LMFDB - Embedded newform 507.4.a.o.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

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

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

chi := DirichletCharacter("507.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	507.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$29.9139683729$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 56x^{7} - 27x^{6} + 945x^{5} + 763x^{4} - 4139x^{3} - 2478x^{2} + 63x + 27 $ x^9 - 56x^7 - 27x^6 + 945x^5 + 763x^4 - 4139x^3 - 2478x^2 + 63*x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$2.37150$ of defining polynomial
Character	$\chi$	$=$	507.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.17344 q^{2} +3.00000 q^{3} +2.07074 q^{4} -6.74147 q^{5} +9.52033 q^{6} +14.1726 q^{7} -18.8162 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+3.17344 q^{2} +3.00000 q^{3} +2.07074 q^{4} -6.74147 q^{5} +9.52033 q^{6} +14.1726 q^{7} -18.8162 q^{8} +9.00000 q^{9} -21.3937 q^{10} -62.4956 q^{11} +6.21221 q^{12} +44.9761 q^{14} -20.2244 q^{15} -76.2779 q^{16} -58.6172 q^{17} +28.5610 q^{18} -64.1652 q^{19} -13.9598 q^{20} +42.5179 q^{21} -198.326 q^{22} +10.9221 q^{23} -56.4485 q^{24} -79.5526 q^{25} +27.0000 q^{27} +29.3478 q^{28} +216.316 q^{29} -64.1810 q^{30} +38.6271 q^{31} -91.5342 q^{32} -187.487 q^{33} -186.018 q^{34} -95.5445 q^{35} +18.6366 q^{36} -423.770 q^{37} -203.625 q^{38} +126.849 q^{40} +366.126 q^{41} +134.928 q^{42} -128.297 q^{43} -129.412 q^{44} -60.6732 q^{45} +34.6605 q^{46} +93.1169 q^{47} -228.834 q^{48} -142.136 q^{49} -252.455 q^{50} -175.852 q^{51} +131.909 q^{53} +85.6829 q^{54} +421.313 q^{55} -266.675 q^{56} -192.496 q^{57} +686.467 q^{58} -386.729 q^{59} -41.8794 q^{60} -621.077 q^{61} +122.581 q^{62} +127.554 q^{63} +319.745 q^{64} -594.979 q^{66} -865.273 q^{67} -121.381 q^{68} +32.7662 q^{69} -303.205 q^{70} +607.506 q^{71} -169.346 q^{72} -980.958 q^{73} -1344.81 q^{74} -238.658 q^{75} -132.869 q^{76} -885.728 q^{77} +1331.91 q^{79} +514.226 q^{80} +81.0000 q^{81} +1161.88 q^{82} -907.633 q^{83} +88.0434 q^{84} +395.166 q^{85} -407.142 q^{86} +648.949 q^{87} +1175.93 q^{88} +1033.67 q^{89} -192.543 q^{90} +22.6167 q^{92} +115.881 q^{93} +295.501 q^{94} +432.568 q^{95} -274.603 q^{96} -1046.17 q^{97} -451.061 q^{98} -562.461 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 6 q^{2} + 27 q^{3} + 44 q^{4} - 33 q^{5} - 18 q^{6} - 83 q^{7} - 87 q^{8} + 81 q^{9}+O(q^{10}) $ 9 * q - 6 * q^2 + 27 * q^3 + 44 * q^4 - 33 * q^5 - 18 * q^6 - 83 * q^7 - 87 * q^8 + 81 * q^9	\( 9 q - 6 q^{2} + 27 q^{3} + 44 q^{4} - 33 q^{5} - 18 q^{6} - 83 q^{7} - 87 q^{8} + 81 q^{9} - 54 q^{10} - 85 q^{11} + 132 q^{12} + 158 q^{14} - 99 q^{15} + 216 q^{16} + 178 q^{17} - 54 q^{18} - 352 q^{19} - 402 q^{20} - 249 q^{21} - 630 q^{22} + 150 q^{23} - 261 q^{24} - 20 q^{25} + 243 q^{27} - 940 q^{28} - 97 q^{29} - 162 q^{30} - 717 q^{31} - 707 q^{32} - 255 q^{33} - 632 q^{34} - 418 q^{35} + 396 q^{36} - 1108 q^{37} - 660 q^{38} - 1506 q^{40} - 334 q^{41} + 474 q^{42} + 242 q^{43} + 307 q^{44} - 297 q^{45} - 979 q^{46} + 184 q^{47} + 648 q^{48} - 38 q^{49} + 2031 q^{50} + 534 q^{51} - 151 q^{53} - 162 q^{54} + 2064 q^{55} + 2276 q^{56} - 1056 q^{57} - 1161 q^{58} - 537 q^{59} - 1206 q^{60} - 1340 q^{61} + 347 q^{62} - 747 q^{63} + 893 q^{64} - 1890 q^{66} - 2308 q^{67} + 2785 q^{68} + 450 q^{69} + 1420 q^{70} - 96 q^{71} - 783 q^{72} - 2505 q^{73} - 1191 q^{74} - 60 q^{75} - 2409 q^{76} - 2142 q^{77} - 1591 q^{79} + 2671 q^{80} + 729 q^{81} + 1517 q^{82} - 1539 q^{83} - 2820 q^{84} - 4296 q^{85} + 3763 q^{86} - 291 q^{87} - 3716 q^{88} + 592 q^{89} - 486 q^{90} + 515 q^{92} - 2151 q^{93} - 692 q^{94} + 4158 q^{95} - 2121 q^{96} - 1445 q^{97} - 1457 q^{98} - 765 q^{99}+O(q^{100}) \) 9 * q - 6 * q^2 + 27 * q^3 + 44 * q^4 - 33 * q^5 - 18 * q^6 - 83 * q^7 - 87 * q^8 + 81 * q^9 - 54 * q^10 - 85 * q^11 + 132 * q^12 + 158 * q^14 - 99 * q^15 + 216 * q^16 + 178 * q^17 - 54 * q^18 - 352 * q^19 - 402 * q^20 - 249 * q^21 - 630 * q^22 + 150 * q^23 - 261 * q^24 - 20 * q^25 + 243 * q^27 - 940 * q^28 - 97 * q^29 - 162 * q^30 - 717 * q^31 - 707 * q^32 - 255 * q^33 - 632 * q^34 - 418 * q^35 + 396 * q^36 - 1108 * q^37 - 660 * q^38 - 1506 * q^40 - 334 * q^41 + 474 * q^42 + 242 * q^43 + 307 * q^44 - 297 * q^45 - 979 * q^46 + 184 * q^47 + 648 * q^48 - 38 * q^49 + 2031 * q^50 + 534 * q^51 - 151 * q^53 - 162 * q^54 + 2064 * q^55 + 2276 * q^56 - 1056 * q^57 - 1161 * q^58 - 537 * q^59 - 1206 * q^60 - 1340 * q^61 + 347 * q^62 - 747 * q^63 + 893 * q^64 - 1890 * q^66 - 2308 * q^67 + 2785 * q^68 + 450 * q^69 + 1420 * q^70 - 96 * q^71 - 783 * q^72 - 2505 * q^73 - 1191 * q^74 - 60 * q^75 - 2409 * q^76 - 2142 * q^77 - 1591 * q^79 + 2671 * q^80 + 729 * q^81 + 1517 * q^82 - 1539 * q^83 - 2820 * q^84 - 4296 * q^85 + 3763 * q^86 - 291 * q^87 - 3716 * q^88 + 592 * q^89 - 486 * q^90 + 515 * q^92 - 2151 * q^93 - 692 * q^94 + 4158 * q^95 - 2121 * q^96 - 1445 * q^97 - 1457 * q^98 - 765 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	3.17344	1.12198	0.560991	−	0.827822i	$-0.310420\pi$
$2$	3.17344	1.12198	0.560991	+	0.827822i	$0.310420\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	2.07074	0.258842
$5$	−6.74147	−0.602975	−0.301488	−	0.953470i	$-0.597483\pi$
$5$	−6.74147	−0.602975	−0.301488	+	0.953470i	$0.597483\pi$
$6$	9.52033	0.647776
$7$	14.1726	0.765251	0.382625	−	0.923904i	$-0.375020\pi$
$7$	14.1726	0.765251	0.382625	+	0.923904i	$0.375020\pi$
$8$	−18.8162	−0.831565
$9$	9.00000	0.333333
$10$	−21.3937	−0.676527
$11$	−62.4956	−1.71301	−0.856507	−	0.516136i	$-0.827370\pi$
$11$	−62.4956	−1.71301	−0.856507	+	0.516136i	$0.827370\pi$
$12$	6.21221	0.149442
$13$	0	0
$13$	0	0
$14$	44.9761	0.858597
$15$	−20.2244	−0.348128
$16$	−76.2779	−1.19184
$17$	−58.6172	−0.836280	−0.418140	−	0.908383i	$-0.637318\pi$
$17$	−58.6172	−0.836280	−0.418140	+	0.908383i	$0.637318\pi$
$18$	28.5610	0.373994
$19$	−64.1652	−0.774764	−0.387382	−	0.921919i	$-0.626620\pi$
$19$	−64.1652	−0.774764	−0.387382	+	0.921919i	$0.626620\pi$
$20$	−13.9598	−0.156075
$21$	42.5179	0.441818
$22$	−198.326	−1.92197
$23$	10.9221	0.0990177	0.0495088	−	0.998774i	$-0.484234\pi$
$23$	10.9221	0.0990177	0.0495088	+	0.998774i	$0.484234\pi$
$24$	−56.4485	−0.480105
$25$	−79.5526	−0.636421
$26$	0	0
$27$	27.0000	0.192450
$28$	29.3478	0.198079
$29$	216.316	1.38514	0.692568	−	0.721353i	$-0.256479\pi$
$29$	216.316	1.38514	0.692568	+	0.721353i	$0.256479\pi$
$30$	−64.1810	−0.390593
$31$	38.6271	0.223795	0.111897	−	0.993720i	$-0.464307\pi$
$31$	38.6271	0.223795	0.111897	+	0.993720i	$0.464307\pi$
$32$	−91.5342	−0.505660
$33$	−187.487	−0.989009
$34$	−186.018	−0.938290
$35$	−95.5445	−0.461427
$36$	18.6366	0.0862807
$37$	−423.770	−1.88290	−0.941452	−	0.337147i	$-0.890538\pi$
$37$	−423.770	−1.88290	−0.941452	+	0.337147i	$0.890538\pi$
$38$	−203.625	−0.869270
$39$	0	0
$40$	126.849	0.501414
$41$	366.126	1.39461	0.697307	−	0.716772i	$-0.254381\pi$
$41$	366.126	1.39461	0.697307	+	0.716772i	$0.254381\pi$
$42$	134.928	0.495711
$43$	−128.297	−0.455001	−0.227501	−	0.973778i	$-0.573055\pi$
$43$	−128.297	−0.455001	−0.227501	+	0.973778i	$0.573055\pi$
$44$	−129.412	−0.443400
$45$	−60.6732	−0.200992
$46$	34.6605	0.111096
$47$	93.1169	0.288989	0.144495	−	0.989506i	$-0.453844\pi$
$47$	93.1169	0.288989	0.144495	+	0.989506i	$0.453844\pi$
$48$	−228.834	−0.688111
$49$	−142.136	−0.414391
$50$	−252.455	−0.714052
$51$	−175.852	−0.482826
$52$	0	0
$53$	131.909	0.341869	0.170934	−	0.985282i	$-0.445321\pi$
$53$	131.909	0.341869	0.170934	+	0.985282i	$0.445321\pi$
$54$	85.6829	0.215925
$55$	421.313	1.03290
$56$	−266.675	−0.636356
$57$	−192.496	−0.447310
$58$	686.467	1.55410
$59$	−386.729	−0.853353	−0.426677	−	0.904404i	$-0.640316\pi$
$59$	−386.729	−0.853353	−0.426677	+	0.904404i	$0.640316\pi$
$60$	−41.8794	−0.0901102
$61$	−621.077	−1.30362	−0.651810	−	0.758382i	$-0.725990\pi$
$61$	−621.077	−1.30362	−0.651810	+	0.758382i	$0.725990\pi$
$62$	122.581	0.251094
$63$	127.554	0.255084
$64$	319.745	0.624502
$65$	0	0
$66$	−594.979	−1.10965
$67$	−865.273	−1.57776	−0.788880	−	0.614547i	$-0.789339\pi$
$67$	−865.273	−1.57776	−0.788880	+	0.614547i	$0.789339\pi$
$68$	−121.381	−0.216464
$69$	32.7662	0.0571679
$70$	−303.205	−0.517713
$71$	607.506	1.01546	0.507730	−	0.861516i	$-0.330485\pi$
$71$	607.506	1.01546	0.507730	+	0.861516i	$0.330485\pi$
$72$	−169.346	−0.277188
$73$	−980.958	−1.57277	−0.786387	−	0.617735i	$-0.788051\pi$
$73$	−980.958	−1.57277	−0.786387	+	0.617735i	$0.788051\pi$
$74$	−1344.81	−2.11258
$75$	−238.658	−0.367438
$76$	−132.869	−0.200541
$77$	−885.728	−1.31088
$78$	0	0
$79$	1331.91	1.89685	0.948425	−	0.317003i	$-0.102676\pi$
$79$	1331.91	1.89685	0.948425	+	0.317003i	$0.102676\pi$
$80$	514.226	0.718652
$81$	81.0000	0.111111
$82$	1161.88	1.56473
$83$	−907.633	−1.20031	−0.600155	−	0.799884i	$-0.704894\pi$
$83$	−907.633	−1.20031	−0.600155	+	0.799884i	$0.704894\pi$
$84$	88.0434	0.114361
$85$	395.166	0.504256
$86$	−407.142	−0.510503
$87$	648.949	0.799708
$88$	1175.93	1.42448
$89$	1033.67	1.23110	0.615552	−	0.788096i	$-0.288933\pi$
$89$	1033.67	1.23110	0.615552	+	0.788096i	$0.288933\pi$
$90$	−192.543	−0.225509
$91$	0	0
$92$	22.6167	0.0256299
$93$	115.881	0.129208
$94$	295.501	0.324240
$95$	432.568	0.467163
$96$	−274.603	−0.291943
$97$	−1046.17	−1.09508	−0.547538	−	0.836781i	$-0.684435\pi$
$97$	−1046.17	−1.09508	−0.547538	+	0.836781i	$0.684435\pi$
$98$	−451.061	−0.464939
$99$	−562.461	−0.571004
$100$	−164.732	−0.164732
$101$	1416.64	1.39566	0.697828	−	0.716265i	$-0.254150\pi$
$101$	1416.64	1.39566	0.697828	+	0.716265i	$0.254150\pi$
$102$	−558.055	−0.541722
$103$	387.629	0.370818	0.185409	−	0.982661i	$-0.440639\pi$
$103$	387.629	0.370818	0.185409	+	0.982661i	$0.440639\pi$
$104$	0	0
$105$	−286.633	−0.266405
$106$	418.605	0.383570
$107$	86.4526	0.0781092	0.0390546	−	0.999237i	$-0.487565\pi$
$107$	86.4526	0.0781092	0.0390546	+	0.999237i	$0.487565\pi$
$108$	55.9099	0.0498142
$109$	940.072	0.826079	0.413039	−	0.910713i	$-0.364467\pi$
$109$	940.072	0.826079	0.413039	+	0.910713i	$0.364467\pi$
$110$	1337.01	1.15890
$111$	−1271.31	−1.08709
$112$	−1081.06	−0.912059
$113$	960.499	0.799612	0.399806	−	0.916600i	$-0.369078\pi$
$113$	960.499	0.799612	0.399806	+	0.916600i	$0.369078\pi$
$114$	−610.874	−0.501873
$115$	−73.6307	−0.0597052
$116$	447.934	0.358531
$117$	0	0
$118$	−1227.26	−0.957447
$119$	−830.760	−0.639964
$120$	380.546	0.289491
$121$	2574.71	1.93441
$122$	−1970.95	−1.46264
$123$	1098.38	0.805181
$124$	79.9866	0.0579275
$125$	1378.99	0.986721
$126$	404.785	0.286199
$127$	2022.18	1.41291	0.706456	−	0.707757i	$-0.250293\pi$
$127$	2022.18	1.41291	0.706456	+	0.707757i	$0.250293\pi$
$128$	1746.97	1.20634
$129$	−384.890	−0.262695
$130$	0	0
$131$	1857.90	1.23912	0.619561	−	0.784948i	$-0.287310\pi$
$131$	1857.90	1.23912	0.619561	+	0.784948i	$0.287310\pi$
$132$	−388.236	−0.255997
$133$	−909.390	−0.592888
$134$	−2745.90	−1.77022
$135$	−182.020	−0.116043
$136$	1102.95	0.695421
$137$	−1894.12	−1.18121	−0.590604	−	0.806961i	$-0.701110\pi$
$137$	−1894.12	−1.18121	−0.590604	+	0.806961i	$0.701110\pi$
$138$	103.982	0.0641413
$139$	−1226.08	−0.748165	−0.374082	−	0.927395i	$-0.622042\pi$
$139$	−1226.08	−0.748165	−0.374082	+	0.927395i	$0.622042\pi$
$140$	−197.847	−0.119437
$141$	279.351	0.166848
$142$	1927.89	1.13933
$143$	0	0
$144$	−686.501	−0.397281
$145$	−1458.29	−0.835203
$146$	−3113.01	−1.76462
$147$	−426.409	−0.239249
$148$	−877.517	−0.487375
$149$	−3195.65	−1.75703	−0.878517	−	0.477711i	$-0.841467\pi$
$149$	−3195.65	−1.75703	−0.878517	+	0.477711i	$0.841467\pi$
$150$	−757.366	−0.412258
$151$	−508.232	−0.273903	−0.136951	−	0.990578i	$-0.543730\pi$
$151$	−508.232	−0.273903	−0.136951	+	0.990578i	$0.543730\pi$
$152$	1207.34	0.644267
$153$	−527.555	−0.278760
$154$	−2810.81	−1.47079
$155$	−260.404	−0.134943
$156$	0	0
$157$	−1243.08	−0.631900	−0.315950	−	0.948776i	$-0.602323\pi$
$157$	−1243.08	−0.631900	−0.315950	+	0.948776i	$0.602323\pi$
$158$	4226.73	2.12823
$159$	395.726	0.197378
$160$	617.075	0.304901
$161$	154.794	0.0757734
$162$	257.049	0.124665
$163$	33.9996	0.0163378	0.00816888	−	0.999967i	$-0.497400\pi$
$163$	33.9996	0.0163378	0.00816888	+	0.999967i	$0.497400\pi$
$164$	758.149	0.360985
$165$	1263.94	0.596348
$166$	−2880.32	−1.34673
$167$	2210.67	1.02435	0.512176	−	0.858880i	$-0.328839\pi$
$167$	2210.67	1.02435	0.512176	+	0.858880i	$0.328839\pi$
$168$	−800.025	−0.367400
$169$	0	0
$170$	1254.04	0.565766
$171$	−577.487	−0.258255
$172$	−265.668	−0.117773
$173$	−661.307	−0.290626	−0.145313	−	0.989386i	$-0.546419\pi$
$173$	−661.307	−0.290626	−0.145313	+	0.989386i	$0.546419\pi$
$174$	2059.40	0.897258
$175$	−1127.47	−0.487021
$176$	4767.04	2.04164
$177$	−1160.19	−0.492684
$178$	3280.28	1.38128
$179$	−2325.05	−0.970850	−0.485425	−	0.874278i	$-0.661335\pi$
$179$	−2325.05	−0.970850	−0.485425	+	0.874278i	$0.661335\pi$
$180$	−125.638	−0.0520251
$181$	2122.20	0.871503	0.435752	−	0.900067i	$-0.356483\pi$
$181$	2122.20	0.871503	0.435752	+	0.900067i	$0.356483\pi$
$182$	0	0
$183$	−1863.23	−0.752645
$184$	−205.511	−0.0823397
$185$	2856.84	1.13534
$186$	367.743	0.144969
$187$	3663.32	1.43256
$188$	192.820	0.0748025
$189$	382.661	0.147273
$190$	1372.73	0.524149
$191$	−2484.37	−0.941166	−0.470583	−	0.882356i	$-0.655956\pi$
$191$	−2484.37	−0.941166	−0.470583	+	0.882356i	$0.655956\pi$
$192$	959.235	0.360556
$193$	266.771	0.0994955	0.0497478	−	0.998762i	$-0.484158\pi$
$193$	266.771	0.0994955	0.0497478	+	0.998762i	$0.484158\pi$
$194$	−3319.95	−1.22865
$195$	0	0
$196$	−294.327	−0.107262
$197$	1231.03	0.445216	0.222608	−	0.974908i	$-0.428543\pi$
$197$	1231.03	0.445216	0.222608	+	0.974908i	$0.428543\pi$
$198$	−1784.94	−0.640656
$199$	−3246.14	−1.15635	−0.578173	−	0.815914i	$-0.696234\pi$
$199$	−3246.14	−1.15635	−0.578173	+	0.815914i	$0.696234\pi$
$200$	1496.88	0.529225
$201$	−2595.82	−0.910921
$202$	4495.63	1.56590
$203$	3065.77	1.05998
$204$	−364.142	−0.124976
$205$	−2468.23	−0.840919
$206$	1230.12	0.416051
$207$	98.2985	0.0330059
$208$	0	0
$209$	4010.05	1.32718
$210$	−909.614	−0.298902
$211$	330.708	0.107900	0.0539500	−	0.998544i	$-0.482819\pi$
$211$	330.708	0.107900	0.0539500	+	0.998544i	$0.482819\pi$
$212$	273.148	0.0884900
$213$	1822.52	0.586277
$214$	274.352	0.0876371
$215$	864.908	0.274355
$216$	−508.037	−0.160035
$217$	547.449	0.171259
$218$	2983.26	0.926845
$219$	−2942.87	−0.908041
$220$	872.427	0.267359
$221$	0	0
$222$	−4034.43	−1.21970
$223$	−5785.86	−1.73744	−0.868722	−	0.495300i	$-0.835058\pi$
$223$	−5785.86	−1.73744	−0.868722	+	0.495300i	$0.835058\pi$
$224$	−1297.28	−0.386957
$225$	−715.973	−0.212140
$226$	3048.09	0.897149
$227$	−2945.35	−0.861189	−0.430595	−	0.902545i	$-0.641696\pi$
$227$	−2945.35	−0.861189	−0.430595	+	0.902545i	$0.641696\pi$
$228$	−398.608	−0.115783
$229$	3541.26	1.02189	0.510945	−	0.859613i	$-0.329296\pi$
$229$	3541.26	1.02189	0.510945	+	0.859613i	$0.329296\pi$
$230$	−233.663	−0.0669882
$231$	−2657.18	−0.756840
$232$	−4070.25	−1.15183
$233$	2340.76	0.658148	0.329074	−	0.944304i	$-0.393263\pi$
$233$	2340.76	0.658148	0.329074	+	0.944304i	$0.393263\pi$
$234$	0	0
$235$	−627.745	−0.174253
$236$	−800.814	−0.220884
$237$	3995.72	1.09515
$238$	−2636.37	−0.718027
$239$	1515.70	0.410218	0.205109	−	0.978739i	$-0.434245\pi$
$239$	1515.70	0.410218	0.205109	+	0.978739i	$0.434245\pi$
$240$	1542.68	0.414914
$241$	−2392.47	−0.639472	−0.319736	−	0.947507i	$-0.603594\pi$
$241$	−2392.47	−0.639472	−0.319736	+	0.947507i	$0.603594\pi$
$242$	8170.68	2.17038
$243$	243.000	0.0641500
$244$	−1286.09	−0.337432
$245$	958.207	0.249868
$246$	3485.64	0.903398
$247$	0	0
$248$	−726.815	−0.186100
$249$	−2722.90	−0.692999
$250$	4376.13	1.10708
$251$	−2198.78	−0.552931	−0.276465	−	0.961024i	$-0.589163\pi$
$251$	−2198.78	−0.552931	−0.276465	+	0.961024i	$0.589163\pi$
$252$	264.130	0.0660263
$253$	−682.581	−0.169619
$254$	6417.29	1.58526
$255$	1185.50	0.291132
$256$	2985.94	0.728988
$257$	−6194.26	−1.50345	−0.751727	−	0.659475i	$-0.770779\pi$
$257$	−6194.26	−1.50345	−0.751727	+	0.659475i	$0.770779\pi$
$258$	−1221.43	−0.294739
$259$	−6005.95	−1.44089
$260$	0	0
$261$	1946.85	0.461712
$262$	5895.92	1.39027
$263$	−4181.74	−0.980445	−0.490222	−	0.871597i	$-0.663084\pi$
$263$	−4181.74	−0.980445	−0.490222	+	0.871597i	$0.663084\pi$
$264$	3527.79	0.822425
$265$	−889.258	−0.206139
$266$	−2885.90	−0.665210
$267$	3101.00	0.710778
$268$	−1791.75	−0.408391
$269$	2767.69	0.627320	0.313660	−	0.949535i	$-0.398445\pi$
$269$	2767.69	0.627320	0.313660	+	0.949535i	$0.398445\pi$
$270$	−577.629	−0.130198
$271$	7191.36	1.61197	0.805986	−	0.591935i	$-0.201636\pi$
$271$	7191.36	1.61197	0.805986	+	0.591935i	$0.201636\pi$
$272$	4471.20	0.996714
$273$	0	0
$274$	−6010.88	−1.32529
$275$	4971.69	1.09020
$276$	67.8501	0.0147974
$277$	1317.27	0.285729	0.142864	−	0.989742i	$-0.454369\pi$
$277$	1317.27	0.285729	0.142864	+	0.989742i	$0.454369\pi$
$278$	−3890.90	−0.839427
$279$	347.644	0.0745983
$280$	1797.78	0.383707
$281$	3948.92	0.838338	0.419169	−	0.907908i	$-0.362321\pi$
$281$	3948.92	0.838338	0.419169	+	0.907908i	$0.362321\pi$
$282$	886.503	0.187200
$283$	−4981.52	−1.04636	−0.523181	−	0.852221i	$-0.675255\pi$
$283$	−4981.52	−1.04636	−0.523181	+	0.852221i	$0.675255\pi$
$284$	1257.99	0.262844
$285$	1297.70	0.269717
$286$	0	0
$287$	5188.97	1.06723
$288$	−823.808	−0.168553
$289$	−1477.03	−0.300636
$290$	−4627.80	−0.937082
$291$	−3138.50	−0.632242
$292$	−2031.31	−0.407100
$293$	3203.02	0.638644	0.319322	−	0.947646i	$-0.396545\pi$
$293$	3203.02	0.638644	0.319322	+	0.947646i	$0.396545\pi$
$294$	−1353.18	−0.268433
$295$	2607.12	0.514551
$296$	7973.74	1.56576
$297$	−1687.38	−0.329670
$298$	−10141.2	−1.97136
$299$	0	0
$300$	−494.197	−0.0951083
$301$	−1818.30	−0.348190
$302$	−1612.84	−0.307314
$303$	4249.93	0.805782
$304$	4894.39	0.923396
$305$	4186.97	0.786051
$306$	−1674.16	−0.312763
$307$	−4795.67	−0.891542	−0.445771	−	0.895147i	$-0.647070\pi$
$307$	−4795.67	−0.891542	−0.445771	+	0.895147i	$0.647070\pi$
$308$	−1834.11	−0.339312
$309$	1162.89	0.214092
$310$	−826.376	−0.151403
$311$	630.213	0.114907	0.0574535	−	0.998348i	$-0.481702\pi$
$311$	630.213	0.114907	0.0574535	+	0.998348i	$0.481702\pi$
$312$	0	0
$313$	−9314.73	−1.68211	−0.841054	−	0.540952i	$-0.818064\pi$
$313$	−9314.73	−1.68211	−0.841054	+	0.540952i	$0.818064\pi$
$314$	−3944.83	−0.708980
$315$	−859.900	−0.153809
$316$	2758.02	0.490984
$317$	−576.333	−0.102114	−0.0510569	−	0.998696i	$-0.516259\pi$
$317$	−576.333	−0.102114	−0.0510569	+	0.998696i	$0.516259\pi$
$318$	1255.81	0.221455
$319$	−13518.8	−2.37275
$320$	−2155.55	−0.376559
$321$	259.358	0.0450964
$322$	491.231	0.0850163
$323$	3761.18	0.647919
$324$	167.730	0.0287602
$325$	0	0
$326$	107.896	0.0183307
$327$	2820.22	0.476937
$328$	−6889.08	−1.15971
$329$	1319.71	0.221149
$330$	4011.03	0.669091
$331$	−1575.95	−0.261699	−0.130849	−	0.991402i	$-0.541770\pi$
$331$	−1575.95	−0.261699	−0.130849	+	0.991402i	$0.541770\pi$
$332$	−1879.47	−0.310691
$333$	−3813.93	−0.627635
$334$	7015.44	1.14930
$335$	5833.22	0.951351
$336$	−3243.18	−0.526577
$337$	9289.32	1.50155	0.750774	−	0.660559i	$-0.229681\pi$
$337$	9289.32	1.50155	0.750774	+	0.660559i	$0.229681\pi$
$338$	0	0
$339$	2881.50	0.461656
$340$	818.284	0.130523
$341$	−2414.03	−0.383363
$342$	−1832.62	−0.289757
$343$	−6875.66	−1.08236
$344$	2414.05	0.378363
$345$	−220.892	−0.0344708
$346$	−2098.62	−0.326077
$347$	−7701.82	−1.19151	−0.595757	−	0.803164i	$-0.703148\pi$
$347$	−7701.82	−1.19151	−0.595757	+	0.803164i	$0.703148\pi$
$348$	1343.80	0.206998
$349$	4972.89	0.762730	0.381365	−	0.924425i	$-0.375454\pi$
$349$	4972.89	0.762730	0.381365	+	0.924425i	$0.375454\pi$
$350$	−3577.96	−0.546429
$351$	0	0
$352$	5720.49	0.866202
$353$	−1575.34	−0.237526	−0.118763	−	0.992923i	$-0.537893\pi$
$353$	−1575.34	−0.237526	−0.118763	+	0.992923i	$0.537893\pi$
$354$	−3681.79	−0.552782
$355$	−4095.49	−0.612298
$356$	2140.45	0.318661
$357$	−2492.28	−0.369483
$358$	−7378.41	−1.08928
$359$	−7567.42	−1.11252	−0.556258	−	0.831010i	$-0.687763\pi$
$359$	−7567.42	−1.11252	−0.556258	+	0.831010i	$0.687763\pi$
$360$	1141.64	0.167138
$361$	−2741.83	−0.399741
$362$	6734.69	0.977810
$363$	7724.12	1.11683
$364$	0	0
$365$	6613.10	0.948344
$366$	−5912.86	−0.844454
$367$	4368.25	0.621310	0.310655	−	0.950523i	$-0.399452\pi$
$367$	4368.25	0.621310	0.310655	+	0.950523i	$0.399452\pi$
$368$	−833.112	−0.118014
$369$	3295.13	0.464872
$370$	9066.01	1.27384
$371$	1869.49	0.261615
$372$	239.960	0.0334445
$373$	−801.944	−0.111322	−0.0556610	−	0.998450i	$-0.517727\pi$
$373$	−801.944	−0.111322	−0.0556610	+	0.998450i	$0.517727\pi$
$374$	11625.3	1.60730
$375$	4136.96	0.569684
$376$	−1752.10	−0.240313
$377$	0	0
$378$	1214.35	0.165237
$379$	−68.0819	−0.00922727	−0.00461363	−	0.999989i	$-0.501469\pi$
$379$	−68.0819	−0.00922727	−0.00461363	+	0.999989i	$0.501469\pi$
$380$	895.734	0.120922
$381$	6066.55	0.815745
$382$	−7884.01	−1.05597
$383$	−1549.01	−0.206659	−0.103330	−	0.994647i	$-0.532950\pi$
$383$	−1549.01	−0.206659	−0.103330	+	0.994647i	$0.532950\pi$
$384$	5240.90	0.696480
$385$	5971.11	0.790431
$386$	846.584	0.111632
$387$	−1154.67	−0.151667
$388$	−2166.34	−0.283451
$389$	7300.51	0.951544	0.475772	−	0.879569i	$-0.342169\pi$
$389$	7300.51	0.951544	0.475772	+	0.879569i	$0.342169\pi$
$390$	0	0
$391$	−640.220	−0.0828065
$392$	2674.46	0.344594
$393$	5573.69	0.715408
$394$	3906.61	0.499524
$395$	−8979.00	−1.14375
$396$	−1164.71	−0.147800
$397$	−6096.27	−0.770688	−0.385344	−	0.922773i	$-0.625917\pi$
$397$	−6096.27	−0.770688	−0.385344	+	0.922773i	$0.625917\pi$
$398$	−10301.4	−1.29740
$399$	−2728.17	−0.342304
$400$	6068.11	0.758513
$401$	−7592.37	−0.945498	−0.472749	−	0.881197i	$-0.656738\pi$
$401$	−7592.37	−0.945498	−0.472749	+	0.881197i	$0.656738\pi$
$402$	−8237.69	−1.02204
$403$	0	0
$404$	2933.49	0.361254
$405$	−546.059	−0.0669973
$406$	9729.05	1.18927
$407$	26483.8	3.22544
$408$	3308.85	0.401502
$409$	7233.86	0.874551	0.437275	−	0.899328i	$-0.355943\pi$
$409$	7233.86	0.874551	0.437275	+	0.899328i	$0.355943\pi$
$410$	−7832.77	−0.943495
$411$	−5682.36	−0.681971
$412$	802.678	0.0959832
$413$	−5480.97	−0.653029
$414$	311.945	0.0370320
$415$	6118.78	0.723757
$416$	0	0
$417$	−3678.25	−0.431953
$418$	12725.6	1.48907
$419$	−5312.55	−0.619416	−0.309708	−	0.950832i	$-0.600231\pi$
$419$	−5312.55	−0.619416	−0.309708	+	0.950832i	$0.600231\pi$
$420$	−593.542	−0.0689569
$421$	−15028.1	−1.73973	−0.869865	−	0.493290i	$-0.835794\pi$
$421$	−15028.1	−1.73973	−0.869865	+	0.493290i	$0.835794\pi$
$422$	1049.48	0.121062
$423$	838.052	0.0963297
$424$	−2482.02	−0.284286
$425$	4663.15	0.532226
$426$	5783.66	0.657791
$427$	−8802.31	−0.997596
$428$	179.020	0.0202179
$429$	0	0
$430$	2744.73	0.307821
$431$	7154.66	0.799600	0.399800	−	0.916602i	$-0.369080\pi$
$431$	7154.66	0.799600	0.399800	+	0.916602i	$0.369080\pi$
$432$	−2059.50	−0.229370
$433$	−9542.58	−1.05909	−0.529546	−	0.848281i	$-0.677638\pi$
$433$	−9542.58	−1.05909	−0.529546	+	0.848281i	$0.677638\pi$
$434$	1737.30	0.192150
$435$	−4374.87	−0.482204
$436$	1946.64	0.213824
$437$	−700.816	−0.0767153
$438$	−9339.04	−1.01881
$439$	7070.70	0.768715	0.384358	−	0.923184i	$-0.374423\pi$
$439$	7070.70	0.768715	0.384358	+	0.923184i	$0.374423\pi$
$440$	−7927.49	−0.858928
$441$	−1279.23	−0.138130
$442$	0	0
$443$	2092.58	0.224428	0.112214	−	0.993684i	$-0.464206\pi$
$443$	2092.58	0.224428	0.112214	+	0.993684i	$0.464206\pi$
$444$	−2632.55	−0.281386
$445$	−6968.42	−0.742326
$446$	−18361.1	−1.94938
$447$	−9586.96	−1.01442
$448$	4531.63	0.477901
$449$	−5842.05	−0.614038	−0.307019	−	0.951703i	$-0.599332\pi$
$449$	−5842.05	−0.614038	−0.307019	+	0.951703i	$0.599332\pi$
$450$	−2272.10	−0.238017
$451$	−22881.3	−2.38899
$452$	1988.94	0.206973
$453$	−1524.69	−0.158138
$454$	−9346.91	−0.966238
$455$	0	0
$456$	3622.03	0.371967
$457$	−5954.40	−0.609486	−0.304743	−	0.952435i	$-0.598571\pi$
$457$	−5954.40	−0.609486	−0.304743	+	0.952435i	$0.598571\pi$
$458$	11238.0	1.14654
$459$	−1582.66	−0.160942
$460$	−152.470	−0.0154542
$461$	1865.94	0.188515	0.0942576	−	0.995548i	$-0.469952\pi$
$461$	1865.94	0.188515	0.0942576	+	0.995548i	$0.469952\pi$
$462$	−8432.42	−0.849160
$463$	6700.05	0.672522	0.336261	−	0.941769i	$-0.390838\pi$
$463$	6700.05	0.672522	0.336261	+	0.941769i	$0.390838\pi$
$464$	−16500.2	−1.65086
$465$	−781.211	−0.0779093
$466$	7428.28	0.738430
$467$	−16585.8	−1.64347	−0.821734	−	0.569871i	$-0.806993\pi$
$467$	−16585.8	−1.64347	−0.821734	+	0.569871i	$0.806993\pi$
$468$	0	0
$469$	−12263.2	−1.20738
$470$	−1992.11	−0.195509
$471$	−3729.23	−0.364828
$472$	7276.77	0.709619
$473$	8017.98	0.779423
$474$	12680.2	1.22873
$475$	5104.51	0.493075
$476$	−1720.29	−0.165649
$477$	1187.18	0.113956
$478$	4809.97	0.460257
$479$	6166.88	0.588250	0.294125	−	0.955767i	$-0.404972\pi$
$479$	6166.88	0.588250	0.294125	+	0.955767i	$0.404972\pi$
$480$	1851.23	0.176034
$481$	0	0
$482$	−7592.38	−0.717476
$483$	464.383	0.0437478
$484$	5331.53	0.500708
$485$	7052.71	0.660303
$486$	771.146	0.0719751
$487$	−5718.51	−0.532095	−0.266047	−	0.963960i	$-0.585718\pi$
$487$	−5718.51	−0.532095	−0.266047	+	0.963960i	$0.585718\pi$
$488$	11686.3	1.08405
$489$	101.999	0.00943261
$490$	3040.82	0.280347
$491$	−21060.9	−1.93578	−0.967888	−	0.251383i	$-0.919115\pi$
$491$	−21060.9	−1.93578	−0.967888	+	0.251383i	$0.919115\pi$
$492$	2274.45	0.208415
$493$	−12679.8	−1.15836
$494$	0	0
$495$	3791.81	0.344302
$496$	−2946.40	−0.266728
$497$	8609.97	0.777082
$498$	−8640.97	−0.777532
$499$	7863.87	0.705481	0.352741	−	0.935721i	$-0.385250\pi$
$499$	7863.87	0.705481	0.352741	+	0.935721i	$0.385250\pi$
$500$	2855.51	0.255405
$501$	6632.01	0.591410
$502$	−6977.70	−0.620378
$503$	−6504.06	−0.576544	−0.288272	−	0.957549i	$-0.593081\pi$
$503$	−6504.06	−0.576544	−0.288272	+	0.957549i	$0.593081\pi$
$504$	−2400.07	−0.212119
$505$	−9550.26	−0.841546
$506$	−2166.13	−0.190309
$507$	0	0
$508$	4187.41	0.365721
$509$	−14799.3	−1.28873	−0.644367	−	0.764717i	$-0.722879\pi$
$509$	−14799.3	−1.28873	−0.644367	+	0.764717i	$0.722879\pi$
$510$	3762.11	0.326645
$511$	−13902.8	−1.20357
$512$	−4500.03	−0.388428
$513$	−1732.46	−0.149103
$514$	−19657.1	−1.68685
$515$	−2613.19	−0.223594
$516$	−797.005	−0.0679965
$517$	−5819.40	−0.495042
$518$	−19059.5	−1.61666
$519$	−1983.92	−0.167793
$520$	0	0
$521$	−6633.65	−0.557822	−0.278911	−	0.960317i	$-0.589974\pi$
$521$	−6633.65	−0.557822	−0.278911	+	0.960317i	$0.589974\pi$
$522$	6178.20	0.518032
$523$	4527.41	0.378527	0.189264	−	0.981926i	$-0.439390\pi$
$523$	4527.41	0.378527	0.189264	+	0.981926i	$0.439390\pi$
$524$	3847.21	0.320737
$525$	−3382.41	−0.281182
$526$	−13270.5	−1.10004
$527$	−2264.21	−0.187155
$528$	14301.1	1.17874
$529$	−12047.7	−0.990195
$530$	−2822.01	−0.231284
$531$	−3480.56	−0.284451
$532$	−1883.11	−0.153464
$533$	0	0
$534$	9840.83	0.797480
$535$	−582.818	−0.0470980
$536$	16281.1	1.31201
$537$	−6975.14	−0.560521
$538$	8783.11	0.703841
$539$	8882.89	0.709858
$540$	−376.915	−0.0300367
$541$	−8685.42	−0.690232	−0.345116	−	0.938560i	$-0.612160\pi$
$541$	−8685.42	−0.690232	−0.345116	+	0.938560i	$0.612160\pi$
$542$	22821.4	1.80860
$543$	6366.61	0.503162
$544$	5365.48	0.422873
$545$	−6337.47	−0.498105
$546$	0	0
$547$	24656.6	1.92731	0.963657	−	0.267142i	$-0.0860793\pi$
$547$	24656.6	1.92731	0.963657	+	0.267142i	$0.0860793\pi$
$548$	−3922.22	−0.305746
$549$	−5589.70	−0.434540
$550$	15777.4	1.22318
$551$	−13880.0	−1.07315
$552$	−616.534	−0.0475388
$553$	18876.6	1.45157
$554$	4180.27	0.320582
$555$	8570.51	0.655492
$556$	−2538.89	−0.193656
$557$	−9215.90	−0.701059	−0.350530	−	0.936552i	$-0.613998\pi$
$557$	−9215.90	−0.701059	−0.350530	+	0.936552i	$0.613998\pi$
$558$	1103.23	0.0836979
$559$	0	0
$560$	7287.93	0.549949
$561$	10990.0	0.827088
$562$	12531.7	0.940600
$563$	−19686.7	−1.47370	−0.736850	−	0.676056i	$-0.763688\pi$
$563$	−19686.7	−1.47370	−0.736850	+	0.676056i	$0.763688\pi$
$564$	578.461	0.0431873
$565$	−6475.17	−0.482146
$566$	−15808.6	−1.17400
$567$	1147.98	0.0850279
$568$	−11430.9	−0.844422
$569$	−3559.36	−0.262243	−0.131121	−	0.991366i	$-0.541858\pi$
$569$	−3559.36	−0.262243	−0.131121	+	0.991366i	$0.541858\pi$
$570$	4118.19	0.302617
$571$	−710.968	−0.0521070	−0.0260535	−	0.999661i	$-0.508294\pi$
$571$	−710.968	−0.0521070	−0.0260535	+	0.999661i	$0.508294\pi$
$572$	0	0
$573$	−7453.11	−0.543383
$574$	16466.9	1.19741
$575$	−868.878	−0.0630169
$576$	2877.70	0.208167
$577$	8041.67	0.580206	0.290103	−	0.956995i	$-0.406310\pi$
$577$	8041.67	0.580206	0.290103	+	0.956995i	$0.406310\pi$
$578$	−4687.26	−0.337308
$579$	800.314	0.0574438
$580$	−3019.73	−0.216186
$581$	−12863.6	−0.918538
$582$	−9959.86	−0.709364
$583$	−8243.72	−0.585626
$584$	18457.9	1.30786
$585$	0	0
$586$	10164.6	0.716546
$587$	14641.3	1.02949	0.514744	−	0.857344i	$-0.327887\pi$
$587$	14641.3	1.02949	0.514744	+	0.857344i	$0.327887\pi$
$588$	−882.980	−0.0619277
$589$	−2478.52	−0.173388
$590$	8273.56	0.577317
$591$	3693.10	0.257045
$592$	32324.3	2.24413
$593$	−5735.76	−0.397200	−0.198600	−	0.980081i	$-0.563639\pi$
$593$	−5735.76	−0.397200	−0.198600	+	0.980081i	$0.563639\pi$
$594$	−5354.81	−0.369883
$595$	5600.55	0.385882
$596$	−6617.35	−0.454794
$597$	−9738.43	−0.667617
$598$	0	0
$599$	−16109.3	−1.09884	−0.549422	−	0.835545i	$-0.685152\pi$
$599$	−16109.3	−1.09884	−0.549422	+	0.835545i	$0.685152\pi$
$600$	4490.63	0.305548
$601$	−21005.4	−1.42567	−0.712837	−	0.701330i	$-0.752590\pi$
$601$	−21005.4	−1.42567	−0.712837	+	0.701330i	$0.752590\pi$
$602$	−5770.28	−0.390663
$603$	−7787.46	−0.525920
$604$	−1052.41	−0.0708975
$605$	−17357.3	−1.16640
$606$	13486.9	0.904073
$607$	7478.16	0.500048	0.250024	−	0.968240i	$-0.419561\pi$
$607$	7478.16	0.500048	0.250024	+	0.968240i	$0.419561\pi$
$608$	5873.31	0.391767
$609$	9197.32	0.611977
$610$	13287.1	0.881934
$611$	0	0
$612$	−1092.43	−0.0721548
$613$	−16435.5	−1.08291	−0.541455	−	0.840730i	$-0.682126\pi$
$613$	−16435.5	−1.08291	−0.541455	+	0.840730i	$0.682126\pi$
$614$	−15218.8	−1.00029
$615$	−7404.68	−0.485505
$616$	16666.0	1.09009
$617$	−1290.89	−0.0842289	−0.0421145	−	0.999113i	$-0.513409\pi$
$617$	−1290.89	−0.0842289	−0.0421145	+	0.999113i	$0.513409\pi$
$618$	3690.36	0.240207
$619$	26719.0	1.73494	0.867470	−	0.497490i	$-0.165745\pi$
$619$	26719.0	1.73494	0.867470	+	0.497490i	$0.165745\pi$
$620$	−539.227	−0.0349289
$621$	294.896	0.0190560
$622$	1999.94	0.128924
$623$	14649.8	0.942103
$624$	0	0
$625$	647.683	0.0414517
$626$	−29559.8	−1.88729
$627$	12030.1	0.766248
$628$	−2574.08	−0.163562
$629$	24840.2	1.57463
$630$	−2728.84	−0.172571
$631$	10697.4	0.674893	0.337447	−	0.941345i	$-0.390437\pi$
$631$	10697.4	0.674893	0.337447	+	0.941345i	$0.390437\pi$
$632$	−25061.4	−1.57735
$633$	992.125	0.0622961
$634$	−1828.96	−0.114570
$635$	−13632.5	−0.851951
$636$	819.444	0.0510897
$637$	0	0
$638$	−42901.2	−2.66219
$639$	5467.56	0.338487
$640$	−11777.1	−0.727393
$641$	23572.8	1.45253	0.726264	−	0.687416i	$-0.241255\pi$
$641$	23572.8	1.45253	0.726264	+	0.687416i	$0.241255\pi$
$642$	823.057	0.0505973
$643$	14000.3	0.858661	0.429331	−	0.903147i	$-0.358750\pi$
$643$	14000.3	0.858661	0.429331	+	0.903147i	$0.358750\pi$
$644$	320.538	0.0196133
$645$	2594.72	0.158399
$646$	11935.9	0.726953
$647$	614.196	0.0373207	0.0186604	−	0.999826i	$-0.494060\pi$
$647$	614.196	0.0373207	0.0186604	+	0.999826i	$0.494060\pi$
$648$	−1524.11	−0.0923962
$649$	24168.9	1.46181
$650$	0	0
$651$	1642.35	0.0988765
$652$	70.4042	0.00422890
$653$	−5333.42	−0.319622	−0.159811	−	0.987148i	$-0.551088\pi$
$653$	−5333.42	−0.319622	−0.159811	+	0.987148i	$0.551088\pi$
$654$	8949.79	0.535114
$655$	−12524.9	−0.747161
$656$	−27927.3	−1.66216
$657$	−8828.62	−0.524258
$658$	4188.03	0.248125
$659$	21396.8	1.26480	0.632398	−	0.774644i	$-0.282071\pi$
$659$	21396.8	1.26480	0.632398	+	0.774644i	$0.282071\pi$
$660$	2617.28	0.154360
$661$	16107.9	0.947841	0.473921	−	0.880568i	$-0.342838\pi$
$661$	16107.9	0.947841	0.473921	+	0.880568i	$0.342838\pi$
$662$	−5001.20	−0.293621
$663$	0	0
$664$	17078.2	0.998136
$665$	6130.63	0.357497
$666$	−12103.3	−0.704194
$667$	2362.62	0.137153
$668$	4577.72	0.265145
$669$	−17357.6	−1.00311
$670$	18511.4	1.06740
$671$	38814.6	2.23312
$672$	−3891.85	−0.223410
$673$	20329.9	1.16443	0.582213	−	0.813036i	$-0.302187\pi$
$673$	20329.9	1.16443	0.582213	+	0.813036i	$0.302187\pi$
$674$	29479.1	1.68471
$675$	−2147.92	−0.122479
$676$	0	0
$677$	−24883.5	−1.41263	−0.706314	−	0.707899i	$-0.749643\pi$
$677$	−24883.5	−1.41263	−0.706314	+	0.707899i	$0.749643\pi$
$678$	9144.26	0.517969
$679$	−14827.0	−0.838007
$680$	−7435.51	−0.419322
$681$	−8836.06	−0.497208
$682$	−7660.78	−0.430127
$683$	258.953	0.0145074	0.00725369	−	0.999974i	$-0.497691\pi$
$683$	258.953	0.0145074	0.00725369	+	0.999974i	$0.497691\pi$
$684$	−1195.82	−0.0668471
$685$	12769.1	0.712240
$686$	−21819.5	−1.21439
$687$	10623.8	0.589989
$688$	9786.20	0.542290
$689$	0	0
$690$	−700.989	−0.0386756
$691$	658.193	0.0362357	0.0181178	−	0.999836i	$-0.494233\pi$
$691$	658.193	0.0362357	0.0181178	+	0.999836i	$0.494233\pi$
$692$	−1369.39	−0.0752262
$693$	−7971.55	−0.436962
$694$	−24441.3	−1.33686
$695$	8265.60	0.451125
$696$	−12210.7	−0.665010
$697$	−21461.2	−1.16629
$698$	15781.2	0.855768
$699$	7022.29	0.379982
$700$	−2334.69	−0.126062
$701$	−8222.16	−0.443005	−0.221503	−	0.975160i	$-0.571096\pi$
$701$	−8222.16	−0.443005	−0.221503	+	0.975160i	$0.571096\pi$
$702$	0	0
$703$	27191.3	1.45881
$704$	−19982.7	−1.06978
$705$	−1883.23	−0.100605
$706$	−4999.25	−0.266500
$707$	20077.6	1.06803
$708$	−2402.44	−0.127527
$709$	−6817.51	−0.361124	−0.180562	−	0.983564i	$-0.557792\pi$
$709$	−6817.51	−0.361124	−0.180562	+	0.983564i	$0.557792\pi$
$710$	−12996.8	−0.686987
$711$	11987.2	0.632283
$712$	−19449.6	−1.02374
$713$	421.888	0.0221596
$714$	−7909.11	−0.414553
$715$	0	0
$716$	−4814.56	−0.251297
$717$	4547.09	0.236840
$718$	−24014.8	−1.24822
$719$	23385.7	1.21299	0.606496	−	0.795087i	$-0.292575\pi$
$719$	23385.7	1.21299	0.606496	+	0.795087i	$0.292575\pi$
$720$	4628.03	0.239551
$721$	5493.73	0.283769
$722$	−8701.03	−0.448502
$723$	−7177.42	−0.369199
$724$	4394.52	0.225582
$725$	−17208.5	−0.881529
$726$	24512.0	1.25307
$727$	20488.6	1.04523	0.522613	−	0.852570i	$-0.324957\pi$
$727$	20488.6	1.04523	0.522613	+	0.852570i	$0.324957\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	20986.3	1.06402
$731$	7520.38	0.380508
$732$	−3858.26	−0.194816
$733$	16993.4	0.856299	0.428149	−	0.903708i	$-0.359166\pi$
$733$	16993.4	0.856299	0.428149	+	0.903708i	$0.359166\pi$
$734$	13862.4	0.697099
$735$	2874.62	0.144261
$736$	−999.742	−0.0500693
$737$	54075.8	2.70272
$738$	10456.9	0.521577
$739$	14014.3	0.697600	0.348800	−	0.937197i	$-0.386589\pi$
$739$	14014.3	0.697600	0.348800	+	0.937197i	$0.386589\pi$
$740$	5915.75	0.293875
$741$	0	0
$742$	5932.73	0.293528
$743$	141.560	0.00698971	0.00349485	−	0.999994i	$-0.498888\pi$
$743$	141.560	0.00698971	0.00349485	+	0.999994i	$0.498888\pi$
$744$	−2180.45	−0.107445
$745$	21543.4	1.05945
$746$	−2544.92	−0.124901
$747$	−8168.70	−0.400103
$748$	7585.76	0.370806
$749$	1225.26	0.0597731
$750$	13128.4	0.639175
$751$	20734.3	1.00746	0.503732	−	0.863860i	$-0.331960\pi$
$751$	20734.3	1.00746	0.503732	+	0.863860i	$0.331960\pi$
$752$	−7102.76	−0.344430
$753$	−6596.34	−0.319235
$754$	0	0
$755$	3426.23	0.165157
$756$	792.391	0.0381203
$757$	17449.8	0.837812	0.418906	−	0.908030i	$-0.362414\pi$
$757$	17449.8	0.837812	0.418906	+	0.908030i	$0.362414\pi$
$758$	−216.054	−0.0103528
$759$	−2047.74	−0.0979293
$760$	−8139.27	−0.388477
$761$	424.121	0.0202028	0.0101014	−	0.999949i	$-0.496785\pi$
$761$	424.121	0.0202028	0.0101014	+	0.999949i	$0.496785\pi$
$762$	19251.9	0.915251
$763$	13323.3	0.632157
$764$	−5144.48	−0.243613
$765$	3556.49	0.168085
$766$	−4915.68	−0.231868
$767$	0	0
$768$	8957.81	0.420882
$769$	−38060.7	−1.78479	−0.892396	−	0.451252i	$-0.850977\pi$
$769$	−38060.7	−1.78479	−0.892396	+	0.451252i	$0.850977\pi$
$770$	18949.0	0.886849
$771$	−18582.8	−0.868019
$772$	552.413	0.0257536
$773$	16683.3	0.776268	0.388134	−	0.921603i	$-0.373120\pi$
$773$	16683.3	0.776268	0.388134	+	0.921603i	$0.373120\pi$
$774$	−3664.28	−0.170168
$775$	−3072.89	−0.142428
$776$	19684.9	0.910627
$777$	−18017.8	−0.831900
$778$	23167.7	1.06761
$779$	−23492.5	−1.08050
$780$	0	0
$781$	−37966.5	−1.73950
$782$	−2031.70	−0.0929073
$783$	5840.54	0.266569
$784$	10841.9	0.493889
$785$	8380.17	0.381020
$786$	17687.8	0.802674
$787$	39105.2	1.77122	0.885609	−	0.464432i	$-0.153741\pi$
$787$	39105.2	1.77122	0.885609	+	0.464432i	$0.153741\pi$
$788$	2549.14	0.115240
$789$	−12545.2	−0.566060
$790$	−28494.3	−1.28327
$791$	13612.8	0.611903
$792$	10583.4	0.474827
$793$	0	0
$794$	−19346.2	−0.864697
$795$	−2667.78	−0.119014
$796$	−6721.90	−0.299311
$797$	−4346.93	−0.193195	−0.0965974	−	0.995324i	$-0.530796\pi$
$797$	−4346.93	−0.193195	−0.0965974	+	0.995324i	$0.530796\pi$
$798$	−8657.69	−0.384059
$799$	−5458.25	−0.241676
$800$	7281.78	0.321812
$801$	9302.99	0.410368
$802$	−24093.9	−1.06083
$803$	61305.6	2.69418
$804$	−5375.26	−0.235784
$805$	−1043.54	−0.0456895
$806$	0	0
$807$	8303.07	0.362183
$808$	−26655.8	−1.16058
$809$	−23030.2	−1.00086	−0.500432	−	0.865776i	$-0.666826\pi$
$809$	−23030.2	−1.00086	−0.500432	+	0.865776i	$0.666826\pi$
$810$	−1732.89	−0.0751697
$811$	−7898.11	−0.341973	−0.170987	−	0.985273i	$-0.554695\pi$
$811$	−7898.11	−0.341973	−0.170987	+	0.985273i	$0.554695\pi$
$812$	6348.41	0.274366
$813$	21574.1	0.930672
$814$	84044.8	3.61888
$815$	−229.207	−0.00985127
$816$	13413.6	0.575453
$817$	8232.18	0.352518
$818$	22956.2	0.981230
$819$	0	0
$820$	−5111.04	−0.217665
$821$	3939.61	0.167471	0.0837353	−	0.996488i	$-0.473315\pi$
$821$	3939.61	0.167471	0.0837353	+	0.996488i	$0.473315\pi$
$822$	−18032.6	−0.765159
$823$	−17599.8	−0.745430	−0.372715	−	0.927946i	$-0.621573\pi$
$823$	−17599.8	−0.745430	−0.372715	+	0.927946i	$0.621573\pi$
$824$	−7293.70	−0.308359
$825$	14915.1	0.629425
$826$	−17393.6	−0.732687
$827$	−12510.6	−0.526042	−0.263021	−	0.964790i	$-0.584719\pi$
$827$	−12510.6	−0.526042	−0.263021	+	0.964790i	$0.584719\pi$
$828$	203.550	0.00854331
$829$	28630.8	1.19950	0.599752	−	0.800186i	$-0.295266\pi$
$829$	28630.8	1.19950	0.599752	+	0.800186i	$0.295266\pi$
$830$	19417.6	0.812042
$831$	3951.80	0.164966
$832$	0	0
$833$	8331.62	0.346547
$834$	−11672.7	−0.484643
$835$	−14903.2	−0.617660
$836$	8303.75	0.343530
$837$	1042.93	0.0430693
$838$	−16859.1	−0.694973
$839$	−21250.3	−0.874426	−0.437213	−	0.899358i	$-0.644034\pi$
$839$	−21250.3	−0.874426	−0.437213	+	0.899358i	$0.644034\pi$
$840$	5393.34	0.221533
$841$	22403.7	0.918600
$842$	−47690.9	−1.95194
$843$	11846.8	0.484015
$844$	684.810	0.0279291
$845$	0	0
$846$	2659.51	0.108080
$847$	36490.4	1.48031
$848$	−10061.7	−0.407454
$849$	−14944.6	−0.604118
$850$	14798.2	0.597147
$851$	−4628.45	−0.186441
$852$	3773.96	0.151753
$853$	−34721.1	−1.39370	−0.696852	−	0.717215i	$-0.745416\pi$
$853$	−34721.1	−1.39370	−0.696852	+	0.717215i	$0.745416\pi$
$854$	−27933.6	−1.11928
$855$	3893.11	0.155721
$856$	−1626.71	−0.0649529
$857$	4898.06	0.195233	0.0976163	−	0.995224i	$-0.468878\pi$
$857$	4898.06	0.195233	0.0976163	+	0.995224i	$0.468878\pi$
$858$	0	0
$859$	−12564.2	−0.499051	−0.249526	−	0.968368i	$-0.580275\pi$
$859$	−12564.2	−0.499051	−0.249526	+	0.968368i	$0.580275\pi$
$860$	1791.00	0.0710145
$861$	15566.9	0.616166
$862$	22704.9	0.897137
$863$	22826.6	0.900380	0.450190	−	0.892933i	$-0.351356\pi$
$863$	22826.6	0.900380	0.450190	+	0.892933i	$0.351356\pi$
$864$	−2471.42	−0.0973143
$865$	4458.18	0.175240
$866$	−30282.8	−1.18828
$867$	−4431.08	−0.173572
$868$	1133.62	0.0443291
$869$	−83238.3	−3.24933
$870$	−13883.4	−0.541024
$871$	0	0
$872$	−17688.6	−0.686939
$873$	−9415.51	−0.365025
$874$	−2224.00	−0.0860731
$875$	19543.9	0.755089
$876$	−6093.92	−0.235039
$877$	−25212.7	−0.970776	−0.485388	−	0.874299i	$-0.661322\pi$
$877$	−25212.7	−0.970776	−0.485388	+	0.874299i	$0.661322\pi$
$878$	22438.5	0.862484
$879$	9609.07	0.368721
$880$	−32136.9	−1.23106
$881$	−18026.2	−0.689352	−0.344676	−	0.938722i	$-0.612011\pi$
$881$	−18026.2	−0.689352	−0.344676	+	0.938722i	$0.612011\pi$
$882$	−4059.55	−0.154980
$883$	−18833.1	−0.717764	−0.358882	−	0.933383i	$-0.616842\pi$
$883$	−18833.1	−0.717764	−0.358882	+	0.933383i	$0.616842\pi$
$884$	0	0
$885$	7821.37	0.297076
$886$	6640.68	0.251803
$887$	38451.4	1.45555	0.727775	−	0.685816i	$-0.240555\pi$
$887$	38451.4	1.45555	0.727775	+	0.685816i	$0.240555\pi$
$888$	23921.2	0.903991
$889$	28659.7	1.08123
$890$	−22113.9	−0.832876
$891$	−5062.15	−0.190335
$892$	−11981.0	−0.449723
$893$	−5974.86	−0.223898
$894$	−30423.7	−1.13816
$895$	15674.2	0.585399
$896$	24759.1	0.923152
$897$	0	0
$898$	−18539.4	−0.688940
$899$	8355.68	0.309986
$900$	−1482.59	−0.0549108
$901$	−7732.11	−0.285898
$902$	−72612.3	−2.68041
$903$	−5454.91	−0.201028
$904$	−18072.9	−0.664929
$905$	−14306.8	−0.525495
$906$	−4838.53	−0.177428
$907$	−5531.31	−0.202496	−0.101248	−	0.994861i	$-0.532284\pi$
$907$	−5531.31	−0.202496	−0.101248	+	0.994861i	$0.532284\pi$
$908$	−6099.05	−0.222912
$909$	12749.8	0.465219
$910$	0	0
$911$	15695.2	0.570806	0.285403	−	0.958408i	$-0.407872\pi$
$911$	15695.2	0.570806	0.285403	+	0.958408i	$0.407872\pi$
$912$	14683.2	0.533123
$913$	56723.1	2.05615
$914$	−18895.9	−0.683832
$915$	12560.9	0.453827
$916$	7333.01	0.264508
$917$	26331.3	0.948240
$918$	−5022.49	−0.180574
$919$	−51503.5	−1.84869	−0.924344	−	0.381561i	$-0.875386\pi$
$919$	−51503.5	−1.84869	−0.924344	+	0.381561i	$0.875386\pi$
$920$	1385.45	0.0496488
$921$	−14387.0	−0.514732
$922$	5921.45	0.211510
$923$	0	0
$924$	−5502.33	−0.195902
$925$	33712.0	1.19832
$926$	21262.2	0.754557
$927$	3488.66	0.123606
$928$	−19800.3	−0.700407
$929$	42927.0	1.51603	0.758014	−	0.652238i	$-0.226170\pi$
$929$	42927.0	1.51603	0.758014	+	0.652238i	$0.226170\pi$
$930$	−2479.13	−0.0874127
$931$	9120.20	0.321055
$932$	4847.10	0.170356
$933$	1890.64	0.0663416
$934$	−52634.1	−1.84394
$935$	−24696.2	−0.863797
$936$	0	0
$937$	44206.6	1.54127	0.770633	−	0.637280i	$-0.219940\pi$
$937$	44206.6	1.54127	0.770633	+	0.637280i	$0.219940\pi$
$938$	−38916.6	−1.35466
$939$	−27944.2	−0.971165
$940$	−1299.89	−0.0451041
$941$	44067.7	1.52664	0.763319	−	0.646022i	$-0.223568\pi$
$941$	44067.7	1.52664	0.763319	+	0.646022i	$0.223568\pi$
$942$	−11834.5	−0.409330
$943$	3998.85	0.138092
$944$	29498.9	1.01706
$945$	−2579.70	−0.0888017
$946$	25444.6	0.874498
$947$	44402.5	1.52364	0.761820	−	0.647789i	$-0.224306\pi$
$947$	44402.5	1.52364	0.761820	+	0.647789i	$0.224306\pi$
$948$	8274.07	0.283470
$949$	0	0
$950$	16198.9	0.553221
$951$	−1729.00	−0.0589554
$952$	15631.7	0.532172
$953$	−10361.7	−0.352202	−0.176101	−	0.984372i	$-0.556349\pi$
$953$	−10361.7	−0.352202	−0.176101	+	0.984372i	$0.556349\pi$
$954$	3767.44	0.127857
$955$	16748.3	0.567500
$956$	3138.60	0.106182
$957$	−40556.5	−1.36991
$958$	19570.2	0.660006
$959$	−26844.7	−0.903920
$960$	−6466.65	−0.217407
$961$	−28298.9	−0.949916
$962$	0	0
$963$	778.073	0.0260364
$964$	−4954.18	−0.165522
$965$	−1798.43	−0.0599933
$966$	1473.69	0.0490842
$967$	−8432.54	−0.280426	−0.140213	−	0.990121i	$-0.544779\pi$
$967$	−8432.54	−0.280426	−0.140213	+	0.990121i	$0.544779\pi$
$968$	−48446.1	−1.60859
$969$	11283.5	0.374076
$970$	22381.4	0.740848
$971$	−36917.1	−1.22011	−0.610055	−	0.792359i	$-0.708853\pi$
$971$	−36917.1	−1.22011	−0.610055	+	0.792359i	$0.708853\pi$
$972$	503.189	0.0166047
$973$	−17376.8	−0.572534
$974$	−18147.4	−0.597001
$975$	0	0
$976$	47374.5	1.55371
$977$	38306.9	1.25440	0.627199	−	0.778859i	$-0.284201\pi$
$977$	38306.9	1.25440	0.627199	+	0.778859i	$0.284201\pi$
$978$	323.687	0.0105832
$979$	−64599.6	−2.10890
$980$	1984.19	0.0646763
$981$	8460.65	0.275360
$982$	−66835.6	−2.17190
$983$	18810.9	0.610350	0.305175	−	0.952296i	$-0.401285\pi$
$983$	18810.9	0.610350	0.305175	+	0.952296i	$0.401285\pi$
$984$	−20667.3	−0.669561
$985$	−8298.97	−0.268454
$986$	−40238.8	−1.29966
$987$	3959.14	0.127681
$988$	0	0
$989$	−1401.26	−0.0450532
$990$	12033.1	0.386300
$991$	−6492.09	−0.208101	−0.104051	−	0.994572i	$-0.533180\pi$
$991$	−6492.09	−0.208101	−0.104051	+	0.994572i	$0.533180\pi$
$992$	−3535.71	−0.113164
$993$	−4727.86	−0.151092
$994$	27323.2	0.871872
$995$	21883.8	0.697249
$996$	−5638.41	−0.179377
$997$	48552.1	1.54229	0.771143	−	0.636662i	$-0.219685\pi$
$997$	48552.1	1.54229	0.771143	+	0.636662i	$0.219685\pi$
$998$	24955.5	0.791537
$999$	−11441.8	−0.362365

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.a.o.1.8	✓	9
3.2	odd	2		1521.4.a.bi.1.2		9
13.5	odd	4		507.4.b.k.337.5		18
13.8	odd	4		507.4.b.k.337.14		18
13.12	even	2		507.4.a.p.1.2	yes	9
39.38	odd	2		1521.4.a.bf.1.8		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.o.1.8	✓	9	1.1	even	1	trivial
507.4.a.p.1.2	yes	9	13.12	even	2
507.4.b.k.337.5		18	13.5	odd	4
507.4.b.k.337.14		18	13.8	odd	4
1521.4.a.bf.1.8		9	39.38	odd	2
1521.4.a.bi.1.2		9	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.a (trivial)

\(f(q)\)	\(=\)	\(q+3.17344 q^{2} +3.00000 q^{3} +2.07074 q^{4} -6.74147 q^{5} +9.52033 q^{6} +14.1726 q^{7} -18.8162 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+3.17344 q^{2} +3.00000 q^{3} +2.07074 q^{4} -6.74147 q^{5} +9.52033 q^{6} +14.1726 q^{7} -18.8162 q^{8} +9.00000 q^{9} -21.3937 q^{10} -62.4956 q^{11} +6.21221 q^{12} +44.9761 q^{14} -20.2244 q^{15} -76.2779 q^{16} -58.6172 q^{17} +28.5610 q^{18} -64.1652 q^{19} -13.9598 q^{20} +42.5179 q^{21} -198.326 q^{22} +10.9221 q^{23} -56.4485 q^{24} -79.5526 q^{25} +27.0000 q^{27} +29.3478 q^{28} +216.316 q^{29} -64.1810 q^{30} +38.6271 q^{31} -91.5342 q^{32} -187.487 q^{33} -186.018 q^{34} -95.5445 q^{35} +18.6366 q^{36} -423.770 q^{37} -203.625 q^{38} +126.849 q^{40} +366.126 q^{41} +134.928 q^{42} -128.297 q^{43} -129.412 q^{44} -60.6732 q^{45} +34.6605 q^{46} +93.1169 q^{47} -228.834 q^{48} -142.136 q^{49} -252.455 q^{50} -175.852 q^{51} +131.909 q^{53} +85.6829 q^{54} +421.313 q^{55} -266.675 q^{56} -192.496 q^{57} +686.467 q^{58} -386.729 q^{59} -41.8794 q^{60} -621.077 q^{61} +122.581 q^{62} +127.554 q^{63} +319.745 q^{64} -594.979 q^{66} -865.273 q^{67} -121.381 q^{68} +32.7662 q^{69} -303.205 q^{70} +607.506 q^{71} -169.346 q^{72} -980.958 q^{73} -1344.81 q^{74} -238.658 q^{75} -132.869 q^{76} -885.728 q^{77} +1331.91 q^{79} +514.226 q^{80} +81.0000 q^{81} +1161.88 q^{82} -907.633 q^{83} +88.0434 q^{84} +395.166 q^{85} -407.142 q^{86} +648.949 q^{87} +1175.93 q^{88} +1033.67 q^{89} -192.543 q^{90} +22.6167 q^{92} +115.881 q^{93} +295.501 q^{94} +432.568 q^{95} -274.603 q^{96} -1046.17 q^{97} -451.061 q^{98} -562.461 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 6 q^{2} + 27 q^{3} + 44 q^{4} - 33 q^{5} - 18 q^{6} - 83 q^{7} - 87 q^{8} + 81 q^{9}+O(q^{10}) \) 9 * q - 6 * q^2 + 27 * q^3 + 44 * q^4 - 33 * q^5 - 18 * q^6 - 83 * q^7 - 87 * q^8 + 81 * q^9	\( 9 q - 6 q^{2} + 27 q^{3} + 44 q^{4} - 33 q^{5} - 18 q^{6} - 83 q^{7} - 87 q^{8} + 81 q^{9} - 54 q^{10} - 85 q^{11} + 132 q^{12} + 158 q^{14} - 99 q^{15} + 216 q^{16} + 178 q^{17} - 54 q^{18} - 352 q^{19} - 402 q^{20} - 249 q^{21} - 630 q^{22} + 150 q^{23} - 261 q^{24} - 20 q^{25} + 243 q^{27} - 940 q^{28} - 97 q^{29} - 162 q^{30} - 717 q^{31} - 707 q^{32} - 255 q^{33} - 632 q^{34} - 418 q^{35} + 396 q^{36} - 1108 q^{37} - 660 q^{38} - 1506 q^{40} - 334 q^{41} + 474 q^{42} + 242 q^{43} + 307 q^{44} - 297 q^{45} - 979 q^{46} + 184 q^{47} + 648 q^{48} - 38 q^{49} + 2031 q^{50} + 534 q^{51} - 151 q^{53} - 162 q^{54} + 2064 q^{55} + 2276 q^{56} - 1056 q^{57} - 1161 q^{58} - 537 q^{59} - 1206 q^{60} - 1340 q^{61} + 347 q^{62} - 747 q^{63} + 893 q^{64} - 1890 q^{66} - 2308 q^{67} + 2785 q^{68} + 450 q^{69} + 1420 q^{70} - 96 q^{71} - 783 q^{72} - 2505 q^{73} - 1191 q^{74} - 60 q^{75} - 2409 q^{76} - 2142 q^{77} - 1591 q^{79} + 2671 q^{80} + 729 q^{81} + 1517 q^{82} - 1539 q^{83} - 2820 q^{84} - 4296 q^{85} + 3763 q^{86} - 291 q^{87} - 3716 q^{88} + 592 q^{89} - 486 q^{90} + 515 q^{92} - 2151 q^{93} - 692 q^{94} + 4158 q^{95} - 2121 q^{96} - 1445 q^{97} - 1457 q^{98} - 765 q^{99}+O(q^{100}) \) 9 * q - 6 * q^2 + 27 * q^3 + 44 * q^4 - 33 * q^5 - 18 * q^6 - 83 * q^7 - 87 * q^8 + 81 * q^9 - 54 * q^10 - 85 * q^11 + 132 * q^12 + 158 * q^14 - 99 * q^15 + 216 * q^16 + 178 * q^17 - 54 * q^18 - 352 * q^19 - 402 * q^20 - 249 * q^21 - 630 * q^22 + 150 * q^23 - 261 * q^24 - 20 * q^25 + 243 * q^27 - 940 * q^28 - 97 * q^29 - 162 * q^30 - 717 * q^31 - 707 * q^32 - 255 * q^33 - 632 * q^34 - 418 * q^35 + 396 * q^36 - 1108 * q^37 - 660 * q^38 - 1506 * q^40 - 334 * q^41 + 474 * q^42 + 242 * q^43 + 307 * q^44 - 297 * q^45 - 979 * q^46 + 184 * q^47 + 648 * q^48 - 38 * q^49 + 2031 * q^50 + 534 * q^51 - 151 * q^53 - 162 * q^54 + 2064 * q^55 + 2276 * q^56 - 1056 * q^57 - 1161 * q^58 - 537 * q^59 - 1206 * q^60 - 1340 * q^61 + 347 * q^62 - 747 * q^63 + 893 * q^64 - 1890 * q^66 - 2308 * q^67 + 2785 * q^68 + 450 * q^69 + 1420 * q^70 - 96 * q^71 - 783 * q^72 - 2505 * q^73 - 1191 * q^74 - 60 * q^75 - 2409 * q^76 - 2142 * q^77 - 1591 * q^79 + 2671 * q^80 + 729 * q^81 + 1517 * q^82 - 1539 * q^83 - 2820 * q^84 - 4296 * q^85 + 3763 * q^86 - 291 * q^87 - 3716 * q^88 + 592 * q^89 - 486 * q^90 + 515 * q^92 - 2151 * q^93 - 692 * q^94 + 4158 * q^95 - 2121 * q^96 - 1445 * q^97 - 1457 * q^98 - 765 * q^99

Introduction

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