LMFDB - Embedded newform 507.4.b.f.337.1

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:	$4$
Coefficient field:	$\Q(i, \sqrt{14})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 49 $ x^4 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$1.87083 - 1.87083i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.f.337.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.74166i q^{2} -3.00000 q^{3} -14.4833 q^{4} -4.51669i q^{5} +14.2250i q^{6} -7.48331i q^{7} +30.7417i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-4.74166i q^{2} -3.00000 q^{3} -14.4833 q^{4} -4.51669i q^{5} +14.2250i q^{6} -7.48331i q^{7} +30.7417i q^{8} +9.00000 q^{9} -21.4166 q^{10} -66.8999i q^{11} +43.4499 q^{12} -35.4833 q^{14} +13.5501i q^{15} +29.8999 q^{16} -96.9666 q^{17} -42.6749i q^{18} -31.4833i q^{19} +65.4166i q^{20} +22.4499i q^{21} -317.216 q^{22} -183.600 q^{23} -92.2250i q^{24} +104.600 q^{25} -27.0000 q^{27} +108.383i q^{28} +112.200 q^{29} +64.2497 q^{30} +77.2831i q^{31} +104.158i q^{32} +200.700i q^{33} +459.783i q^{34} -33.7998 q^{35} -130.350 q^{36} +54.7664i q^{37} -149.283 q^{38} +138.850 q^{40} -451.716i q^{41} +106.450 q^{42} +113.434 q^{43} +968.932i q^{44} -40.6502i q^{45} +870.566i q^{46} -42.2670i q^{47} -89.6997 q^{48} +287.000 q^{49} -495.975i q^{50} +290.900 q^{51} -530.999 q^{53} +128.025i q^{54} -302.166 q^{55} +230.049 q^{56} +94.4499i q^{57} -532.015i q^{58} +219.666i q^{59} -196.250i q^{60} +822.865 q^{61} +366.450 q^{62} -67.3498i q^{63} +733.082 q^{64} +951.649 q^{66} +872.082i q^{67} +1404.40 q^{68} +550.799 q^{69} +160.267i q^{70} +100.299i q^{71} +276.675i q^{72} -165.634i q^{73} +259.684 q^{74} -313.799 q^{75} +455.983i q^{76} -500.633 q^{77} -545.266 q^{79} -135.048i q^{80} +81.0000 q^{81} -2141.88 q^{82} +454.534i q^{83} -325.150i q^{84} +437.968i q^{85} -537.864i q^{86} -336.601 q^{87} +2056.61 q^{88} -230.915i q^{89} -192.749 q^{90} +2659.13 q^{92} -231.849i q^{93} -200.415 q^{94} -142.200 q^{95} -312.475i q^{96} +1089.16i q^{97} -1360.86i q^{98} -602.099i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} - 28 q^{4} + 36 q^{9}+O(q^{10}) $ 4 * q - 12 * q^3 - 28 * q^4 + 36 * q^9	$ 4 q - 12 q^{3} - 28 q^{4} + 36 q^{9} + 64 q^{10} + 84 q^{12} - 112 q^{14} - 60 q^{16} - 328 q^{17} - 760 q^{22} - 16 q^{23} - 300 q^{25} - 108 q^{27} + 808 q^{29} - 192 q^{30} + 224 q^{35} - 252 q^{36} - 208 q^{38} + 1184 q^{40} + 336 q^{42} + 1232 q^{43} + 180 q^{48} + 1148 q^{49} + 984 q^{51} - 328 q^{53} + 288 q^{55} + 112 q^{56} + 1256 q^{61} + 1376 q^{62} + 388 q^{64} + 2280 q^{66} + 2744 q^{68} + 48 q^{69} + 1368 q^{74} + 900 q^{75} - 1344 q^{77} - 864 q^{79} + 324 q^{81} - 5664 q^{82} - 2424 q^{87} + 3048 q^{88} + 576 q^{90} + 5488 q^{92} + 1144 q^{94} - 928 q^{95}+O(q^{100}) $ 4 * q - 12 * q^3 - 28 * q^4 + 36 * q^9 + 64 * q^10 + 84 * q^12 - 112 * q^14 - 60 * q^16 - 328 * q^17 - 760 * q^22 - 16 * q^23 - 300 * q^25 - 108 * q^27 + 808 * q^29 - 192 * q^30 + 224 * q^35 - 252 * q^36 - 208 * q^38 + 1184 * q^40 + 336 * q^42 + 1232 * q^43 + 180 * q^48 + 1148 * q^49 + 984 * q^51 - 328 * q^53 + 288 * q^55 + 112 * q^56 + 1256 * q^61 + 1376 * q^62 + 388 * q^64 + 2280 * q^66 + 2744 * q^68 + 48 * q^69 + 1368 * q^74 + 900 * q^75 - 1344 * q^77 - 864 * q^79 + 324 * q^81 - 5664 * q^82 - 2424 * q^87 + 3048 * q^88 + 576 * q^90 + 5488 * q^92 + 1144 * q^94 - 928 * 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 $

$2$	− 4.74166i	− 1.67643i	−0.545341	−	0.838215i	$-0.683600\pi$
$2$	− 4.74166i	− 1.67643i	0.545341	−	0.838215i	$-0.316400\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−14.4833	−1.81041
$5$	− 4.51669i	− 0.403985i	−0.979387	−	0.201992i	$-0.935258\pi$
$5$	− 4.51669i	− 0.403985i	0.979387	−	0.201992i	$-0.0647416\pi$
$6$	14.2250i	0.967887i
$7$	− 7.48331i	− 0.404061i	−0.979379	−	0.202031i	$-0.935246\pi$
$7$	− 7.48331i	− 0.404061i	0.979379	−	0.202031i	$-0.0647540\pi$
$8$	30.7417i	1.35860i
$9$	9.00000	0.333333
$10$	−21.4166	−0.677252
$11$	− 66.8999i	− 1.83373i	−0.399193	−	0.916867i	$-0.630710\pi$
$11$	− 66.8999i	− 1.83373i	0.399193	−	0.916867i	$-0.369290\pi$
$12$	43.4499	1.04524
$13$	0	0
$13$	0	0
$14$	−35.4833	−0.677380
$15$	13.5501i	0.233241i
$16$	29.8999	0.467186
$17$	−96.9666	−1.38340	−0.691702	−	0.722183i	$-0.743139\pi$
$17$	−96.9666	−1.38340	−0.691702	+	0.722183i	$0.743139\pi$
$18$	− 42.6749i	− 0.558810i
$19$	− 31.4833i	− 0.380146i	−0.981770	−	0.190073i	$-0.939128\pi$
$19$	− 31.4833i	− 0.380146i	0.981770	−	0.190073i	$-0.0608724\pi$
$20$	65.4166i	0.731380i
$21$	22.4499i	0.233285i
$22$	−317.216	−3.07413
$23$	−183.600	−1.66448	−0.832242	−	0.554412i	$-0.812943\pi$
$23$	−183.600	−1.66448	−0.832242	+	0.554412i	$0.812943\pi$
$24$	− 92.2250i	− 0.784389i
$25$	104.600	0.836796
$26$	0	0
$27$	−27.0000	−0.192450
$28$	108.383i	0.731518i
$29$	112.200	0.718450	0.359225	−	0.933251i	$-0.383041\pi$
$29$	112.200	0.718450	0.359225	+	0.933251i	$0.383041\pi$
$30$	64.2497	0.391011
$31$	77.2831i	0.447757i	0.974617	+	0.223878i	$0.0718718\pi$
$31$	77.2831i	0.447757i	−0.974617	+	0.223878i	$0.928128\pi$
$32$	104.158i	0.575398i
$33$	200.700i	1.05871i
$34$	459.783i	2.31918i
$35$	−33.7998	−0.163234
$36$	−130.350	−0.603471
$37$	54.7664i	0.243339i	0.992571	+	0.121669i	$0.0388248\pi$
$37$	54.7664i	0.243339i	−0.992571	+	0.121669i	$0.961175\pi$
$38$	−149.283	−0.637287
$39$	0	0
$40$	138.850	0.548854
$41$	− 451.716i	− 1.72064i	−0.509756	−	0.860319i	$-0.670264\pi$
$41$	− 451.716i	− 1.72064i	0.509756	−	0.860319i	$-0.329736\pi$
$42$	106.450	0.391085
$43$	113.434	0.402291	0.201145	−	0.979561i	$-0.435534\pi$
$43$	113.434	0.402291	0.201145	+	0.979561i	$0.435534\pi$
$44$	968.932i	3.31982i
$45$	− 40.6502i	− 0.134662i
$46$	870.566i	2.79039i
$47$	− 42.2670i	− 0.131176i	−0.997847	−	0.0655880i	$-0.979108\pi$
$47$	− 42.2670i	− 0.131176i	0.997847	−	0.0655880i	$-0.0208923\pi$
$48$	−89.6997	−0.269730
$49$	287.000	0.836735
$50$	− 495.975i	− 1.40283i
$51$	290.900	0.798708
$52$	0	0
$53$	−530.999	−1.37619	−0.688097	−	0.725618i	$-0.741554\pi$
$53$	−530.999	−1.37619	−0.688097	+	0.725618i	$0.741554\pi$
$54$	128.025i	0.322629i
$55$	−302.166	−0.740800
$56$	230.049	0.548958
$57$	94.4499i	0.219477i
$58$	− 532.015i	− 1.20443i
$59$	219.666i	0.484714i	0.970187	+	0.242357i	$0.0779205\pi$
$59$	219.666i	0.484714i	−0.970187	+	0.242357i	$0.922080\pi$
$60$	− 196.250i	− 0.422262i
$61$	822.865	1.72717	0.863583	−	0.504207i	$-0.168215\pi$
$61$	822.865	1.72717	0.863583	+	0.504207i	$0.168215\pi$
$62$	366.450	0.750632
$63$	− 67.3498i	− 0.134687i
$64$	733.082	1.43180
$65$	0	0
$66$	951.649	1.77485
$67$	872.082i	1.59018i	0.606495	+	0.795088i	$0.292575\pi$
$67$	872.082i	1.59018i	−0.606495	+	0.795088i	$0.707425\pi$
$68$	1404.40	2.50453
$69$	550.799	0.960991
$70$	160.267i	0.273651i
$71$	100.299i	0.167653i	0.996480	+	0.0838263i	$0.0267141\pi$
$71$	100.299i	0.167653i	−0.996480	+	0.0838263i	$0.973286\pi$
$72$	276.675i	0.452867i
$73$	− 165.634i	− 0.265562i	−0.991145	−	0.132781i	$-0.957609\pi$
$73$	− 165.634i	− 0.265562i	0.991145	−	0.132781i	$-0.0423906\pi$
$74$	259.684	0.407941
$75$	−313.799	−0.483125
$76$	455.983i	0.688221i
$77$	−500.633	−0.740940
$78$	0	0
$79$	−545.266	−0.776547	−0.388273	−	0.921544i	$-0.626928\pi$
$79$	−545.266	−0.776547	−0.388273	+	0.921544i	$0.626928\pi$
$80$	− 135.048i	− 0.188736i
$81$	81.0000	0.111111
$82$	−2141.88	−2.88453
$83$	454.534i	0.601103i	0.953766	+	0.300552i	$0.0971708\pi$
$83$	454.534i	0.601103i	−0.953766	+	0.300552i	$0.902829\pi$
$84$	− 325.150i	− 0.422342i
$85$	437.968i	0.558874i
$86$	− 537.864i	− 0.674412i
$87$	−336.601	−0.414797
$88$	2056.61	2.49132
$89$	− 230.915i	− 0.275022i	−0.990500	−	0.137511i	$-0.956090\pi$
$89$	− 230.915i	− 0.275022i	0.990500	−	0.137511i	$-0.0439102\pi$
$90$	−192.749	−0.225751
$91$	0	0
$92$	2659.13	3.01341
$93$	− 231.849i	− 0.258512i
$94$	−200.415	−0.219907
$95$	−142.200	−0.153573
$96$	− 312.475i	− 0.332206i
$97$	1089.16i	1.14008i	0.821616	+	0.570041i	$0.193073\pi$
$97$	1089.16i	1.14008i	−0.821616	+	0.570041i	$0.806927\pi$
$98$	− 1360.86i	− 1.40273i
$99$	− 602.099i	− 0.611245i
$100$	−1514.95	−1.51495
$101$	77.2336	0.0760894	0.0380447	−	0.999276i	$-0.487887\pi$
$101$	77.2336	0.0760894	0.0380447	+	0.999276i	$0.487887\pi$
$102$	− 1379.35i	− 1.33898i
$103$	−1351.36	−1.29275	−0.646377	−	0.763018i	$-0.723717\pi$
$103$	−1351.36	−1.29275	−0.646377	+	0.763018i	$0.723717\pi$
$104$	0	0
$105$	101.399	0.0942434
$106$	2517.81i	2.30709i
$107$	1133.67	1.02426	0.512129	−	0.858908i	$-0.328857\pi$
$107$	1133.67	1.02426	0.512129	+	0.858908i	$0.328857\pi$
$108$	391.049	0.348414
$109$	− 1017.66i	− 0.894262i	−0.894469	−	0.447131i	$-0.852446\pi$
$109$	− 1017.66i	− 0.894262i	0.894469	−	0.447131i	$-0.147554\pi$
$110$	1432.77i	1.24190i
$111$	− 164.299i	− 0.140492i
$112$	− 223.750i	− 0.188772i
$113$	1570.06	1.30707	0.653536	−	0.756895i	$-0.273285\pi$
$113$	1570.06	1.30707	0.653536	+	0.756895i	$0.273285\pi$
$114$	447.849	0.367938
$115$	829.261i	0.672426i
$116$	−1625.03	−1.30069
$117$	0	0
$118$	1041.58	0.812588
$119$	725.632i	0.558979i
$120$	−416.551	−0.316881
$121$	−3144.60	−2.36258
$122$	− 3901.75i	− 2.89547i
$123$	1355.15i	0.993411i
$124$	− 1119.32i	− 0.810625i
$125$	− 1037.03i	− 0.742037i
$126$	−319.350	−0.225793
$127$	1248.16	0.872099	0.436050	−	0.899923i	$-0.356377\pi$
$127$	1248.16	0.872099	0.436050	+	0.899923i	$0.356377\pi$
$128$	− 2642.76i	− 1.82491i
$129$	−340.301	−0.232263
$130$	0	0
$131$	−1274.80	−0.850227	−0.425113	−	0.905140i	$-0.639766\pi$
$131$	−1274.80	−0.850227	−0.425113	+	0.905140i	$0.639766\pi$
$132$	− 2906.80i	− 1.91670i
$133$	−235.600	−0.153602
$134$	4135.11	2.66582
$135$	121.951i	0.0777469i
$136$	− 2980.91i	− 1.87950i
$137$	874.915i	0.545613i	0.962069	+	0.272807i	$0.0879519\pi$
$137$	874.915i	0.545613i	−0.962069	+	0.272807i	$0.912048\pi$
$138$	− 2611.70i	− 1.61103i
$139$	310.334	0.189368	0.0946840	−	0.995507i	$-0.469816\pi$
$139$	310.334	0.189368	0.0946840	+	0.995507i	$0.469816\pi$
$140$	489.533	0.295522
$141$	126.801i	0.0757345i
$142$	475.585	0.281058
$143$	0	0
$144$	269.099	0.155729
$145$	− 506.773i	− 0.290243i
$146$	−785.380	−0.445195
$147$	−861.000	−0.483089
$148$	− 793.199i	− 0.440544i
$149$	− 5.08064i	− 0.00279344i	−0.999999	−	0.00139672i	$-0.999555\pi$
$149$	− 5.08064i	− 0.00279344i	0.999999	−	0.00139672i	$-0.000444590\pi$
$150$	1487.93i	0.809924i
$151$	6.54894i	0.00352944i	0.999998	+	0.00176472i	$0.000561728\pi$
$151$	6.54894i	0.00352944i	−0.999998	+	0.00176472i	$0.999438\pi$
$152$	967.849	0.516467
$153$	−872.700	−0.461135
$154$	2373.83i	1.24213i
$155$	349.063	0.180887
$156$	0	0
$157$	−2297.60	−1.16795	−0.583975	−	0.811772i	$-0.698503\pi$
$157$	−2297.60	−1.16795	−0.583975	+	0.811772i	$0.698503\pi$
$158$	2585.46i	1.30183i
$159$	1593.00	0.794546
$160$	470.450	0.232452
$161$	1373.93i	0.672553i
$162$	− 384.074i	− 0.186270i
$163$	− 1085.49i	− 0.521606i	−0.965392	−	0.260803i	$-0.916013\pi$
$163$	− 1085.49i	− 0.521606i	0.965392	−	0.260803i	$-0.0839873\pi$
$164$	6542.34i	3.11507i
$165$	906.497	0.427701
$166$	2155.24	1.00771
$167$	− 109.066i	− 0.0505374i	−0.999681	−	0.0252687i	$-0.991956\pi$
$167$	− 109.066i	− 0.0505374i	0.999681	−	0.0252687i	$-0.00804414\pi$
$168$	−690.148	−0.316941
$169$	0	0
$170$	2076.69	0.936912
$171$	− 283.350i	− 0.126715i
$172$	−1642.90	−0.728313
$173$	1889.17	0.830236	0.415118	−	0.909768i	$-0.363740\pi$
$173$	1889.17	0.830236	0.415118	+	0.909768i	$0.363740\pi$
$174$	1596.05i	0.695379i
$175$	− 782.751i	− 0.338117i
$176$	− 2000.30i	− 0.856694i
$177$	− 658.999i	− 0.279850i
$178$	−1094.92	−0.461054
$179$	−3427.86	−1.43134	−0.715672	−	0.698437i	$-0.753879\pi$
$179$	−3427.86	−1.43134	−0.715672	+	0.698437i	$0.753879\pi$
$180$	588.749i	0.243793i
$181$	−208.403	−0.0855826	−0.0427913	−	0.999084i	$-0.513625\pi$
$181$	−208.403	−0.0855826	−0.0427913	+	0.999084i	$0.513625\pi$
$182$	0	0
$183$	−2468.60	−0.997180
$184$	− 5644.15i	− 2.26137i
$185$	247.363	0.0983052
$186$	−1099.35	−0.433378
$187$	6487.06i	2.53679i
$188$	612.166i	0.237483i
$189$	202.049i	0.0777616i
$190$	674.265i	0.257454i
$191$	957.735	0.362824	0.181412	−	0.983407i	$-0.441933\pi$
$191$	957.735	0.362824	0.181412	+	0.983407i	$0.441933\pi$
$192$	−2199.25	−0.826650
$193$	− 512.730i	− 0.191228i	−0.995418	−	0.0956142i	$-0.969518\pi$
$193$	− 512.730i	− 0.191228i	0.995418	−	0.0956142i	$-0.0304815\pi$
$194$	5164.45	1.91127
$195$	0	0
$196$	−4156.71	−1.51484
$197$	3870.35i	1.39975i	0.714265	+	0.699876i	$0.246761\pi$
$197$	3870.35i	1.39975i	−0.714265	+	0.699876i	$0.753239\pi$
$198$	−2854.95	−1.02471
$199$	−2305.83	−0.821388	−0.410694	−	0.911773i	$-0.634714\pi$
$199$	−2305.83	−0.821388	−0.410694	+	0.911773i	$0.634714\pi$
$200$	3215.56i	1.13687i
$201$	− 2616.25i	− 0.918088i
$202$	− 366.215i	− 0.127558i
$203$	− 839.630i	− 0.290298i
$204$	−4213.19	−1.44599
$205$	−2040.26	−0.695111
$206$	6407.70i	2.16721i
$207$	−1652.40	−0.554828
$208$	0	0
$209$	−2106.23	−0.697086
$210$	− 480.801i	− 0.157992i
$211$	−3672.40	−1.19819	−0.599096	−	0.800677i	$-0.704473\pi$
$211$	−3672.40	−1.19819	−0.599096	+	0.800677i	$0.704473\pi$
$212$	7690.62	2.49148
$213$	− 300.898i	− 0.0967942i
$214$	− 5375.46i	− 1.71710i
$215$	− 512.345i	− 0.162519i
$216$	− 830.025i	− 0.261463i
$217$	578.334	0.180921
$218$	−4825.41	−1.49917
$219$	496.902i	0.153322i
$220$	4376.36	1.34116
$221$	0	0
$222$	−779.051	−0.235525
$223$	− 5087.05i	− 1.52760i	−0.645455	−	0.763798i	$-0.723332\pi$
$223$	− 5087.05i	− 1.52760i	0.645455	−	0.763798i	$-0.276668\pi$
$224$	779.449	0.232496
$225$	941.396	0.278932
$226$	− 7444.71i	− 2.19122i
$227$	2625.83i	0.767763i	0.923382	+	0.383882i	$0.125413\pi$
$227$	2625.83i	0.767763i	−0.923382	+	0.383882i	$0.874587\pi$
$228$	− 1367.95i	− 0.397345i
$229$	1678.73i	0.484425i	0.970223	+	0.242213i	$0.0778732\pi$
$229$	1678.73i	0.484425i	−0.970223	+	0.242213i	$0.922127\pi$
$230$	3932.07	1.12727
$231$	1501.90	0.427782
$232$	3449.22i	0.976088i
$233$	−648.506	−0.182339	−0.0911696	−	0.995835i	$-0.529061\pi$
$233$	−648.506	−0.182339	−0.0911696	+	0.995835i	$0.529061\pi$
$234$	0	0
$235$	−190.907	−0.0529931
$236$	− 3181.50i	− 0.877533i
$237$	1635.80	0.448340
$238$	3440.70	0.937089
$239$	− 5219.69i	− 1.41269i	−0.707866	−	0.706347i	$-0.750342\pi$
$239$	− 5219.69i	− 1.41269i	0.707866	−	0.706347i	$-0.249658\pi$
$240$	405.145i	0.108967i
$241$	6103.56i	1.63139i	0.578483	+	0.815695i	$0.303645\pi$
$241$	6103.56i	1.63139i	−0.578483	+	0.815695i	$0.696355\pi$
$242$	14910.6i	3.96070i
$243$	−243.000	−0.0641500
$244$	−11917.8	−3.12689
$245$	− 1296.29i	− 0.338028i
$246$	6425.64	1.66538
$247$	0	0
$248$	−2375.81	−0.608323
$249$	− 1363.60i	− 0.347047i
$250$	−4917.24	−1.24397
$251$	−6423.40	−1.61530	−0.807652	−	0.589660i	$-0.799262\pi$
$251$	−6423.40	−1.61530	−0.807652	+	0.589660i	$0.799262\pi$
$252$	975.449i	0.243839i
$253$	12282.8i	3.05222i
$254$	− 5918.36i	− 1.46201i
$255$	− 1313.90i	− 0.322666i
$256$	−6666.39	−1.62754
$257$	1230.23	0.298597	0.149299	−	0.988792i	$-0.452298\pi$
$257$	1230.23	0.298597	0.149299	+	0.988792i	$0.452298\pi$
$258$	1613.59i	0.389372i
$259$	409.834	0.0983238
$260$	0	0
$261$	1009.80	0.239483
$262$	6044.66i	1.42534i
$263$	−514.992	−0.120744	−0.0603722	−	0.998176i	$-0.519229\pi$
$263$	−514.992	−0.120744	−0.0603722	+	0.998176i	$0.519229\pi$
$264$	−6169.84	−1.43836
$265$	2398.35i	0.555961i
$266$	1117.13i	0.257503i
$267$	692.745i	0.158784i
$268$	− 12630.6i	− 2.87888i
$269$	−5132.60	−1.16335	−0.581673	−	0.813423i	$-0.697602\pi$
$269$	−5132.60	−1.16335	−0.581673	+	0.813423i	$0.697602\pi$
$270$	578.247	0.130337
$271$	− 4300.00i	− 0.963862i	−0.876209	−	0.481931i	$-0.839936\pi$
$271$	− 4300.00i	− 0.963862i	0.876209	−	0.481931i	$-0.160064\pi$
$272$	−2899.29	−0.646306
$273$	0	0
$274$	4148.55	0.914682
$275$	− 6997.70i	− 1.53446i
$276$	−7977.39	−1.73979
$277$	1812.80	0.393215	0.196607	−	0.980482i	$-0.437008\pi$
$277$	1812.80	0.393215	0.196607	+	0.980482i	$0.437008\pi$
$278$	− 1471.50i	− 0.317462i
$279$	695.548i	0.149252i
$280$	− 1039.06i	− 0.221771i
$281$	− 4073.08i	− 0.864696i	−0.901707	−	0.432348i	$-0.857685\pi$
$281$	− 4073.08i	− 0.864696i	0.901707	−	0.432348i	$-0.142315\pi$
$282$	601.246	0.126963
$283$	−6346.29	−1.33303	−0.666516	−	0.745491i	$-0.732215\pi$
$283$	−6346.29	−1.33303	−0.666516	+	0.745491i	$0.732215\pi$
$284$	− 1452.67i	− 0.303520i
$285$	426.601	0.0886654
$286$	0	0
$287$	−3380.33	−0.695243
$288$	937.424i	0.191799i
$289$	4489.53	0.913806
$290$	−2402.94	−0.486572
$291$	− 3267.49i	− 0.658226i
$292$	2398.93i	0.480777i
$293$	− 8390.97i	− 1.67306i	−0.547923	−	0.836529i	$-0.684581\pi$
$293$	− 8390.97i	− 1.67306i	0.547923	−	0.836529i	$-0.315419\pi$
$294$	4082.57i	0.809864i
$295$	992.164	0.195817
$296$	−1683.61	−0.330601
$297$	1806.30i	0.352902i
$298$	−24.0907	−0.00468300
$299$	0	0
$300$	4544.84	0.874656
$301$	− 848.861i	− 0.162550i
$302$	31.0528	0.00591685
$303$	−231.701	−0.0439302
$304$	− 941.348i	− 0.177599i
$305$	− 3716.62i	− 0.697748i
$306$	4138.04i	0.773059i
$307$	4005.27i	0.744603i	0.928112	+	0.372301i	$0.121431\pi$
$307$	4005.27i	0.744603i	−0.928112	+	0.372301i	$0.878569\pi$
$308$	7250.82	1.34141
$309$	4054.09	0.746372
$310$	− 1655.14i	− 0.303244i
$311$	−5836.53	−1.06418	−0.532088	−	0.846689i	$-0.678593\pi$
$311$	−5836.53	−1.06418	−0.532088	+	0.846689i	$0.678593\pi$
$312$	0	0
$313$	1763.19	0.318407	0.159204	−	0.987246i	$-0.449107\pi$
$313$	1763.19	0.318407	0.159204	+	0.987246i	$0.449107\pi$
$314$	10894.4i	1.95798i
$315$	−304.198	−0.0544115
$316$	7897.26	1.40587
$317$	6106.35i	1.08191i	0.841050	+	0.540957i	$0.181938\pi$
$317$	6106.35i	1.08191i	−0.841050	+	0.540957i	$0.818062\pi$
$318$	− 7553.44i	− 1.33200i
$319$	− 7506.18i	− 1.31745i
$320$	− 3311.10i	− 0.578425i
$321$	−3401.00	−0.591356
$322$	6514.72	1.12749
$323$	3052.83i	0.525895i
$324$	−1173.15	−0.201157
$325$	0	0
$326$	−5147.00	−0.874436
$327$	3052.99i	0.516302i
$328$	13886.5	2.33766
$329$	−316.297	−0.0530031
$330$	− 4298.30i	− 0.717011i
$331$	− 7490.38i	− 1.24383i	−0.783084	−	0.621916i	$-0.786354\pi$
$331$	− 7490.38i	− 1.24383i	0.783084	−	0.621916i	$-0.213646\pi$
$332$	− 6583.16i	− 1.08825i
$333$	492.898i	0.0811130i
$334$	−517.152	−0.0847224
$335$	3938.92	0.642406
$336$	671.251i	0.108987i
$337$	−9462.46	−1.52953	−0.764767	−	0.644307i	$-0.777146\pi$
$337$	−9462.46	−1.52953	−0.764767	+	0.644307i	$0.777146\pi$
$338$	0	0
$339$	−4710.19	−0.754639
$340$	− 6343.22i	− 1.01179i
$341$	5170.23	0.821066
$342$	−1343.55	−0.212429
$343$	− 4714.49i	− 0.742153i
$344$	3487.14i	0.546553i
$345$	− 2487.78i	− 0.388226i
$346$	− 8957.79i	− 1.39183i
$347$	11460.3	1.77297	0.886487	−	0.462753i	$-0.153138\pi$
$347$	11460.3	1.77297	0.886487	+	0.462753i	$0.153138\pi$
$348$	4875.09	0.750955
$349$	− 3673.96i	− 0.563503i	−0.959487	−	0.281751i	$-0.909085\pi$
$349$	− 3673.96i	− 0.563503i	0.959487	−	0.281751i	$-0.0909153\pi$
$350$	−3711.54	−0.566829
$351$	0	0
$352$	6968.17	1.05513
$353$	− 3388.65i	− 0.510935i	−0.966818	−	0.255467i	$-0.917771\pi$
$353$	− 3388.65i	− 0.510935i	0.966818	−	0.255467i	$-0.0822293\pi$
$354$	−3124.75	−0.469148
$355$	453.020	0.0677290
$356$	3344.41i	0.497903i
$357$	− 2176.90i	− 0.322727i
$358$	16253.7i	2.39955i
$359$	− 9673.98i	− 1.42221i	−0.703086	−	0.711105i	$-0.748195\pi$
$359$	− 9673.98i	− 1.42221i	0.703086	−	0.711105i	$-0.251805\pi$
$360$	1249.65	0.182951
$361$	5867.80	0.855489
$362$	988.174i	0.143473i
$363$	9433.79	1.36404
$364$	0	0
$365$	−748.117	−0.107283
$366$	11705.2i	1.67170i
$367$	−8715.98	−1.23970	−0.619851	−	0.784720i	$-0.712807\pi$
$367$	−8715.98	−1.23970	−0.619851	+	0.784720i	$0.712807\pi$
$368$	−5489.61	−0.777624
$369$	− 4065.44i	− 0.573546i
$370$	− 1172.91i	− 0.164802i
$371$	3973.63i	0.556067i
$372$	3357.95i	0.468014i
$373$	4667.99	0.647987	0.323994	−	0.946059i	$-0.394974\pi$
$373$	4667.99	0.647987	0.323994	+	0.946059i	$0.394974\pi$
$374$	30759.4	4.25276
$375$	3111.09i	0.428416i
$376$	1299.36	0.178216
$377$	0	0
$378$	958.049	0.130362
$379$	− 10862.7i	− 1.47225i	−0.676848	−	0.736123i	$-0.736655\pi$
$379$	− 10862.7i	− 1.47225i	0.676848	−	0.736123i	$-0.263345\pi$
$380$	2059.53	0.278031
$381$	−3744.49	−0.503507
$382$	− 4541.25i	− 0.608248i
$383$	− 10054.2i	− 1.34137i	−0.741742	−	0.670686i	$-0.766000\pi$
$383$	− 10054.2i	− 1.34137i	0.741742	−	0.670686i	$-0.234000\pi$
$384$	7928.27i	1.05361i
$385$	2261.20i	0.299329i
$386$	−2431.19	−0.320581
$387$	1020.90	0.134097
$388$	− 15774.7i	− 2.06402i
$389$	−6418.50	−0.836584	−0.418292	−	0.908313i	$-0.637371\pi$
$389$	−6418.50	−0.836584	−0.418292	+	0.908313i	$0.637371\pi$
$390$	0	0
$391$	17803.0	2.30265
$392$	8822.86i	1.13679i
$393$	3824.40	0.490879
$394$	18351.9	2.34658
$395$	2462.79i	0.313713i
$396$	8720.39i	1.10661i
$397$	− 12019.9i	− 1.51955i	−0.650186	−	0.759775i	$-0.725309\pi$
$397$	− 12019.9i	− 1.51955i	0.650186	−	0.759775i	$-0.274691\pi$
$398$	10933.5i	1.37700i
$399$	706.799	0.0886822
$400$	3127.52	0.390939
$401$	3599.80i	0.448293i	0.974555	+	0.224147i	$0.0719594\pi$
$401$	3599.80i	0.448293i	−0.974555	+	0.224147i	$0.928041\pi$
$402$	−12405.3	−1.53911
$403$	0	0
$404$	−1118.60	−0.137753
$405$	− 365.852i	− 0.0448872i
$406$	−3981.24	−0.486664
$407$	3663.87	0.446219
$408$	8942.74i	1.08513i
$409$	48.0968i	0.00581475i	0.999996	+	0.00290737i	$0.000925447\pi$
$409$	48.0968i	0.00581475i	−0.999996	+	0.00290737i	$0.999075\pi$
$410$	9674.20i	1.16530i
$411$	− 2624.74i	− 0.315010i
$412$	19572.2	2.34042
$413$	1643.83	0.195854
$414$	7835.10i	0.930130i
$415$	2052.99	0.242837
$416$	0	0
$417$	−931.001	−0.109332
$418$	9987.02i	1.16862i
$419$	723.462	0.0843518	0.0421759	−	0.999110i	$-0.486571\pi$
$419$	723.462	0.0843518	0.0421759	+	0.999110i	$0.486571\pi$
$420$	−1468.60	−0.170620
$421$	14845.5i	1.71859i	0.511481	+	0.859295i	$0.329097\pi$
$421$	14845.5i	1.71859i	−0.511481	+	0.859295i	$0.670903\pi$
$422$	17413.3i	2.00868i
$423$	− 380.403i	− 0.0437253i
$424$	− 16323.8i	− 1.86970i
$425$	−10142.7	−1.15763
$426$	−1426.75	−0.162269
$427$	− 6157.76i	− 0.697880i
$428$	−16419.2	−1.85433
$429$	0	0
$430$	−2429.36	−0.272452
$431$	− 1103.11i	− 0.123283i	−0.998098	−	0.0616417i	$-0.980366\pi$
$431$	− 1103.11i	− 0.123283i	0.998098	−	0.0616417i	$-0.0196336\pi$
$432$	−807.297	−0.0899099
$433$	−8893.53	−0.987057	−0.493528	−	0.869730i	$-0.664293\pi$
$433$	−8893.53	−0.987057	−0.493528	+	0.869730i	$0.664293\pi$
$434$	− 2742.26i	− 0.303301i
$435$	1520.32i	0.167572i
$436$	14739.1i	1.61898i
$437$	5780.32i	0.632747i
$438$	2356.14	0.257034
$439$	10901.7	1.18521	0.592607	−	0.805492i	$-0.298099\pi$
$439$	10901.7	1.18521	0.592607	+	0.805492i	$0.298099\pi$
$440$	− 9289.08i	− 1.00645i
$441$	2583.00	0.278912
$442$	0	0
$443$	−3781.37	−0.405550	−0.202775	−	0.979225i	$-0.564996\pi$
$443$	−3781.37	−0.405550	−0.202775	+	0.979225i	$0.564996\pi$
$444$	2379.60i	0.254348i
$445$	−1042.97	−0.111105
$446$	−24121.0	−2.56091
$447$	15.2419i	0.00161279i
$448$	− 5485.88i	− 0.578535i
$449$	106.834i	0.0112289i	0.999984	+	0.00561447i	$0.00178715\pi$
$449$	106.834i	0.0112289i	−0.999984	+	0.00561447i	$0.998213\pi$
$450$	− 4463.78i	− 0.467610i
$451$	−30219.7	−3.15519
$452$	−22739.7	−2.36634
$453$	− 19.6468i	− 0.00203772i
$454$	12450.8	1.28710
$455$	0	0
$456$	−2903.55	−0.298182
$457$	1237.64i	0.126684i	0.997992	+	0.0633419i	$0.0201759\pi$
$457$	1237.64i	0.126684i	−0.997992	+	0.0633419i	$0.979824\pi$
$458$	7959.95	0.812105
$459$	2618.10	0.266236
$460$	− 12010.5i	− 1.21737i
$461$	− 8790.90i	− 0.888141i	−0.895992	−	0.444071i	$-0.853534\pi$
$461$	− 8790.90i	− 0.888141i	0.895992	−	0.444071i	$-0.146466\pi$
$462$	− 7121.49i	− 0.717146i
$463$	− 3861.55i	− 0.387606i	−0.981040	−	0.193803i	$-0.937918\pi$
$463$	− 3861.55i	− 0.387606i	0.981040	−	0.193803i	$-0.0620823\pi$
$464$	3354.77	0.335650
$465$	−1047.19	−0.104435
$466$	3074.99i	0.305679i
$467$	8991.38	0.890945	0.445473	−	0.895296i	$-0.353036\pi$
$467$	8991.38	0.890945	0.445473	+	0.895296i	$0.353036\pi$
$468$	0	0
$469$	6526.06	0.642528
$470$	905.214i	0.0888391i
$471$	6892.79	0.674316
$472$	−6752.91	−0.658533
$473$	− 7588.71i	− 0.737694i
$474$	− 7756.39i	− 0.751609i
$475$	− 3293.14i	− 0.318105i
$476$	− 10509.6i	− 1.01198i
$477$	−4778.99	−0.458731
$478$	−24750.0	−2.36828
$479$	4179.82i	0.398707i	0.979928	+	0.199354i	$0.0638842\pi$
$479$	4179.82i	0.398707i	−0.979928	+	0.199354i	$0.936116\pi$
$480$	−1411.35	−0.134206
$481$	0	0
$482$	28941.0	2.73491
$483$	− 4121.80i	− 0.388299i
$484$	45544.2	4.27725
$485$	4919.41	0.460575
$486$	1152.22i	0.107543i
$487$	18443.8i	1.71616i	0.513519	+	0.858078i	$0.328342\pi$
$487$	18443.8i	1.71616i	−0.513519	+	0.858078i	$0.671658\pi$
$488$	25296.2i	2.34653i
$489$	3256.46i	0.301149i
$490$	−6146.56	−0.566680
$491$	−8093.26	−0.743877	−0.371939	−	0.928257i	$-0.621307\pi$
$491$	−8093.26	−0.743877	−0.371939	+	0.928257i	$0.621307\pi$
$492$	− 19627.0i	− 1.79849i
$493$	−10879.7	−0.993907
$494$	0	0
$495$	−2719.49	−0.246933
$496$	2310.76i	0.209185i
$497$	750.571	0.0677418
$498$	−6465.73	−0.581800
$499$	10941.6i	0.981591i	0.871275	+	0.490796i	$0.163294\pi$
$499$	10941.6i	0.981591i	−0.871275	+	0.490796i	$0.836706\pi$
$500$	15019.6i	1.34340i
$501$	327.197i	0.0291778i
$502$	30457.5i	2.70794i
$503$	−9260.11	−0.820851	−0.410425	−	0.911894i	$-0.634620\pi$
$503$	−9260.11	−0.820851	−0.410425	+	0.911894i	$0.634620\pi$
$504$	2070.45	0.182986
$505$	− 348.840i	− 0.0307389i
$506$	58240.8	5.11684
$507$	0	0
$508$	−18077.5	−1.57886
$509$	9996.40i	0.870497i	0.900310	+	0.435248i	$0.143339\pi$
$509$	9996.40i	0.870497i	−0.900310	+	0.435248i	$0.856661\pi$
$510$	−6230.08	−0.540927
$511$	−1239.49	−0.107303
$512$	10467.7i	0.903538i
$513$	850.049i	0.0731591i
$514$	− 5833.32i	− 0.500577i
$515$	6103.68i	0.522253i
$516$	4928.69	0.420491
$517$	−2827.66	−0.240542
$518$	− 1943.29i	− 0.164833i
$519$	−5667.51	−0.479337
$520$	0	0
$521$	11427.9	0.960973	0.480486	−	0.877002i	$-0.340460\pi$
$521$	11427.9	0.960973	0.480486	+	0.877002i	$0.340460\pi$
$522$	− 4788.14i	− 0.401477i
$523$	−4810.47	−0.402193	−0.201097	−	0.979571i	$-0.564451\pi$
$523$	−4810.47	−0.402193	−0.201097	+	0.979571i	$0.564451\pi$
$524$	18463.3	1.53926
$525$	2348.25i	0.195212i
$526$	2441.92i	0.202419i
$527$	− 7493.88i	− 0.619428i
$528$	6000.90i	0.494613i
$529$	21541.8	1.77051
$530$	11372.2	0.932030
$531$	1977.00i	0.161571i
$532$	3412.26	0.278083
$533$	0	0
$534$	3284.76	0.266190
$535$	− 5120.41i	− 0.413785i
$536$	−26809.2	−2.16042
$537$	10283.6	0.826386
$538$	24337.0i	1.95027i
$539$	− 19200.3i	− 1.53435i
$540$	− 1766.25i	− 0.140754i
$541$	2411.99i	0.191681i	0.995397	+	0.0958406i	$0.0305539\pi$
$541$	2411.99i	0.191681i	−0.995397	+	0.0958406i	$0.969446\pi$
$542$	−20389.1	−1.61585
$543$	625.208	0.0494111
$544$	− 10099.9i	− 0.796008i
$545$	−4596.47	−0.361268
$546$	0	0
$547$	−4396.34	−0.343646	−0.171823	−	0.985128i	$-0.554966\pi$
$547$	−4396.34	−0.343646	−0.171823	+	0.985128i	$0.554966\pi$
$548$	− 12671.7i	− 0.987786i
$549$	7405.79	0.575722
$550$	−33180.7	−2.57242
$551$	− 3532.43i	− 0.273116i
$552$	16932.5i	1.30560i
$553$	4080.40i	0.313772i
$554$	− 8595.67i	− 0.659197i
$555$	−742.088	−0.0567565
$556$	−4494.66	−0.342835
$557$	− 17488.0i	− 1.33032i	−0.746701	−	0.665160i	$-0.768363\pi$
$557$	− 17488.0i	− 1.33032i	0.746701	−	0.665160i	$-0.231637\pi$
$558$	3298.05	0.250211
$559$	0	0
$560$	−1010.61	−0.0762608
$561$	− 19461.2i	− 1.46462i
$562$	−19313.2	−1.44960
$563$	6881.77	0.515154	0.257577	−	0.966258i	$-0.417076\pi$
$563$	6881.77	0.515154	0.257577	+	0.966258i	$0.417076\pi$
$564$	− 1836.50i	− 0.137111i
$565$	− 7091.49i	− 0.528037i
$566$	30092.0i	2.23473i
$567$	− 606.148i	− 0.0448957i
$568$	−3083.36	−0.227773
$569$	−14733.5	−1.08552	−0.542758	−	0.839889i	$-0.682620\pi$
$569$	−14733.5	−1.08552	−0.542758	+	0.839889i	$0.682620\pi$
$570$	− 2022.79i	− 0.148641i
$571$	−4488.51	−0.328964	−0.164482	−	0.986380i	$-0.552595\pi$
$571$	−4488.51	−0.328964	−0.164482	+	0.986380i	$0.552595\pi$
$572$	0	0
$573$	−2873.21	−0.209476
$574$	16028.4i	1.16553i
$575$	−19204.4	−1.39284
$576$	6597.74	0.477267
$577$	10552.2i	0.761338i	0.924711	+	0.380669i	$0.124306\pi$
$577$	10552.2i	0.761338i	−0.924711	+	0.380669i	$0.875694\pi$
$578$	− 21287.8i	− 1.53193i
$579$	1538.19i	0.110406i
$580$	7339.75i	0.525460i
$581$	3401.42	0.242882
$582$	−15493.3	−1.10347
$583$	35523.8i	2.52357i
$584$	5091.86	0.360793
$585$	0	0
$586$	−39787.1	−2.80476
$587$	1637.20i	0.115118i	0.998342	+	0.0575591i	$0.0183318\pi$
$587$	1637.20i	0.115118i	−0.998342	+	0.0575591i	$0.981668\pi$
$588$	12470.1	0.874591
$589$	2433.13	0.170213
$590$	− 4704.50i	− 0.328273i
$591$	− 11611.1i	− 0.808147i
$592$	1637.51i	0.113685i
$593$	14024.0i	0.971155i	0.874194	+	0.485577i	$0.161391\pi$
$593$	14024.0i	0.971155i	−0.874194	+	0.485577i	$0.838609\pi$
$594$	8564.84	0.591616
$595$	3277.45	0.225819
$596$	73.5845i	0.00505728i
$597$	6917.50	0.474229
$598$	0	0
$599$	365.860	0.0249560	0.0124780	−	0.999922i	$-0.496028\pi$
$599$	365.860	0.0249560	0.0124780	+	0.999922i	$0.496028\pi$
$600$	− 9646.69i	− 0.656374i
$601$	−10128.9	−0.687465	−0.343733	−	0.939068i	$-0.611691\pi$
$601$	−10128.9	−0.687465	−0.343733	+	0.939068i	$0.611691\pi$
$602$	−4025.01	−0.272503
$603$	7848.74i	0.530058i
$604$	− 94.8504i	− 0.00638975i
$605$	14203.1i	0.954446i
$606$	1098.65i	0.0736459i
$607$	2324.97	0.155465	0.0777327	−	0.996974i	$-0.475232\pi$
$607$	2324.97	0.155465	0.0777327	+	0.996974i	$0.475232\pi$
$608$	3279.25	0.218735
$609$	2518.89i	0.167603i
$610$	−17623.0	−1.16973
$611$	0	0
$612$	12639.6	0.834845
$613$	− 633.133i	− 0.0417162i	−0.999782	−	0.0208581i	$-0.993360\pi$
$613$	− 633.133i	− 0.0417162i	0.999782	−	0.0208581i	$-0.00663982\pi$
$614$	18991.6	1.24827
$615$	6120.77	0.401323
$616$	− 15390.3i	− 1.00664i
$617$	2981.85i	0.194562i	0.995257	+	0.0972810i	$0.0310146\pi$
$617$	2981.85i	0.194562i	−0.995257	+	0.0972810i	$0.968985\pi$
$618$	− 19223.1i	− 1.25124i
$619$	15158.6i	0.984292i	0.870513	+	0.492146i	$0.163787\pi$
$619$	15158.6i	0.984292i	−0.870513	+	0.492146i	$0.836213\pi$
$620$	−5055.60	−0.327480
$621$	4957.19	0.320330
$622$	27674.8i	1.78402i
$623$	−1728.01	−0.111126
$624$	0	0
$625$	8391.01	0.537025
$626$	− 8360.45i	− 0.533787i
$627$	6318.69	0.402463
$628$	33276.8	2.11447
$629$	− 5310.51i	− 0.336636i
$630$	1442.40i	0.0912170i
$631$	5562.30i	0.350922i	0.984486	+	0.175461i	$0.0561415\pi$
$631$	5562.30i	0.350922i	−0.984486	+	0.175461i	$0.943858\pi$
$632$	− 16762.4i	− 1.05502i
$633$	11017.2	0.691776
$634$	28954.2	1.81375
$635$	− 5637.56i	− 0.352315i
$636$	−23071.9	−1.43846
$637$	0	0
$638$	−35591.7	−2.20861
$639$	902.693i	0.0558842i
$640$	−11936.5	−0.737237
$641$	24140.7	1.48752	0.743761	−	0.668446i	$-0.233040\pi$
$641$	24140.7	1.48752	0.743761	+	0.668446i	$0.233040\pi$
$642$	16126.4i	0.991366i
$643$	1749.69i	0.107311i	0.998560	+	0.0536555i	$0.0170873\pi$
$643$	1749.69i	0.107311i	−0.998560	+	0.0536555i	$0.982913\pi$
$644$	− 19899.1i	− 1.21760i
$645$	1537.03i	0.0938305i
$646$	14475.5	0.881626
$647$	−1489.51	−0.0905083	−0.0452542	−	0.998976i	$-0.514410\pi$
$647$	−1489.51	−0.0905083	−0.0452542	+	0.998976i	$0.514410\pi$
$648$	2490.07i	0.150956i
$649$	14695.7	0.888836
$650$	0	0
$651$	−1735.00	−0.104455
$652$	15721.4i	0.944323i
$653$	10668.2	0.639327	0.319663	−	0.947531i	$-0.396430\pi$
$653$	10668.2	0.639327	0.319663	+	0.947531i	$0.396430\pi$
$654$	14476.2	0.865544
$655$	5757.86i	0.343478i
$656$	− 13506.3i	− 0.803858i
$657$	− 1490.71i	− 0.0885205i
$658$	1499.77i	0.0888559i
$659$	−13933.1	−0.823608	−0.411804	−	0.911272i	$-0.635101\pi$
$659$	−13933.1	−0.823608	−0.411804	+	0.911272i	$0.635101\pi$
$660$	−13129.1	−0.774317
$661$	− 2349.69i	− 0.138264i	−0.997608	−	0.0691320i	$-0.977977\pi$
$661$	− 2349.69i	− 0.138264i	0.997608	−	0.0691320i	$-0.0220230\pi$
$662$	−35516.8	−2.08520
$663$	0	0
$664$	−13973.1	−0.816660
$665$	1064.13i	0.0620529i
$666$	2337.15	0.135980
$667$	−20599.9	−1.19585
$668$	1579.63i	0.0914937i
$669$	15261.1i	0.881958i
$670$	− 18677.0i	− 1.07695i
$671$	− 55049.6i	− 3.16716i
$672$	−2338.35	−0.134232
$673$	32421.0	1.85697	0.928483	−	0.371374i	$-0.121113\pi$
$673$	32421.0	1.85697	0.928483	+	0.371374i	$0.121113\pi$
$674$	44867.7i	2.56415i
$675$	−2824.19	−0.161042
$676$	0	0
$677$	−1071.74	−0.0608421	−0.0304211	−	0.999537i	$-0.509685\pi$
$677$	−1071.74	−0.0608421	−0.0304211	+	0.999537i	$0.509685\pi$
$678$	22334.1i	1.26510i
$679$	8150.56	0.460663
$680$	−13463.9	−0.759287
$681$	− 7877.48i	− 0.443268i
$682$	− 24515.5i	− 1.37646i
$683$	− 305.487i	− 0.0171144i	−0.999963	−	0.00855721i	$-0.997276\pi$
$683$	− 305.487i	− 0.0171144i	0.999963	−	0.00855721i	$-0.00272388\pi$
$684$	4103.84i	0.229407i
$685$	3951.72	0.220419
$686$	−22354.5	−1.24417
$687$	− 5036.18i	− 0.279683i
$688$	3391.66	0.187944
$689$	0	0
$690$	−11796.2	−0.650833
$691$	− 2180.81i	− 0.120061i	−0.998197	−	0.0600303i	$-0.980880\pi$
$691$	− 2180.81i	− 0.120061i	0.998197	−	0.0600303i	$-0.0191197\pi$
$692$	−27361.4	−1.50307
$693$	−4505.70	−0.246980
$694$	− 54340.9i	− 2.97227i
$695$	− 1401.68i	− 0.0765018i
$696$	− 10347.7i	− 0.563545i
$697$	43801.4i	2.38034i
$698$	−17420.6	−0.944672
$699$	1945.52	0.105274
$700$	11336.8i	0.612132i
$701$	15168.3	0.817259	0.408629	−	0.912700i	$-0.366007\pi$
$701$	15168.3	0.817259	0.408629	+	0.912700i	$0.366007\pi$
$702$	0	0
$703$	1724.23	0.0925043
$704$	− 49043.1i	− 2.62554i
$705$	572.720	0.0305956
$706$	−16067.8	−0.856545
$707$	− 577.963i	− 0.0307448i
$708$	9544.49i	0.506644i
$709$	− 8988.17i	− 0.476104i	−0.971252	−	0.238052i	$-0.923491\pi$
$709$	− 8988.17i	− 0.476104i	0.971252	−	0.238052i	$-0.0765089\pi$
$710$	− 2148.07i	− 0.113543i
$711$	−4907.39	−0.258849
$712$	7098.71	0.373645
$713$	− 14189.1i	− 0.745284i
$714$	−10322.1	−0.541029
$715$	0	0
$716$	49646.8	2.59132
$717$	15659.1i	0.815619i
$718$	−45870.7	−2.38423
$719$	−8448.18	−0.438198	−0.219099	−	0.975703i	$-0.570312\pi$
$719$	−8448.18	−0.438198	−0.219099	+	0.975703i	$0.570312\pi$
$720$	− 1215.44i	− 0.0629120i
$721$	10112.7i	0.522352i
$722$	− 27823.1i	− 1.43417i
$723$	− 18310.7i	− 0.941883i
$724$	3018.36	0.154940
$725$	11736.1	0.601197
$726$	− 44731.8i	− 2.28671i
$727$	8624.18	0.439963	0.219982	−	0.975504i	$-0.429400\pi$
$727$	8624.18	0.439963	0.219982	+	0.975504i	$0.429400\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	3547.31i	0.179852i
$731$	−10999.3	−0.556530
$732$	35753.5	1.80531
$733$	− 31124.2i	− 1.56835i	−0.620541	−	0.784174i	$-0.713087\pi$
$733$	− 31124.2i	− 1.56835i	0.620541	−	0.784174i	$-0.286913\pi$
$734$	41328.2i	2.07827i
$735$	3888.87i	0.195161i
$736$	− 19123.4i	− 0.957742i
$737$	58342.2	2.91596
$738$	−19276.9	−0.961509
$739$	− 17671.1i	− 0.879626i	−0.898089	−	0.439813i	$-0.855045\pi$
$739$	− 17671.1i	− 0.879626i	0.898089	−	0.439813i	$-0.144955\pi$
$740$	−3582.63	−0.177973
$741$	0	0
$742$	18841.6	0.932206
$743$	21331.1i	1.05325i	0.850099	+	0.526623i	$0.176542\pi$
$743$	21331.1i	1.05325i	−0.850099	+	0.526623i	$0.823458\pi$
$744$	7127.43	0.351215
$745$	−22.9477	−0.00112851
$746$	− 22134.0i	− 1.08630i
$747$	4090.81i	0.200368i
$748$	− 93954.1i	− 4.59265i
$749$	− 8483.58i	− 0.413863i
$750$	14751.7	0.718208
$751$	−11712.9	−0.569122	−0.284561	−	0.958658i	$-0.591848\pi$
$751$	−11712.9	−0.569122	−0.284561	+	0.958658i	$0.591848\pi$
$752$	− 1263.78i	− 0.0612835i
$753$	19270.2	0.932596
$754$	0	0
$755$	29.5795	0.00142584
$756$	− 2926.35i	− 0.140781i
$757$	−16610.9	−0.797537	−0.398768	−	0.917052i	$-0.630562\pi$
$757$	−16610.9	−0.797537	−0.398768	+	0.917052i	$0.630562\pi$
$758$	−51507.4	−2.46812
$759$	− 36848.4i	− 1.76220i
$760$	− 4371.47i	− 0.208645i
$761$	− 29365.5i	− 1.39882i	−0.714723	−	0.699408i	$-0.753447\pi$
$761$	− 29365.5i	− 1.39882i	0.714723	−	0.699408i	$-0.246553\pi$
$762$	17755.1i	0.844093i
$763$	−7615.50	−0.361336
$764$	−13871.2	−0.656861
$765$	3941.71i	0.186291i
$766$	−47673.5	−2.24871
$767$	0	0
$768$	19999.2	0.939659
$769$	28599.9i	1.34114i	0.741844	+	0.670572i	$0.233951\pi$
$769$	28599.9i	1.34114i	−0.741844	+	0.670572i	$0.766049\pi$
$770$	10721.8	0.501803
$771$	−3690.68	−0.172395
$772$	7426.03i	0.346203i
$773$	− 13491.8i	− 0.627772i	−0.949461	−	0.313886i	$-0.898369\pi$
$773$	− 13491.8i	− 0.627772i	0.949461	−	0.313886i	$-0.101631\pi$
$774$	− 4840.78i	− 0.224804i
$775$	8083.78i	0.374681i
$776$	−33482.7	−1.54892
$777$	−1229.50	−0.0567673
$778$	30434.3i	1.40247i
$779$	−14221.5	−0.654093
$780$	0	0
$781$	6710.01	0.307430
$782$	− 84415.9i	− 3.86024i
$783$	−3029.41	−0.138266
$784$	8581.27	0.390911
$785$	10377.5i	0.471834i
$786$	− 18134.0i	− 0.822923i
$787$	25876.7i	1.17205i	0.810293	+	0.586025i	$0.199308\pi$
$787$	25876.7i	1.17205i	−0.810293	+	0.586025i	$0.800692\pi$
$788$	− 56055.5i	− 2.53413i
$789$	1544.98	0.0697118
$790$	11677.7	0.525918
$791$	− 11749.3i	− 0.528137i
$792$	18509.5	0.830438
$793$	0	0
$794$	−56994.2	−2.54742
$795$	− 7195.06i	− 0.320984i
$796$	33396.1	1.48705
$797$	−21936.4	−0.974938	−0.487469	−	0.873140i	$-0.662080\pi$
$797$	−21936.4	−0.974938	−0.487469	+	0.873140i	$0.662080\pi$
$798$	− 3351.40i	− 0.148669i
$799$	4098.49i	0.181469i
$800$	10894.9i	0.481491i
$801$	− 2078.23i	− 0.0916739i
$802$	17069.0	0.751531
$803$	−11080.9	−0.486969
$804$	37891.9i	1.66212i
$805$	6205.62	0.271701
$806$	0	0
$807$	15397.8	0.671658
$808$	2374.29i	0.103375i
$809$	5583.23	0.242640	0.121320	−	0.992613i	$-0.461287\pi$
$809$	5583.23	0.242640	0.121320	+	0.992613i	$0.461287\pi$
$810$	−1734.74	−0.0752502
$811$	− 12925.4i	− 0.559647i	−0.960052	−	0.279823i	$-0.909724\pi$
$811$	− 12925.4i	− 0.559647i	0.960052	−	0.279823i	$-0.0902759\pi$
$812$	12160.6i	0.525559i
$813$	12900.0i	0.556486i
$814$	− 17372.8i	− 0.748054i
$815$	−4902.80	−0.210721
$816$	8697.87	0.373145
$817$	− 3571.27i	− 0.152929i
$818$	228.058	0.00974801
$819$	0	0
$820$	29549.7	1.25844
$821$	10153.2i	0.431608i	0.976437	+	0.215804i	$0.0692372\pi$
$821$	10153.2i	0.431608i	−0.976437	+	0.215804i	$0.930763\pi$
$822$	−12445.6	−0.528092
$823$	−3282.93	−0.139047	−0.0695235	−	0.997580i	$-0.522148\pi$
$823$	−3282.93	−0.139047	−0.0695235	+	0.997580i	$0.522148\pi$
$824$	− 41543.1i	− 1.75634i
$825$	20993.1i	0.885922i
$826$	− 7794.49i	− 0.328335i
$827$	− 17689.3i	− 0.743795i	−0.928274	−	0.371897i	$-0.878707\pi$
$827$	− 17689.3i	− 0.743795i	0.928274	−	0.371897i	$-0.121293\pi$
$828$	23932.2	1.00447
$829$	−38181.5	−1.59964	−0.799818	−	0.600243i	$-0.795070\pi$
$829$	−38181.5	−1.59964	−0.799818	+	0.600243i	$0.795070\pi$
$830$	− 9734.56i	− 0.407098i
$831$	−5438.40	−0.227023
$832$	0	0
$833$	−27829.4	−1.15754
$834$	4414.49i	0.183287i
$835$	−492.615	−0.0204163
$836$	30505.2	1.26201
$837$	− 2086.64i	− 0.0861708i
$838$	− 3430.41i	− 0.141410i
$839$	43895.2i	1.80623i	0.429395	+	0.903117i	$0.358727\pi$
$839$	43895.2i	1.80623i	−0.429395	+	0.903117i	$0.641273\pi$
$840$	3117.18i	0.128039i
$841$	−11800.1	−0.483829
$842$	70392.4	2.88109
$843$	12219.2i	0.499233i
$844$	53188.5	2.16922
$845$	0	0
$846$	−1803.74	−0.0733024
$847$	23532.0i	0.954627i
$848$	−15876.8	−0.642938
$849$	19038.9	0.769626
$850$	48093.0i	1.94068i
$851$	− 10055.1i	− 0.405034i
$852$	4358.00i	0.175238i
$853$	− 19955.2i	− 0.800998i	−0.916297	−	0.400499i	$-0.868837\pi$
$853$	− 19955.2i	− 0.800998i	0.916297	−	0.400499i	$-0.131163\pi$
$854$	−29198.0	−1.16995
$855$	−1279.80	−0.0511910
$856$	34850.8i	1.39156i
$857$	−26030.4	−1.03755	−0.518776	−	0.854910i	$-0.673612\pi$
$857$	−26030.4	−1.03755	−0.518776	+	0.854910i	$0.673612\pi$
$858$	0	0
$859$	−45617.4	−1.81193	−0.905964	−	0.423354i	$-0.860853\pi$
$859$	−45617.4	−1.81193	−0.905964	+	0.423354i	$0.860853\pi$
$860$	7420.45i	0.294227i
$861$	10141.0	0.401399
$862$	−5230.59	−0.206676
$863$	2010.93i	0.0793195i	0.999213	+	0.0396597i	$0.0126274\pi$
$863$	2010.93i	0.0793195i	−0.999213	+	0.0396597i	$0.987373\pi$
$864$	− 2812.27i	− 0.110735i
$865$	− 8532.78i	− 0.335403i
$866$	42170.1i	1.65473i
$867$	−13468.6	−0.527586
$868$	−8376.19	−0.327542
$869$	36478.2i	1.42398i
$870$	7208.83	0.280922
$871$	0	0
$872$	31284.7	1.21495
$873$	9802.48i	0.380027i
$874$	27408.3	1.06076
$875$	−7760.41	−0.299828
$876$	− 7196.79i	− 0.277576i
$877$	36767.3i	1.41567i	0.706376	+	0.707836i	$0.250329\pi$
$877$	36767.3i	1.41567i	−0.706376	+	0.707836i	$0.749671\pi$
$878$	− 51692.0i	− 1.98693i
$879$	25172.9i	0.965940i
$880$	−9034.72	−0.346091
$881$	−35401.2	−1.35380	−0.676899	−	0.736076i	$-0.736677\pi$
$881$	−35401.2	−1.35380	−0.676899	+	0.736076i	$0.736677\pi$
$882$	− 12247.7i	− 0.467575i
$883$	11928.0	0.454596	0.227298	−	0.973825i	$-0.427011\pi$
$883$	11928.0	0.454596	0.227298	+	0.973825i	$0.427011\pi$
$884$	0	0
$885$	−2976.49	−0.113055
$886$	17930.0i	0.679875i
$887$	−32939.3	−1.24689	−0.623447	−	0.781866i	$-0.714268\pi$
$887$	−32939.3	−1.24689	−0.623447	+	0.781866i	$0.714268\pi$
$888$	5050.83	0.190872
$889$	− 9340.40i	− 0.352381i
$890$	4945.41i	0.186259i
$891$	− 5418.89i	− 0.203748i
$892$	73677.3i	2.76558i
$893$	−1330.70	−0.0498660
$894$	72.2720	0.00270373
$895$	15482.6i	0.578241i
$896$	−19776.6	−0.737376
$897$	0	0
$898$	506.569	0.0188245
$899$	8671.18i	0.321691i
$900$	−13634.5	−0.504983
$901$	51489.2	1.90383
$902$	143292.i	5.28946i
$903$	2546.58i	0.0938483i
$904$	48266.4i	1.77579i
$905$	941.289i	0.0345740i
$906$	−93.1585	−0.00341610
$907$	−1500.89	−0.0549461	−0.0274731	−	0.999623i	$-0.508746\pi$
$907$	−1500.89	−0.0549461	−0.0274731	+	0.999623i	$0.508746\pi$
$908$	− 38030.7i	− 1.38997i
$909$	695.102	0.0253631
$910$	0	0
$911$	19728.2	0.717481	0.358740	−	0.933437i	$-0.383206\pi$
$911$	19728.2	0.717481	0.358740	+	0.933437i	$0.383206\pi$
$912$	2824.04i	0.102537i
$913$	30408.3	1.10226
$914$	5868.48	0.212376
$915$	11149.9i	0.402845i
$916$	− 24313.5i	− 0.877011i
$917$	9539.72i	0.343543i
$918$	− 12414.1i	− 0.446326i
$919$	19992.0	0.717602	0.358801	−	0.933414i	$-0.383186\pi$
$919$	19992.0	0.717602	0.358801	+	0.933414i	$0.383186\pi$
$920$	−25492.9	−0.913560
$921$	− 12015.8i	− 0.429897i
$922$	−41683.4	−1.48891
$923$	0	0
$924$	−21752.5	−0.774463
$925$	5728.54i	0.203625i
$926$	−18310.2	−0.649794
$927$	−12162.3	−0.430918
$928$	11686.6i	0.413395i
$929$	1923.62i	0.0679354i	0.999423	+	0.0339677i	$0.0108143\pi$
$929$	1923.62i	0.0679354i	−0.999423	+	0.0339677i	$0.989186\pi$
$930$	4965.42i	0.175078i
$931$	− 9035.71i	− 0.318081i
$932$	9392.52	0.330110
$933$	17509.6	0.614403
$934$	− 42634.0i	− 1.49361i
$935$	29300.0	1.02483
$936$	0	0
$937$	−10252.9	−0.357468	−0.178734	−	0.983897i	$-0.557200\pi$
$937$	−10252.9	−0.357468	−0.178734	+	0.983897i	$0.557200\pi$
$938$	− 30944.4i	− 1.07715i
$939$	−5289.58	−0.183833
$940$	2764.96	0.0959394
$941$	− 11043.0i	− 0.382563i	−0.981535	−	0.191282i	$-0.938736\pi$
$941$	− 11043.0i	− 0.382563i	0.981535	−	0.191282i	$-0.0612644\pi$
$942$	− 32683.2i	− 1.13044i
$943$	82934.8i	2.86398i
$944$	6568.00i	0.226451i
$945$	912.594	0.0314145
$946$	−35983.1	−1.23669
$947$	32105.2i	1.10167i	0.834615	+	0.550833i	$0.185690\pi$
$947$	32105.2i	1.10167i	−0.834615	+	0.550833i	$0.814310\pi$
$948$	−23691.8	−0.811680
$949$	0	0
$950$	−15614.9	−0.533280
$951$	− 18319.0i	− 0.624643i
$952$	−22307.1	−0.759431
$953$	9473.37	0.322007	0.161003	−	0.986954i	$-0.448527\pi$
$953$	9473.37	0.322007	0.161003	+	0.986954i	$0.448527\pi$
$954$	22660.3i	0.769031i
$955$	− 4325.79i	− 0.146575i
$956$	75598.4i	2.55756i
$957$	22518.5i	0.760628i
$958$	19819.3	0.668404
$959$	6547.26	0.220461
$960$	9933.30i	0.333954i
$961$	23818.3	0.799514
$962$	0	0
$963$	10203.0	0.341419
$964$	− 88399.8i	− 2.95349i
$965$	−2315.84	−0.0772534
$966$	−19544.2	−0.650956
$967$	5310.75i	0.176610i	0.996093	+	0.0883052i	$0.0281451\pi$
$967$	5310.75i	0.176610i	−0.996093	+	0.0883052i	$0.971855\pi$
$968$	− 96670.1i	− 3.20981i
$969$	− 9158.49i	− 0.303626i
$970$	− 23326.2i	− 0.772122i
$971$	24271.2	0.802164	0.401082	−	0.916042i	$-0.368634\pi$
$971$	24271.2	0.802164	0.401082	+	0.916042i	$0.368634\pi$
$972$	3519.45	0.116138
$973$	− 2322.32i	− 0.0765163i
$974$	87454.2	2.87702
$975$	0	0
$976$	24603.6	0.806907
$977$	49602.5i	1.62428i	0.583460	+	0.812142i	$0.301699\pi$
$977$	49602.5i	1.62428i	−0.583460	+	0.812142i	$0.698301\pi$
$978$	15441.0	0.504856
$979$	−15448.2	−0.504317
$980$	18774.6i	0.611971i
$981$	− 9158.98i	− 0.298087i
$982$	38375.5i	1.24706i
$983$	47385.7i	1.53751i	0.639545	+	0.768753i	$0.279123\pi$
$983$	47385.7i	1.53751i	−0.639545	+	0.768753i	$0.720877\pi$
$984$	−41659.5	−1.34965
$985$	17481.2	0.565478
$986$	51587.7i	1.66621i
$987$	948.891	0.0306014
$988$	0	0
$989$	−20826.4	−0.669607
$990$	12894.9i	0.413966i
$991$	−8947.33	−0.286802	−0.143401	−	0.989665i	$-0.545804\pi$
$991$	−8947.33	−0.286802	−0.143401	+	0.989665i	$0.545804\pi$
$992$	−8049.67	−0.257638
$993$	22471.1i	0.718127i
$994$	− 3558.95i	− 0.113564i
$995$	10414.7i	0.331828i
$996$	19749.5i	0.628299i
$997$	−14908.6	−0.473582	−0.236791	−	0.971561i	$-0.576096\pi$
$997$	−14908.6	−0.473582	−0.236791	+	0.971561i	$0.576096\pi$
$998$	51881.4	1.64557
$999$	− 1478.69i	− 0.0468306i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.f.337.1		4
13.5	odd	4		507.4.a.f.1.1		2
13.8	odd	4		39.4.a.b.1.2	✓	2
13.12	even	2	inner	507.4.b.f.337.4		4
39.5	even	4		1521.4.a.s.1.2		2
39.8	even	4		117.4.a.c.1.1		2
52.47	even	4		624.4.a.r.1.1		2
65.34	odd	4		975.4.a.j.1.1		2
91.34	even	4		1911.4.a.h.1.2		2
104.21	odd	4		2496.4.a.bc.1.2		2
104.99	even	4		2496.4.a.s.1.2		2
156.47	odd	4		1872.4.a.t.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.a.b.1.2	✓	2	13.8	odd	4
117.4.a.c.1.1		2	39.8	even	4
507.4.a.f.1.1		2	13.5	odd	4
507.4.b.f.337.1		4	1.1	even	1	trivial
507.4.b.f.337.4		4	13.12	even	2	inner
624.4.a.r.1.1		2	52.47	even	4
975.4.a.j.1.1		2	65.34	odd	4
1521.4.a.s.1.2		2	39.5	even	4
1872.4.a.t.1.2		2	156.47	odd	4
1911.4.a.h.1.2		2	91.34	even	4
2496.4.a.s.1.2		2	104.99	even	4
2496.4.a.bc.1.2		2	104.21	odd	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)

\(f(q)\)	\(=\)	\(q-4.74166i q^{2} -3.00000 q^{3} -14.4833 q^{4} -4.51669i q^{5} +14.2250i q^{6} -7.48331i q^{7} +30.7417i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-4.74166i q^{2} -3.00000 q^{3} -14.4833 q^{4} -4.51669i q^{5} +14.2250i q^{6} -7.48331i q^{7} +30.7417i q^{8} +9.00000 q^{9} -21.4166 q^{10} -66.8999i q^{11} +43.4499 q^{12} -35.4833 q^{14} +13.5501i q^{15} +29.8999 q^{16} -96.9666 q^{17} -42.6749i q^{18} -31.4833i q^{19} +65.4166i q^{20} +22.4499i q^{21} -317.216 q^{22} -183.600 q^{23} -92.2250i q^{24} +104.600 q^{25} -27.0000 q^{27} +108.383i q^{28} +112.200 q^{29} +64.2497 q^{30} +77.2831i q^{31} +104.158i q^{32} +200.700i q^{33} +459.783i q^{34} -33.7998 q^{35} -130.350 q^{36} +54.7664i q^{37} -149.283 q^{38} +138.850 q^{40} -451.716i q^{41} +106.450 q^{42} +113.434 q^{43} +968.932i q^{44} -40.6502i q^{45} +870.566i q^{46} -42.2670i q^{47} -89.6997 q^{48} +287.000 q^{49} -495.975i q^{50} +290.900 q^{51} -530.999 q^{53} +128.025i q^{54} -302.166 q^{55} +230.049 q^{56} +94.4499i q^{57} -532.015i q^{58} +219.666i q^{59} -196.250i q^{60} +822.865 q^{61} +366.450 q^{62} -67.3498i q^{63} +733.082 q^{64} +951.649 q^{66} +872.082i q^{67} +1404.40 q^{68} +550.799 q^{69} +160.267i q^{70} +100.299i q^{71} +276.675i q^{72} -165.634i q^{73} +259.684 q^{74} -313.799 q^{75} +455.983i q^{76} -500.633 q^{77} -545.266 q^{79} -135.048i q^{80} +81.0000 q^{81} -2141.88 q^{82} +454.534i q^{83} -325.150i q^{84} +437.968i q^{85} -537.864i q^{86} -336.601 q^{87} +2056.61 q^{88} -230.915i q^{89} -192.749 q^{90} +2659.13 q^{92} -231.849i q^{93} -200.415 q^{94} -142.200 q^{95} -312.475i q^{96} +1089.16i q^{97} -1360.86i q^{98} -602.099i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} - 28 q^{4} + 36 q^{9}+O(q^{10}) \) 4 * q - 12 * q^3 - 28 * q^4 + 36 * q^9	\( 4 q - 12 q^{3} - 28 q^{4} + 36 q^{9} + 64 q^{10} + 84 q^{12} - 112 q^{14} - 60 q^{16} - 328 q^{17} - 760 q^{22} - 16 q^{23} - 300 q^{25} - 108 q^{27} + 808 q^{29} - 192 q^{30} + 224 q^{35} - 252 q^{36} - 208 q^{38} + 1184 q^{40} + 336 q^{42} + 1232 q^{43} + 180 q^{48} + 1148 q^{49} + 984 q^{51} - 328 q^{53} + 288 q^{55} + 112 q^{56} + 1256 q^{61} + 1376 q^{62} + 388 q^{64} + 2280 q^{66} + 2744 q^{68} + 48 q^{69} + 1368 q^{74} + 900 q^{75} - 1344 q^{77} - 864 q^{79} + 324 q^{81} - 5664 q^{82} - 2424 q^{87} + 3048 q^{88} + 576 q^{90} + 5488 q^{92} + 1144 q^{94} - 928 q^{95}+O(q^{100}) \) 4 * q - 12 * q^3 - 28 * q^4 + 36 * q^9 + 64 * q^10 + 84 * q^12 - 112 * q^14 - 60 * q^16 - 328 * q^17 - 760 * q^22 - 16 * q^23 - 300 * q^25 - 108 * q^27 + 808 * q^29 - 192 * q^30 + 224 * q^35 - 252 * q^36 - 208 * q^38 + 1184 * q^40 + 336 * q^42 + 1232 * q^43 + 180 * q^48 + 1148 * q^49 + 984 * q^51 - 328 * q^53 + 288 * q^55 + 112 * q^56 + 1256 * q^61 + 1376 * q^62 + 388 * q^64 + 2280 * q^66 + 2744 * q^68 + 48 * q^69 + 1368 * q^74 + 900 * q^75 - 1344 * q^77 - 864 * q^79 + 324 * q^81 - 5664 * q^82 - 2424 * q^87 + 3048 * q^88 + 576 * q^90 + 5488 * q^92 + 1144 * q^94 - 928 * q^95

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