+	0.851771i	$0.324471\pi$
$457$	16642.8i	1.70354i	−0.523915	+	0.851771i	$0.675529\pi$
$458$	6795.50	0.693303
$459$	−1303.06	−0.132509
$460$	7532.21i	0.763459i
$461$	11729.7i	1.18505i	0.805553	+	0.592523i	$0.201868\pi$
$461$	11729.7i	1.18505i	−0.805553	+	0.592523i	$0.798132\pi$
$462$	6179.35i	0.622271i
$463$	− 3564.93i	− 0.357832i	−0.983864	−	0.178916i	$-0.942741\pi$
$463$	− 3564.93i	− 0.357832i	0.983864	−	0.178916i	$-0.0572591\pi$
$464$	−3320.56	−0.332227
$465$	−946.015	−0.0943450
$466$	− 8629.95i	− 0.857886i
$467$	−1134.81	−0.112447	−0.0562233	−	0.998418i	$-0.517906\pi$
$467$	−1134.81	−0.112447	−0.0562233	+	0.998418i	$0.517906\pi$
$468$	0	0
$469$	−13207.9	−1.30039
$470$	7784.40i	0.763974i
$471$	−1552.88	−0.151917
$472$	−6656.25	−0.649107
$473$	− 4524.05i	− 0.439780i
$474$	661.207i	0.0640722i
$475$	− 3364.40i	− 0.324988i
$476$	8453.65i	0.814018i
$477$	−1370.49	−0.131552
$478$	3131.36	0.299635
$479$	− 19372.6i	− 1.84793i	−0.382481	−	0.923963i	$-0.624930\pi$
$479$	− 19372.6i	− 1.84793i	0.382481	−	0.923963i	$-0.375070\pi$
$480$	−5538.66	−0.526675
$481$	0	0
$482$	−6880.29	−0.650184
$483$	11739.1i	1.10589i
$484$	−5095.97	−0.478585
$485$	−2110.00	−0.197547
$486$	− 356.263i	− 0.0332519i
$487$	9045.95i	0.841707i	0.907129	+	0.420853i	$0.138269\pi$
$487$	9045.95i	0.841707i	−0.907129	+	0.420853i	$0.861731\pi$
$488$	− 1998.51i	− 0.185385i
$489$	− 1830.56i	− 0.169286i
$490$	−7991.86	−0.736807
$491$	−14403.3	−1.32385	−0.661927	−	0.749568i	$-0.730261\pi$
$491$	−14403.3	−1.32385	−0.661927	+	0.749568i	$0.730261\pi$
$492$	− 4244.43i	− 0.388930i
$493$	−9408.40	−0.859499
$494$	0	0
$495$	4160.20	0.377751
$496$	545.273i	0.0493619i
$497$	10330.6	0.932378
$498$	−1486.48	−0.133756
$499$	9319.75i	0.836091i	0.908426	+	0.418045i	$0.137285\pi$
$499$	9319.75i	0.836091i	−0.908426	+	0.418045i	$0.862715\pi$
$500$	− 8815.64i	− 0.788495i
$501$	− 8949.03i	− 0.798030i
$502$	5722.02i	0.508737i
$503$	2745.98	0.243414	0.121707	−	0.992566i	$-0.461163\pi$
$503$	2745.98	0.243414	0.121707	+	0.992566i	$0.461163\pi$
$504$	−5471.67	−0.483586
$505$	15712.7i	1.38457i
$506$	8991.69	0.789979
$507$	0	0
$508$	14584.5	1.27378
$509$	1182.11i	0.102939i	0.998675	+	0.0514697i	$0.0163905\pi$
$509$	1182.11i	0.102939i	−0.998675	+	0.0514697i	$0.983609\pi$
$510$	−2090.97	−0.181548
$511$	−23168.4	−2.00570
$512$	5959.48i	0.514403i
$513$	3248.11i	0.279547i
$514$	6056.22i	0.519705i
$515$	− 15468.3i	− 1.32352i
$516$	1692.13	0.144364
$517$	−25293.7	−2.15167
$518$	1423.28i	0.120724i
$519$	2934.64	0.248201
$520$	0	0
$521$	10858.8	0.913115	0.456558	−	0.889694i	$-0.349082\pi$
$521$	10858.8	0.913115	0.456558	+	0.889694i	$0.349082\pi$
$522$	− 2572.29i	− 0.215682i
$523$	−10161.7	−0.849602	−0.424801	−	0.905287i	$-0.639656\pi$
$523$	−10161.7	−0.849602	−0.424801	+	0.905287i	$0.639656\pi$
$524$	4356.09	0.363162
$525$	− 2511.94i	− 0.208819i
$526$	9312.93i	0.771983i
$527$	1544.96i	0.127703i
$528$	− 2397.89i	− 0.197642i
$529$	4914.73	0.403939
$530$	−2199.16	−0.180237
$531$	2950.13i	0.241101i
$532$	21072.1	1.71728
$533$	0	0
$534$	747.331	0.0605621
$535$	66.3239i	0.00535969i
$536$	−8958.14	−0.721889
$537$	5558.82	0.446706
$538$	266.132i	0.0213267i
$539$	− 25967.8i	− 2.07516i
$540$	1556.04i	0.124002i
$541$	− 9573.04i	− 0.760771i	−0.924828	−	0.380386i	$-0.875791\pi$
$541$	− 9573.04i	− 0.760771i	0.924828	−	0.380386i	$-0.124209\pi$
$542$	−5073.23	−0.402055
$543$	2558.33	0.202189
$544$	9045.34i	0.712896i
$545$	5343.15	0.419955
$546$	0	0
$547$	15958.9	1.24745	0.623724	−	0.781645i	$-0.285619\pi$
$547$	15958.9	1.24745	0.623724	+	0.781645i	$0.285619\pi$
$548$	− 1299.80i	− 0.101322i
$549$	−885.762	−0.0688586
$550$	−1924.05	−0.149167
$551$	23452.0i	1.81323i
$552$	7961.93i	0.613917i
$553$	− 4500.89i	− 0.346107i
$554$	9438.67i	0.723846i
$555$	958.210	0.0732861
$556$	−4546.72	−0.346806
$557$	− 2145.09i	− 0.163178i	−0.996666	−	0.0815892i	$-0.974000\pi$
$557$	− 2145.09i	− 0.163178i	0.996666	−	0.0815892i	$-0.0259996\pi$
$558$	−422.399	−0.0320459
$559$	0	0
$560$	5023.47	0.379072
$561$	− 6794.14i	− 0.511317i
$562$	4360.57	0.327294
$563$	22318.6	1.67072	0.835362	−	0.549701i	$-0.185258\pi$
$563$	22318.6	1.67072	0.835362	+	0.549701i	$0.185258\pi$
$564$	− 9460.59i	− 0.706317i
$565$	14212.6i	1.05828i
$566$	− 4450.67i	− 0.330523i
$567$	2425.11i	0.179621i
$568$	7006.67	0.517594
$569$	19753.0	1.45534	0.727671	−	0.685927i	$-0.240603\pi$
$569$	19753.0	1.45534	0.727671	+	0.685927i	$0.240603\pi$
$570$	5212.09i	0.383001i
$571$	10640.6	0.779850	0.389925	−	0.920847i	$-0.372501\pi$
$571$	10640.6	0.779850	0.389925	+	0.920847i	$0.372501\pi$
$572$	0	0
$573$	−13323.9	−0.971401
$574$	− 10614.8i	− 0.771870i
$575$	−3655.17	−0.265098
$576$	−1246.64	−0.0901791
$577$	− 6547.89i	− 0.472430i	−0.971701	−	0.236215i	$-0.924093\pi$
$577$	− 6547.89i	− 0.472430i	0.971701	−	0.236215i	$-0.0759070\pi$
$578$	− 3788.13i	− 0.272605i
$579$	7447.29i	0.534540i
$580$	11234.9i	0.804319i
$581$	10118.6	0.722529
$582$	−942.123	−0.0671001
$583$	− 7145.69i	− 0.507623i
$584$	−15713.8	−1.11343
$585$	0	0
$586$	2867.32	0.202130
$587$	− 5900.33i	− 0.414877i	−0.978248	−	0.207439i	$-0.933487\pi$
$587$	− 5900.33i	− 0.414877i	0.978248	−	0.207439i	$-0.0665127\pi$
$588$	9712.72	0.681201
$589$	3851.08	0.269408
$590$	4733.94i	0.330328i
$591$	3780.69i	0.263142i
$592$	− 552.303i	− 0.0383437i
$593$	− 15261.5i	− 1.05686i	−0.848978	−	0.528428i	$-0.822782\pi$
$593$	− 15261.5i	− 1.05686i	0.848978	−	0.528428i	$-0.177218\pi$
$594$	1857.54	0.128310
$595$	14233.4	0.980693
$596$	− 10404.0i	− 0.715041i
$597$	−16562.9	−1.13547
$598$	0	0
$599$	−18900.6	−1.28925	−0.644623	−	0.764501i	$-0.722986\pi$
$599$	−18900.6	−1.28925	−0.644623	+	0.764501i	$0.722986\pi$
$600$	− 1703.70i	− 0.115922i
$601$	−18507.0	−1.25610	−0.628049	−	0.778174i	$-0.716146\pi$
$601$	−18507.0	−1.25610	−0.628049	+	0.778174i	$0.716146\pi$
$602$	4231.81	0.286505
$603$	3970.36i	0.268135i
$604$	6821.72i	0.459556i
$605$	8580.08i	0.576578i
$606$	7015.79i	0.470292i
$607$	−6408.50	−0.428522	−0.214261	−	0.976776i	$-0.568734\pi$
$607$	−6408.50	−0.428522	−0.214261	+	0.976776i	$0.568734\pi$
$608$	22547.0	1.50395
$609$	17509.8i	1.16508i
$610$	−1421.34	−0.0943418
$611$	0	0
$612$	2541.21	0.167847
$613$	3507.26i	0.231088i	0.993302	+	0.115544i	$0.0368612\pi$
$613$	3507.26i	0.231088i	−0.993302	+	0.115544i	$0.963139\pi$
$614$	1506.51	0.0990192
$615$	−7146.33	−0.468566
$616$	− 28529.1i	− 1.86602i
$617$	− 14507.8i	− 0.946614i	−0.880897	−	0.473307i	$-0.843060\pi$
$617$	− 14507.8i	− 0.946614i	0.880897	−	0.473307i	$-0.156940\pi$
$618$	− 6906.64i	− 0.449556i
$619$	4750.85i	0.308486i	0.988033	+	0.154243i	$0.0492939\pi$
$619$	4750.85i	0.308486i	−0.988033	+	0.154243i	$0.950706\pi$
$620$	1844.90	0.119505
$621$	3528.82	0.228030
$622$	− 4992.98i	− 0.321865i
$623$	−5087.14	−0.327146
$624$	0	0
$625$	−11347.0	−0.726209
$626$	7056.65i	0.450544i
$627$	−16935.5	−1.07869
$628$	3028.40	0.192431
$629$	− 1564.88i	− 0.0991986i
$630$	3891.46i	0.246095i
$631$	4825.19i	0.304418i	0.988348	+	0.152209i	$0.0486387\pi$
$631$	4825.19i	0.304418i	−0.988348	+	0.152209i	$0.951361\pi$
$632$	− 3052.69i	− 0.192135i
$633$	13583.4	0.852908
$634$	1673.30	0.104819
$635$	− 24555.8i	− 1.53460i
$636$	2672.70	0.166635
$637$	0	0
$638$	13411.9	0.832258
$639$	− 3105.44i	− 0.192252i
$640$	12769.3	0.788675
$641$	5911.89	0.364284	0.182142	−	0.983272i	$-0.441697\pi$
$641$	5911.89	0.364284	0.182142	+	0.983272i	$0.441697\pi$
$642$	29.6139i	0.00182051i
$643$	− 23408.3i	− 1.43567i	−0.696216	−	0.717833i	$-0.745134\pi$
$643$	− 23408.3i	− 1.43567i	0.696216	−	0.717833i	$-0.254866\pi$
$644$	− 22893.3i	− 1.40081i
$645$	− 2849.04i	− 0.173924i
$646$	8512.02	0.518422
$647$	16398.7	0.996442	0.498221	−	0.867050i	$-0.333987\pi$
$647$	16398.7	0.996442	0.498221	+	0.867050i	$0.333987\pi$
$648$	1644.81i	0.0997134i
$649$	−15381.9	−0.930342
$650$	0	0
$651$	2875.30	0.173106
$652$	3569.93i	0.214431i
$653$	27529.6	1.64979	0.824897	−	0.565283i	$-0.191233\pi$
$653$	27529.6	1.64979	0.824897	+	0.565283i	$0.191233\pi$
$654$	2385.74	0.142645
$655$	− 7334.34i	− 0.437521i
$656$	4119.07i	0.245157i
$657$	6964.55i	0.413566i
$658$	− 23659.8i	− 1.40175i
$659$	−24179.7	−1.42930	−0.714650	−	0.699482i	$-0.753414\pi$
$659$	−24179.7	−1.42930	−0.714650	+	0.699482i	$0.753414\pi$
$660$	−8113.14	−0.478490
$661$	4525.04i	0.266269i	0.991098	+	0.133134i	$0.0425042\pi$
$661$	4525.04i	0.266269i	−0.991098	+	0.133134i	$0.957496\pi$
$662$	11219.4	0.658695
$663$	0	0
$664$	6862.84	0.401099
$665$	− 35479.1i	− 2.06890i
$666$	427.844	0.0248928
$667$	25478.8	1.47908
$668$	17452.2i	1.01085i
$669$	13443.5i	0.776916i
$670$	6371.05i	0.367366i
$671$	− 4618.34i	− 0.265706i
$672$	16834.1	0.966354
$673$	−3287.18	−0.188279	−0.0941393	−	0.995559i	$-0.530010\pi$
$673$	−3287.18	−0.188279	−0.0941393	+	0.995559i	$0.530010\pi$
$674$	3670.66i	0.209775i
$675$	−755.102	−0.0430576
$676$	0	0
$677$	−9724.21	−0.552041	−0.276020	−	0.961152i	$-0.589016\pi$
$677$	−9724.21	−0.552041	−0.276020	+	0.961152i	$0.589016\pi$
$678$	6345.97i	0.359462i
$679$	6413.10	0.362463
$680$	9653.69	0.544415
$681$	− 17279.3i	− 0.972309i
$682$	− 2202.38i	− 0.123656i
$683$	14548.7i	0.815065i	0.913191	+	0.407532i	$0.133611\pi$
$683$	14548.7i	0.815065i	−0.913191	+	0.407532i	$0.866389\pi$
$684$	− 6334.40i	− 0.354096i
$685$	−2188.46	−0.122068
$686$	9234.52	0.513959
$687$	13905.2i	0.772224i
$688$	−1642.15	−0.0909979
$689$	0	0
$690$	5662.54	0.312419
$691$	− 6728.96i	− 0.370451i	−0.982696	−	0.185226i	$-0.940698\pi$
$691$	− 6728.96i	− 0.370451i	0.982696	−	0.185226i	$-0.0593016\pi$
$692$	−5723.07	−0.314391
$693$	−12644.4	−0.693106
$694$	6591.76i	0.360548i
$695$	7655.31i	0.417816i
$696$	11875.9i	0.646774i
$697$	11670.9i	0.634241i
$698$	−5245.15	−0.284429
$699$	17659.0	0.955542
$700$	4898.74i	0.264507i
$701$	29159.8	1.57111	0.785557	−	0.618789i	$-0.212376\pi$
$701$	29159.8	1.57111	0.785557	+	0.618789i	$0.212376\pi$
$702$	0	0
$703$	−3900.73	−0.209273
$704$	− 6499.92i	− 0.347976i
$705$	−15928.8	−0.850939
$706$	−11795.0	−0.628770
$707$	− 47757.0i	− 2.54044i
$708$	− 5753.29i	− 0.305398i
$709$	− 20489.0i	− 1.08531i	−0.839957	−	0.542653i	$-0.817420\pi$
$709$	− 20489.0i	− 1.08531i	0.839957	−	0.542653i	$-0.182580\pi$
$710$	− 4983.16i	− 0.263401i
$711$	−1352.99	−0.0713658
$712$	−3450.31	−0.181609
$713$	− 4183.91i	− 0.219760i
$714$	6355.26	0.333109
$715$	0	0
$716$	−10840.7	−0.565833
$717$	6407.53i	0.333743i
$718$	3185.70	0.165584
$719$	17990.1	0.933127	0.466563	−	0.884488i	$-0.345492\pi$
$719$	17990.1	0.933127	0.466563	+	0.884488i	$0.345492\pi$
$720$	− 1510.08i	− 0.0781631i
$721$	47014.0i	2.42842i
$722$	− 11161.6i	− 0.575337i
$723$	− 14078.8i	− 0.724197i
$724$	−4989.21	−0.256108
$725$	−5452.00	−0.279286
$726$	3831.04i	0.195844i
$727$	37652.7	1.92086	0.960428	−	0.278528i	$-0.0898465\pi$
$727$	37652.7	1.92086	0.960428	+	0.278528i	$0.0898465\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	11175.7i	0.566618i
$731$	−4652.84	−0.235419
$732$	1727.40	0.0872219
$733$	− 4524.26i	− 0.227977i	−0.993482	−	0.113989i	$-0.963637\pi$
$733$	− 4524.26i	− 0.227977i	0.993482	−	0.113989i	$-0.0363627\pi$
$734$	11234.4i	0.564944i
$735$	− 16353.3i	− 0.820680i
$736$	− 24495.6i	− 1.22679i
$737$	−20701.3	−1.03466
$738$	−3190.86	−0.159156
$739$	818.302i	0.0407331i	0.999793	+	0.0203665i	$0.00648332\pi$
$739$	818.302i	0.0407331i	−0.999793	+	0.0203665i	$0.993517\pi$
$740$	−1868.68	−0.0928300
$741$	0	0
$742$	6684.10	0.330702
$743$	− 39002.2i	− 1.92578i	−0.269901	−	0.962888i	$-0.586991\pi$
$743$	− 39002.2i	− 1.92578i	0.269901	−	0.962888i	$-0.413009\pi$
$744$	1950.15	0.0960969
$745$	−17517.2	−0.861449
$746$	15456.4i	0.758579i
$747$	− 3041.69i	− 0.148982i
$748$	13249.8i	0.647675i
$749$	− 201.584i	− 0.00983407i
$750$	−6627.39	−0.322664
$751$	22377.7	1.08732	0.543658	−	0.839307i	$-0.317039\pi$
$751$	22377.7	1.08732	0.543658	+	0.839307i	$0.317039\pi$
$752$	9181.18i	0.445217i
$753$	−11708.6	−0.566649
$754$	0	0
$755$	11485.7	0.553653
$756$	− 4729.40i	− 0.227522i
$757$	−34512.6	−1.65704	−0.828521	−	0.559958i	$-0.810817\pi$
$757$	−34512.6	−1.65704	−0.828521	+	0.559958i	$0.810817\pi$
$758$	8027.70	0.384669
$759$	18399.2i	0.879905i
$760$	− 24063.4i	− 1.14852i
$761$	19975.6i	0.951529i	0.879573	+	0.475764i	$0.157828\pi$
$761$	19975.6i	0.951529i	−0.879573	+	0.475764i	$0.842172\pi$
$762$	− 10964.3i	− 0.521251i
$763$	−16239.9	−0.770542
$764$	25984.0	1.23046
$765$	− 4278.63i	− 0.202215i
$766$	1184.73	0.0558827
$767$	0	0
$768$	9025.91	0.424081
$769$	33064.9i	1.55052i	0.631642	+	0.775260i	$0.282381\pi$
$769$	33064.9i	1.55052i	−0.631642	+	0.775260i	$0.717619\pi$
$770$	−20290.0	−0.949610
$771$	−12392.5	−0.578865
$772$	− 14523.6i	− 0.677091i
$773$	18564.0i	0.863779i	0.901927	+	0.431890i	$0.142153\pi$
$773$	18564.0i	0.863779i	−0.901927	+	0.431890i	$0.857847\pi$
$774$	− 1272.10i	− 0.0590761i
$775$	895.279i	0.0414960i
$776$	4349.64	0.201215
$777$	−2912.37	−0.134467
$778$	10351.1i	0.476997i
$779$	29091.6	1.33802
$780$	0	0
$781$	16191.7	0.741848
$782$	− 9247.66i	− 0.422885i
$783$	5263.54	0.240234
$784$	−9425.86	−0.429385
$785$	− 5098.91i	− 0.231832i
$786$	− 3274.81i	− 0.148611i
$787$	36436.8i	1.65036i	0.564871	+	0.825179i	$0.308926\pi$
$787$	36436.8i	1.65036i	−0.564871	+	0.825179i	$0.691074\pi$
$788$	− 7373.04i	− 0.333317i
$789$	−19056.5	−0.859861
$790$	−2171.08	−0.0977768
$791$	− 43197.5i	− 1.94175i
$792$	−8576.00	−0.384766
$793$	0	0
$794$	2081.71	0.0930441
$795$	− 4500.02i	− 0.200754i
$796$	32300.7	1.43827
$797$	−12365.6	−0.549576	−0.274788	−	0.961505i	$-0.588608\pi$
$797$	−12365.6	−0.549576	−0.274788	+	0.961505i	$0.588608\pi$
$798$	− 15841.6i	− 0.702738i
$799$	26013.7i	1.15181i
$800$	5241.61i	0.231649i
$801$	1529.22i	0.0674561i
$802$	−15644.4	−0.688806
$803$	−36313.0	−1.59584
$804$	− 7742.92i	− 0.339641i
$805$	−38545.4	−1.68763
$806$	0	0
$807$	−544.571	−0.0237544
$808$	− 32390.9i	− 1.41028i
$809$	−8093.75	−0.351744	−0.175872	−	0.984413i	$-0.556274\pi$
$809$	−8093.75	−0.351744	−0.175872	+	0.984413i	$0.556274\pi$
$810$	1169.79	0.0507437
$811$	− 15984.7i	− 0.692105i	−0.938215	−	0.346052i	$-0.887522\pi$
$811$	− 15984.7i	− 0.692105i	0.938215	−	0.346052i	$-0.112478\pi$
$812$	− 34147.3i	− 1.47578i
$813$	− 10381.1i	− 0.447822i
$814$	2230.77i	0.0960546i
$815$	6010.68	0.258337
$816$	−2466.16	−0.105800
$817$	11598.0i	0.496649i
$818$	9312.03	0.398029
$819$	0	0
$820$	13936.6	0.593523
$821$	26861.9i	1.14188i	0.820990	+	0.570942i	$0.193422\pi$
$821$	26861.9i	1.14188i	−0.820990	+	0.570942i	$0.806578\pi$
$822$	−977.156	−0.0414626
$823$	5205.94	0.220495	0.110248	−	0.993904i	$-0.464836\pi$
$823$	5205.94	0.220495	0.110248	+	0.993904i	$0.464836\pi$
$824$	31886.9i	1.34810i
$825$	− 3937.08i	− 0.166147i
$826$	− 14388.3i	− 0.606092i
$827$	46621.5i	1.96032i	0.198200	+	0.980162i	$0.436491\pi$
$827$	46621.5i	1.96032i	−0.198200	+	0.980162i	$0.563509\pi$
$828$	−6881.84	−0.288841
$829$	41829.7	1.75248	0.876241	−	0.481874i	$-0.160044\pi$
$829$	41829.7	1.75248	0.876241	+	0.481874i	$0.160044\pi$
$830$	− 4880.87i	− 0.204118i
$831$	−19313.8	−0.806244
$832$	0	0
$833$	−26707.0	−1.11086
$834$	3418.12i	0.141918i
$835$	29384.3	1.21783
$836$	33027.4	1.36636
$837$	− 864.331i	− 0.0356937i
$838$	8236.37i	0.339524i
$839$	12685.1i	0.521975i	0.965342	+	0.260988i	$0.0840482\pi$
$839$	12685.1i	0.521975i	−0.965342	+	0.260988i	$0.915952\pi$
$840$	− 17966.3i	− 0.737972i
$841$	13614.9	0.558238
$842$	−2225.97	−0.0911069
$843$	8922.78i	0.364551i
$844$	−26490.1	−1.08036
$845$	0	0
$846$	−7112.25	−0.289036
$847$	− 26078.2i	− 1.05792i
$848$	−2593.77	−0.105036
$849$	9107.16	0.368147
$850$	1978.83i	0.0798509i
$851$	4237.85i	0.170707i
$852$	6056.17i	0.243522i
$853$	− 37493.3i	− 1.50498i	−0.658606	−	0.752488i	$-0.728854\pi$
$853$	− 37493.3i	− 1.50498i	0.658606	−	0.752488i	$-0.271146\pi$
$854$	4320.01	0.173100
$855$	−10665.2	−0.426599
$856$	− 136.723i	− 0.00545921i
$857$	11826.3	0.471386	0.235693	−	0.971828i	$-0.424264\pi$
$857$	11826.3	0.471386	0.235693	+	0.971828i	$0.424264\pi$
$858$	0	0
$859$	−36498.7	−1.44973	−0.724866	−	0.688890i	$-0.758099\pi$
$859$	−36498.7	−1.44973	−0.724866	+	0.688890i	$0.758099\pi$
$860$	5556.14i	0.220306i
$861$	21720.5	0.859734
$862$	−7286.72	−0.287920
$863$	− 2292.79i	− 0.0904372i	−0.998977	−	0.0452186i	$-0.985602\pi$
$863$	− 2292.79i	− 0.0904372i	0.998977	−	0.0452186i	$-0.0143984\pi$
$864$	− 5060.42i	− 0.199258i
$865$	9635.92i	0.378764i
$866$	4836.24i	0.189772i
$867$	7751.44	0.303636
$868$	−5607.37	−0.219270
$869$	− 7054.45i	− 0.275381i
$870$	8446.16	0.329140
$871$	0	0
$872$	−11014.6	−0.427753
$873$	− 1927.81i	− 0.0747383i
$874$	−23051.3	−0.892132
$875$	45113.2	1.74298
$876$	− 13582.1i	− 0.523856i
$877$	20800.3i	0.800885i	0.916322	+	0.400442i	$0.131144\pi$
$877$	20800.3i	0.800885i	−0.916322	+	0.400442i	$0.868856\pi$
$878$	8867.68i	0.340854i
$879$	5867.23i	0.225139i
$880$	7873.52	0.301610
$881$	19255.9	0.736377	0.368188	−	0.929751i	$-0.379978\pi$
$881$	19255.9	0.736377	0.368188	+	0.929751i	$0.379978\pi$
$882$	− 7301.80i	− 0.278758i
$883$	−1744.49	−0.0664857	−0.0332429	−	0.999447i	$-0.510583\pi$
$883$	−1744.49	−0.0664857	−0.0332429	+	0.999447i	$0.510583\pi$
$884$	0	0
$885$	−9686.80	−0.367930
$886$	10002.7i	0.379284i
$887$	2970.70	0.112454	0.0562268	−	0.998418i	$-0.482093\pi$
$887$	2970.70	0.112454	0.0562268	+	0.998418i	$0.482093\pi$
$888$	−1975.29	−0.0746470
$889$	74634.6i	2.81571i
$890$	2453.87i	0.0924202i
$891$	3800.99i	0.142916i
$892$	− 26217.3i	− 0.984104i
$893$	64843.6	2.42991
$894$	−7821.48	−0.292606
$895$	18252.5i	0.681691i
$896$	−38810.9	−1.44708
$897$	0	0
$898$	−601.703	−0.0223598
$899$	− 6240.66i	− 0.231521i
$900$	1472.59	0.0545402
$901$	−7349.12	−0.271736
$902$	− 16637.1i	− 0.614140i
$903$	8659.32i	0.319119i
$904$	− 29298.4i	− 1.07793i
$905$	8400.32i	0.308548i
$906$	5128.41	0.188058
$907$	−33003.3	−1.20822	−0.604111	−	0.796900i	$-0.706472\pi$
$907$	−33003.3	−1.20822	−0.604111	+	0.796900i	$0.706472\pi$
$908$	33697.7i	1.23160i
$909$	−14356.0	−0.523827
$910$	0	0
$911$	14977.0	0.544686	0.272343	−	0.962200i	$-0.412201\pi$
$911$	14977.0	0.544686	0.272343	+	0.962200i	$0.412201\pi$
$912$	6147.31i	0.223199i
$913$	15859.3	0.574881
$914$	−24400.1	−0.883022
$915$	− 2908.41i	− 0.105081i
$916$	− 27117.7i	− 0.978161i
$917$	22291.9i	0.802773i
$918$	− 1910.42i	− 0.0686856i
$919$	11425.4	0.410107	0.205054	−	0.978751i	$-0.434263\pi$
$919$	11425.4	0.410107	0.205054	+	0.978751i	$0.434263\pi$
$920$	−26143.1	−0.936862
$921$	3082.69i	0.110291i
$922$	−17196.9	−0.614263
$923$	0	0
$924$	24659.0	0.877944
$925$	− 906.820i	− 0.0322336i
$926$	5226.55	0.185481
$927$	14132.7	0.500731
$928$	− 36537.3i	− 1.29245i
$929$	− 11954.2i	− 0.422179i	−0.977467	−	0.211090i	$-0.932299\pi$
$929$	− 11954.2i	− 0.422179i	0.977467	−	0.211090i	$-0.0677012\pi$
$930$	− 1386.95i	− 0.0489033i
$931$	66571.7i	2.34350i
$932$	−34438.2	−1.21037
$933$	10216.8	0.358504
$934$	− 1663.74i	− 0.0582861i
$935$	22308.7	0.780290
$936$	0	0
$937$	42546.4	1.48338	0.741692	−	0.670740i	$-0.234024\pi$
$937$	42546.4	1.48338	0.741692	+	0.670740i	$0.234024\pi$
$938$	− 19364.1i	− 0.674051i
$939$	−14439.6	−0.501831
$940$	31064.0	1.07787
$941$	− 20665.1i	− 0.715903i	−0.933740	−	0.357951i	$-0.883475\pi$
$941$	− 20665.1i	− 0.715903i	0.933740	−	0.357951i	$-0.116525\pi$
$942$	− 2276.68i	− 0.0787456i
$943$	− 31605.9i	− 1.09144i
$944$	5583.37i	0.192503i
$945$	−7962.88	−0.274108
$946$	6632.72	0.227958
$947$	− 9493.73i	− 0.325771i	−0.986645	−	0.162885i	$-0.947920\pi$
$947$	− 9493.73i	− 0.325771i	0.986645	−	0.162885i	$-0.0520801\pi$
$948$	2638.58	0.0903976
$949$	0	0
$950$	4932.56	0.168456
$951$	3423.99i	0.116751i
$952$	−29341.3	−0.998904
$953$	−53334.1	−1.81287	−0.906433	−	0.422349i	$-0.861206\pi$
$953$	−53334.1	−1.81287	−0.906433	+	0.422349i	$0.861206\pi$
$954$	− 2009.28i	− 0.0681894i
$955$	− 43749.2i	− 1.48240i
$956$	− 12495.9i	− 0.422746i
$957$	27443.9i	0.926997i
$958$	28402.2	0.957864
$959$	6651.58	0.223974
$960$	− 4093.35i	− 0.137617i
$961$	28766.2	0.965601
$962$	0	0
$963$	−60.5972	−0.00202774
$964$	27456.1i	0.917326i
$965$	−24453.3	−0.815730
$966$	−17210.6	−0.573233
$967$	42110.1i	1.40038i	0.713956	+	0.700191i	$0.246902\pi$
$967$	42110.1i	1.40038i	−0.713956	+	0.700191i	$0.753098\pi$
$968$	− 17687.3i	− 0.587285i
$969$	17417.7i	0.577436i
$970$	− 3093.47i	− 0.102397i
$971$	−13827.7	−0.457006	−0.228503	−	0.973543i	$-0.573383\pi$
$971$	−13827.7	−0.457006	−0.228503	+	0.973543i	$0.573383\pi$
$972$	−1421.68	−0.0469141
$973$	− 23267.4i	− 0.766618i
$974$	−13262.3	−0.436295
$975$	0	0
$976$	−1676.38	−0.0549791
$977$	− 1133.89i	− 0.0371302i	−0.999828	−	0.0185651i	$-0.994090\pi$
$977$	− 1133.89i	− 0.0371302i	0.999828	−	0.0185651i	$-0.00590979\pi$
$978$	2683.79	0.0877487
$979$	−7973.31	−0.260294
$980$	31891.9i	1.03954i
$981$	4881.80i	0.158883i
$982$	− 21116.7i	− 0.686214i
$983$	26250.2i	0.851729i	0.904787	+	0.425865i	$0.140030\pi$
$983$	26250.2i	0.851729i	−0.904787	+	0.425865i	$0.859970\pi$
$984$	14731.7	0.477267
$985$	−12414.0	−0.401565
$986$	− 13793.7i	− 0.445517i
$987$	48413.7	1.56132
$988$	0	0
$989$	12600.3	0.405124
$990$	6099.27i	0.195806i
$991$	−29360.4	−0.941133	−0.470566	−	0.882365i	$-0.655950\pi$
$991$	−29360.4	−0.941133	−0.470566	+	0.882365i	$0.655950\pi$
$992$	−5999.84	−0.192031
$993$	22957.7i	0.733676i
$994$	15145.7i	0.483293i
$995$	− 54384.5i	− 1.73277i
$996$	5931.86i	0.188713i
$997$	16835.9	0.534803	0.267401	−	0.963585i	$-0.413835\pi$
$997$	16835.9	0.534803	0.267401	+	0.963585i	$0.413835\pi$
$998$	−13663.7	−0.433384
$999$	875.474i	0.0277265i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.h.337.5		8
13.5	odd	4		507.4.a.i.1.3		4
13.7	odd	12		39.4.e.c.16.3	✓	8
13.8	odd	4		507.4.a.m.1.2		4
13.11	odd	12		39.4.e.c.22.3	yes	8
13.12	even	2	inner	507.4.b.h.337.4		8
39.5	even	4		1521.4.a.bb.1.2		4
39.8	even	4		1521.4.a.v.1.3		4
39.11	even	12		117.4.g.e.100.2		8
39.20	even	12		117.4.g.e.55.2		8
52.7	even	12		624.4.q.i.289.4		8
52.11	even	12		624.4.q.i.529.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.e.c.16.3	✓	8	13.7	odd	12
39.4.e.c.22.3	yes	8	13.11	odd	12
117.4.g.e.55.2		8	39.20	even	12
117.4.g.e.100.2		8	39.11	even	12
507.4.a.i.1.3		4	13.5	odd	4
507.4.a.m.1.2		4	13.8	odd	4
507.4.b.h.337.4		8	13.12	even	2	inner
507.4.b.h.337.5		8	1.1	even	1	trivial
624.4.q.i.289.4		8	52.7	even	12
624.4.q.i.529.4		8	52.11	even	12
1521.4.a.v.1.3		4	39.8	even	4
1521.4.a.bb.1.2		4	39.5	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)

\(f(q)\)	\(=\)	\(q+1.46610i q^{2} -3.00000 q^{3} +5.85055 q^{4} -9.85055i q^{5} -4.39830i q^{6} +29.9396i q^{7} +20.3063i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+1.46610i q^{2} -3.00000 q^{3} +5.85055 q^{4} -9.85055i q^{5} -4.39830i q^{6} +29.9396i q^{7} +20.3063i q^{8} +9.00000 q^{9} +14.4419 q^{10} +46.9257i q^{11} -17.5516 q^{12} -43.8945 q^{14} +29.5516i q^{15} +17.0333 q^{16} +48.2616 q^{17} +13.1949i q^{18} -120.300i q^{19} -57.6311i q^{20} -89.8188i q^{21} -68.7979 q^{22} -130.697 q^{23} -60.9189i q^{24} +27.9667 q^{25} -27.0000 q^{27} +175.163i q^{28} -194.946 q^{29} -43.3257 q^{30} +32.0123i q^{31} +187.423i q^{32} -140.777i q^{33} +70.7565i q^{34} +294.921 q^{35} +52.6549 q^{36} -32.4250i q^{37} +176.372 q^{38} +200.028 q^{40} +241.825i q^{41} +131.684 q^{42} -96.4087 q^{43} +274.541i q^{44} -88.6549i q^{45} -191.615i q^{46} +539.015i q^{47} -51.0998 q^{48} -553.380 q^{49} +41.0021i q^{50} -144.785 q^{51} -152.277 q^{53} -39.5847i q^{54} +462.244 q^{55} -607.963 q^{56} +360.901i q^{57} -285.810i q^{58} +327.792i q^{59} +172.893i q^{60} -98.4180 q^{61} -46.9332 q^{62} +269.456i q^{63} -138.515 q^{64} +206.394 q^{66} +441.151i q^{67} +282.357 q^{68} +392.091 q^{69} +432.385i q^{70} -345.049i q^{71} +182.757i q^{72} +773.839i q^{73} +47.5383 q^{74} -83.9002 q^{75} -703.822i q^{76} -1404.94 q^{77} -150.332 q^{79} -167.787i q^{80} +81.0000 q^{81} -354.540 q^{82} -337.966i q^{83} -525.489i q^{84} -475.403i q^{85} -141.345i q^{86} +584.837 q^{87} -952.889 q^{88} +169.913i q^{89} +129.977 q^{90} -764.649 q^{92} -96.0368i q^{93} -790.250 q^{94} -1185.02 q^{95} -562.269i q^{96} -214.201i q^{97} -811.311i q^{98} +422.332i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{3} - 44 q^{4} + 72 q^{9}+O(q^{10}) \) 8 * q - 24 * q^3 - 44 * q^4 + 72 * q^9	\( 8 q - 24 q^{3} - 44 q^{4} + 72 q^{9} + 124 q^{10} + 132 q^{12} + 80 q^{14} + 244 q^{16} - 196 q^{17} + 440 q^{22} - 208 q^{23} + 116 q^{25} - 216 q^{27} + 388 q^{29} - 372 q^{30} + 176 q^{35} - 396 q^{36} - 664 q^{38} - 1996 q^{40} - 240 q^{42} - 900 q^{43} - 732 q^{48} - 2140 q^{49} + 588 q^{51} + 524 q^{53} + 408 q^{55} - 4328 q^{56} - 1856 q^{61} - 5560 q^{62} - 2052 q^{64} - 1320 q^{66} + 3572 q^{68} + 624 q^{69} + 2316 q^{74} - 348 q^{75} - 5016 q^{77} - 1492 q^{79} + 648 q^{81} - 3468 q^{82} - 1164 q^{87} - 6120 q^{88} + 1116 q^{90} + 664 q^{92} - 1544 q^{94} - 4408 q^{95}+O(q^{100}) \) 8 * q - 24 * q^3 - 44 * q^4 + 72 * q^9 + 124 * q^10 + 132 * q^12 + 80 * q^14 + 244 * q^16 - 196 * q^17 + 440 * q^22 - 208 * q^23 + 116 * q^25 - 216 * q^27 + 388 * q^29 - 372 * q^30 + 176 * q^35 - 396 * q^36 - 664 * q^38 - 1996 * q^40 - 240 * q^42 - 900 * q^43 - 732 * q^48 - 2140 * q^49 + 588 * q^51 + 524 * q^53 + 408 * q^55 - 4328 * q^56 - 1856 * q^61 - 5560 * q^62 - 2052 * q^64 - 1320 * q^66 + 3572 * q^68 + 624 * q^69 + 2316 * q^74 - 348 * q^75 - 5016 * q^77 - 1492 * q^79 + 648 * q^81 - 3468 * q^82 - 1164 * q^87 - 6120 * q^88 + 1116 * q^90 + 664 * q^92 - 1544 * q^94 - 4408 * q^95

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