LMFDB - Embedded newform 507.4.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("507.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$29.9139683729$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 56x^{7} - 27x^{6} + 945x^{5} + 763x^{4} - 4139x^{3} - 2478x^{2} + 63x + 27 $ x^9 - 56x^7 - 27x^6 + 945x^5 + 763x^4 - 4139x^3 - 2478x^2 + 63*x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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	$-0.689580\pi$
$2$	−3.17344	−1.12198	−0.560991	+	0.827822i	$0.689580\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	2.07074	0.258842
$5$	6.74147	0.602975	0.301488	−	0.953470i	$-0.402517\pi$
$5$	6.74147	0.602975	0.301488	+	0.953470i	$0.402517\pi$
$6$	−9.52033	−0.647776
$7$	−14.1726	−0.765251	−0.382625	−	0.923904i	$-0.624980\pi$
$7$	−14.1726	−0.765251	−0.382625	+	0.923904i	$0.624980\pi$
$8$	18.8162	0.831565
$9$	9.00000	0.333333
$10$	−21.3937	−0.676527
$11$	62.4956	1.71301	0.856507	−	0.516136i	$-0.172630\pi$
$11$	62.4956	1.71301	0.856507	+	0.516136i	$0.172630\pi$
$12$	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	$-0.373380\pi$
$19$	64.1652	0.774764	0.387382	+	0.921919i	$0.373380\pi$
$20$	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	$-0.535693\pi$
$31$	−38.6271	−0.223795	−0.111897	+	0.993720i	$0.535693\pi$
$32$	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	$-0.109462\pi$
$37$	423.770	1.88290	0.941452	+	0.337147i	$0.109462\pi$
$38$	−203.625	−0.869270
$39$	0	0
$40$	126.849	0.501414
$41$	−366.126	−1.39461	−0.697307	−	0.716772i	$-0.745619\pi$
$41$	−366.126	−1.39461	−0.697307	+	0.716772i	$0.745619\pi$
$42$	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	$-0.546156\pi$
$47$	−93.1169	−0.288989	−0.144495	+	0.989506i	$0.546156\pi$
$48$	−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	$-0.359684\pi$
$59$	386.729	0.853353	0.426677	+	0.904404i	$0.359684\pi$
$60$	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	$-0.210661\pi$
$67$	865.273	1.57776	0.788880	+	0.614547i	$0.210661\pi$
$68$	−121.381	−0.216464
$69$	32.7662	0.0571679
$70$	303.205	0.517713
$71$	−607.506	−1.01546	−0.507730	−	0.861516i	$-0.669515\pi$
$71$	−607.506	−1.01546	−0.507730	+	0.861516i	$0.669515\pi$
$72$	169.346	0.277188
$73$	980.958	1.57277	0.786387	−	0.617735i	$-0.211949\pi$
$73$	980.958	1.57277	0.786387	+	0.617735i	$0.211949\pi$
$74$	−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	$-0.295106\pi$
$83$	907.633	1.20031	0.600155	+	0.799884i	$0.295106\pi$
$84$	−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	$-0.711067\pi$
$89$	−1033.67	−1.23110	−0.615552	+	0.788096i	$0.711067\pi$
$90$	−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	$-0.315565\pi$
$97$	1046.17	1.09508	0.547538	+	0.836781i	$0.315565\pi$
$98$	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	$-0.635533\pi$
$109$	−940.072	−0.826079	−0.413039	+	0.910713i	$0.635533\pi$
$110$	−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	$-0.298890\pi$
$137$	1894.12	1.18121	0.590604	+	0.806961i	$0.298890\pi$
$138$	−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	$-0.158533\pi$
$149$	3195.65	1.75703	0.878517	+	0.477711i	$0.158533\pi$
$150$	757.366	0.412258
$151$	508.232	0.273903	0.136951	−	0.990578i	$-0.456270\pi$
$151$	508.232	0.273903	0.136951	+	0.990578i	$0.456270\pi$
$152$	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	$-0.502600\pi$
$163$	−33.9996	−0.0163378	−0.00816888	+	0.999967i	$0.502600\pi$
$164$	−758.149	−0.360985
$165$	1263.94	0.596348
$166$	−2880.32	−1.34673
$167$	−2210.67	−1.02435	−0.512176	−	0.858880i	$-0.671161\pi$
$167$	−2210.67	−1.02435	−0.512176	+	0.858880i	$0.671161\pi$
$168$	−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	$-0.515842\pi$
$193$	−266.771	−0.0994955	−0.0497478	+	0.998762i	$0.515842\pi$
$194$	−3319.95	−1.22865
$195$	0	0
$196$	−294.327	−0.107262
$197$	−1231.03	−0.445216	−0.222608	−	0.974908i	$-0.571457\pi$
$197$	−1231.03	−0.445216	−0.222608	+	0.974908i	$0.571457\pi$
$198$	−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	$-0.164942\pi$
$223$	5785.86	1.73744	0.868722	+	0.495300i	$0.164942\pi$
$224$	−1297.28	−0.386957
$225$	−715.973	−0.212140
$226$	−3048.09	−0.897149
$227$	2945.35	0.861189	0.430595	−	0.902545i	$-0.358304\pi$
$227$	2945.35	0.861189	0.430595	+	0.902545i	$0.358304\pi$
$228$	398.608	0.115783
$229$	−3541.26	−1.02189	−0.510945	−	0.859613i	$-0.670704\pi$
$229$	−3541.26	−1.02189	−0.510945	+	0.859613i	$0.670704\pi$
$230$	−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	$-0.565755\pi$
$239$	−1515.70	−0.410218	−0.205109	+	0.978739i	$0.565755\pi$
$240$	−1542.68	−0.414914
$241$	2392.47	0.639472	0.319736	−	0.947507i	$-0.396406\pi$
$241$	2392.47	0.639472	0.319736	+	0.947507i	$0.396406\pi$
$242$	−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	$-0.798364\pi$
$271$	−7191.36	−1.61197	−0.805986	+	0.591935i	$0.798364\pi$
$272$	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	$-0.637679\pi$
$281$	−3948.92	−0.838338	−0.419169	+	0.907908i	$0.637679\pi$
$282$	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	$-0.603455\pi$
$293$	−3203.02	−0.638644	−0.319322	+	0.947646i	$0.603455\pi$
$294$	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	$-0.352930\pi$
$307$	4795.67	0.891542	0.445771	+	0.895147i	$0.352930\pi$
$308$	−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	$-0.483741\pi$
$317$	576.333	0.102114	0.0510569	+	0.998696i	$0.483741\pi$
$318$	−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	$-0.458230\pi$
$331$	1575.95	0.261699	0.130849	+	0.991402i	$0.458230\pi$
$332$	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	$-0.624546\pi$
$349$	−4972.89	−0.762730	−0.381365	+	0.924425i	$0.624546\pi$
$350$	−3577.96	−0.546429
$351$	0	0
$352$	5720.49	0.866202
$353$	1575.34	0.237526	0.118763	−	0.992923i	$-0.462107\pi$
$353$	1575.34	0.237526	0.118763	+	0.992923i	$0.462107\pi$
$354$	−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	$-0.312237\pi$
$359$	7567.42	1.11252	0.556258	+	0.831010i	$0.312237\pi$
$360$	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	$-0.498531\pi$
$379$	68.0819	0.00922727	0.00461363	+	0.999989i	$0.498531\pi$
$380$	895.734	0.120922
$381$	6066.55	0.815745
$382$	7884.01	1.05597
$383$	1549.01	0.206659	0.103330	−	0.994647i	$-0.467050\pi$
$383$	1549.01	0.206659	0.103330	+	0.994647i	$0.467050\pi$
$384$	−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	$-0.374083\pi$
$397$	6096.27	0.770688	0.385344	+	0.922773i	$0.374083\pi$
$398$	10301.4	1.29740
$399$	−2728.17	−0.342304
$400$	6068.11	0.758513
$401$	7592.37	0.945498	0.472749	−	0.881197i	$-0.343262\pi$
$401$	7592.37	0.945498	0.472749	+	0.881197i	$0.343262\pi$
$402$	−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	$-0.644057\pi$
$409$	−7233.86	−0.874551	−0.437275	+	0.899328i	$0.644057\pi$
$410$	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	$-0.164206\pi$
$421$	15028.1	1.73973	0.869865	+	0.493290i	$0.164206\pi$
$422$	−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	$-0.630920\pi$
$431$	−7154.66	−0.799600	−0.399800	+	0.916602i	$0.630920\pi$
$432$	−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	$-0.400668\pi$
$449$	5842.05	0.614038	0.307019	+	0.951703i	$0.400668\pi$
$450$	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	$-0.401429\pi$
$457$	5954.40	0.609486	0.304743	+	0.952435i	$0.401429\pi$
$458$	11238.0	1.14654
$459$	−1582.66	−0.160942
$460$	152.470	0.0154542
$461$	−1865.94	−0.188515	−0.0942576	−	0.995548i	$-0.530048\pi$
$461$	−1865.94	−0.188515	−0.0942576	+	0.995548i	$0.530048\pi$
$462$	8432.42	0.849160
$463$	−6700.05	−0.672522	−0.336261	−	0.941769i	$-0.609162\pi$
$463$	−6700.05	−0.672522	−0.336261	+	0.941769i	$0.609162\pi$
$464$	−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	$-0.595028\pi$
$479$	−6166.88	−0.588250	−0.294125	+	0.955767i	$0.595028\pi$
$480$	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	$-0.414282\pi$
$487$	5718.51	0.532095	0.266047	+	0.963960i	$0.414282\pi$
$488$	−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	$-0.614750\pi$
$499$	−7863.87	−0.705481	−0.352741	+	0.935721i	$0.614750\pi$
$500$	−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	$-0.277121\pi$
$509$	14799.3	1.28873	0.644367	+	0.764717i	$0.277121\pi$
$510$	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	$-0.387840\pi$
$541$	8685.42	0.690232	0.345116	+	0.938560i	$0.387840\pi$
$542$	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	$-0.386002\pi$
$557$	9215.90	0.701059	0.350530	+	0.936552i	$0.386002\pi$
$558$	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	$-0.593690\pi$
$577$	−8041.67	−0.580206	−0.290103	+	0.956995i	$0.593690\pi$
$578$	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	$-0.672113\pi$
$587$	−14641.3	−1.02949	−0.514744	+	0.857344i	$0.672113\pi$
$588$	−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	$-0.436361\pi$
$593$	5735.76	0.397200	0.198600	+	0.980081i	$0.436361\pi$
$594$	−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	$-0.317874\pi$
$613$	16435.5	1.08291	0.541455	+	0.840730i	$0.317874\pi$
$614$	−15218.8	−1.00029
$615$	−7404.68	−0.485505
$616$	−16666.0	−1.09009
$617$	1290.89	0.0842289	0.0421145	−	0.999113i	$-0.486591\pi$
$617$	1290.89	0.0842289	0.0421145	+	0.999113i	$0.486591\pi$
$618$	−3690.36	−0.240207
$619$	−26719.0	−1.73494	−0.867470	−	0.497490i	$-0.834255\pi$
$619$	−26719.0	−1.73494	−0.867470	+	0.497490i	$0.834255\pi$
$620$	−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	$-0.609563\pi$
$631$	−10697.4	−0.674893	−0.337447	+	0.941345i	$0.609563\pi$
$632$	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	$-0.641250\pi$
$643$	−14000.3	−0.858661	−0.429331	+	0.903147i	$0.641250\pi$
$644$	−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	$-0.657162\pi$
$661$	−16107.9	−0.947841	−0.473921	+	0.880568i	$0.657162\pi$
$662$	−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	$-0.502309\pi$
$683$	−258.953	−0.0145074	−0.00725369	+	0.999974i	$0.502309\pi$
$684$	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	$-0.505767\pi$
$691$	−658.193	−0.0362357	−0.0181178	+	0.999836i	$0.505767\pi$
$692$	−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	$-0.442208\pi$
$709$	6817.51	0.361124	0.180562	+	0.983564i	$0.442208\pi$
$710$	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	$-0.640834\pi$
$733$	−16993.4	−0.856299	−0.428149	+	0.903708i	$0.640834\pi$
$734$	−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	$-0.613411\pi$
$739$	−14014.3	−0.697600	−0.348800	+	0.937197i	$0.613411\pi$
$740$	5915.75	0.293875
$741$	0	0
$742$	5932.73	0.293528
$743$	−141.560	−0.00698971	−0.00349485	−	0.999994i	$-0.501112\pi$
$743$	−141.560	−0.00698971	−0.00349485	+	0.999994i	$0.501112\pi$
$744$	−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	$-0.503215\pi$
$761$	−424.121	−0.0202028	−0.0101014	+	0.999949i	$0.503215\pi$
$762$	−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	$-0.149023\pi$
$769$	38060.7	1.78479	0.892396	+	0.451252i	$0.149023\pi$
$770$	18949.0	0.886849
$771$	−18582.8	−0.868019
$772$	−552.413	−0.0257536
$773$	−16683.3	−0.776268	−0.388134	−	0.921603i	$-0.626880\pi$
$773$	−16683.3	−0.776268	−0.388134	+	0.921603i	$0.626880\pi$
$774$	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	$-0.846259\pi$
$787$	−39105.2	−1.77122	−0.885609	+	0.464432i	$0.846259\pi$
$788$	−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	$-0.445305\pi$
$811$	7898.11	0.341973	0.170987	+	0.985273i	$0.445305\pi$
$812$	−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	$-0.526685\pi$
$821$	−3939.61	−0.167471	−0.0837353	+	0.996488i	$0.526685\pi$
$822$	−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	$-0.415281\pi$
$827$	12510.6	0.526042	0.263021	+	0.964790i	$0.415281\pi$
$828$	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	$-0.355966\pi$
$839$	21250.3	0.874426	0.437213	+	0.899358i	$0.355966\pi$
$840$	−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	$-0.254584\pi$
$853$	34721.1	1.39370	0.696852	+	0.717215i	$0.254584\pi$
$854$	−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	$-0.648644\pi$
$863$	−22826.6	−0.900380	−0.450190	+	0.892933i	$0.648644\pi$
$864$	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	$-0.338678\pi$
$877$	25212.7	0.970776	0.485388	+	0.874299i	$0.338678\pi$
$878$	−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	$-0.773830\pi$
$929$	−42927.0	−1.51603	−0.758014	+	0.652238i	$0.773830\pi$
$930$	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	$-0.776432\pi$
$941$	−44067.7	−1.52664	−0.763319	+	0.646022i	$0.776432\pi$
$942$	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	$-0.775694\pi$
$947$	−44402.5	−1.52364	−0.761820	+	0.647789i	$0.775694\pi$
$948$	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	$-0.455221\pi$
$967$	8432.54	0.280426	0.140213	+	0.990121i	$0.455221\pi$
$968$	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	$-0.715799\pi$
$977$	−38306.9	−1.25440	−0.627199	+	0.778859i	$0.715799\pi$
$978$	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	$-0.598715\pi$
$983$	−18810.9	−0.610350	−0.305175	+	0.952296i	$0.598715\pi$
$984$	−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.p.1.2	yes	9
3.2	odd	2		1521.4.a.bf.1.8		9
13.5	odd	4		507.4.b.k.337.14		18
13.8	odd	4		507.4.b.k.337.5		18
13.12	even	2		507.4.a.o.1.8	✓	9
39.38	odd	2		1521.4.a.bi.1.2		9

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-3.17344 q^{2} +3.00000 q^{3} +2.07074 q^{4} +6.74147 q^{5} -9.52033 q^{6} -14.1726 q^{7} +18.8162 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-3.17344 q^{2} +3.00000 q^{3} +2.07074 q^{4} +6.74147 q^{5} -9.52033 q^{6} -14.1726 q^{7} +18.8162 q^{8} +9.00000 q^{9} -21.3937 q^{10} +62.4956 q^{11} +6.21221 q^{12} +44.9761 q^{14} +20.2244 q^{15} -76.2779 q^{16} -58.6172 q^{17} -28.5610 q^{18} +64.1652 q^{19} +13.9598 q^{20} -42.5179 q^{21} -198.326 q^{22} +10.9221 q^{23} +56.4485 q^{24} -79.5526 q^{25} +27.0000 q^{27} -29.3478 q^{28} +216.316 q^{29} -64.1810 q^{30} -38.6271 q^{31} +91.5342 q^{32} +187.487 q^{33} +186.018 q^{34} -95.5445 q^{35} +18.6366 q^{36} +423.770 q^{37} -203.625 q^{38} +126.849 q^{40} -366.126 q^{41} +134.928 q^{42} -128.297 q^{43} +129.412 q^{44} +60.6732 q^{45} -34.6605 q^{46} -93.1169 q^{47} -228.834 q^{48} -142.136 q^{49} +252.455 q^{50} -175.852 q^{51} +131.909 q^{53} -85.6829 q^{54} +421.313 q^{55} -266.675 q^{56} +192.496 q^{57} -686.467 q^{58} +386.729 q^{59} +41.8794 q^{60} -621.077 q^{61} +122.581 q^{62} -127.554 q^{63} +319.745 q^{64} -594.979 q^{66} +865.273 q^{67} -121.381 q^{68} +32.7662 q^{69} +303.205 q^{70} -607.506 q^{71} +169.346 q^{72} +980.958 q^{73} -1344.81 q^{74} -238.658 q^{75} +132.869 q^{76} -885.728 q^{77} +1331.91 q^{79} -514.226 q^{80} +81.0000 q^{81} +1161.88 q^{82} +907.633 q^{83} -88.0434 q^{84} -395.166 q^{85} +407.142 q^{86} +648.949 q^{87} +1175.93 q^{88} -1033.67 q^{89} -192.543 q^{90} +22.6167 q^{92} -115.881 q^{93} +295.501 q^{94} +432.568 q^{95} +274.603 q^{96} +1046.17 q^{97} +451.061 q^{98} +562.461 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 6 q^{2} + 27 q^{3} + 44 q^{4} + 33 q^{5} + 18 q^{6} + 83 q^{7} + 87 q^{8} + 81 q^{9}+O(q^{10}) \) 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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