LMFDB - Embedded newform 507.4.a.o.1.7

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			1.7
Root			$5.06791$ 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)

$f(q)$	$=$	$q+2.82093 q^{2} +3.00000 q^{3} -0.0423641 q^{4} +3.41089 q^{5} +8.46278 q^{6} -13.3442 q^{7} -22.6869 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.82093 q^{2} +3.00000 q^{3} -0.0423641 q^{4} +3.41089 q^{5} +8.46278 q^{6} -13.3442 q^{7} -22.6869 q^{8} +9.00000 q^{9} +9.62187 q^{10} -35.4529 q^{11} -0.127092 q^{12} -37.6430 q^{14} +10.2327 q^{15} -63.6593 q^{16} +69.6526 q^{17} +25.3884 q^{18} -12.4014 q^{19} -0.144499 q^{20} -40.0325 q^{21} -100.010 q^{22} -126.251 q^{23} -68.0608 q^{24} -113.366 q^{25} +27.0000 q^{27} +0.565314 q^{28} -179.060 q^{29} +28.8656 q^{30} -255.935 q^{31} +1.91716 q^{32} -106.359 q^{33} +196.485 q^{34} -45.5155 q^{35} -0.381277 q^{36} -207.235 q^{37} -34.9833 q^{38} -77.3825 q^{40} +117.701 q^{41} -112.929 q^{42} +553.224 q^{43} +1.50193 q^{44} +30.6980 q^{45} -356.146 q^{46} -62.9185 q^{47} -190.978 q^{48} -164.933 q^{49} -319.797 q^{50} +208.958 q^{51} -147.031 q^{53} +76.1651 q^{54} -120.926 q^{55} +302.738 q^{56} -37.2041 q^{57} -505.115 q^{58} -274.087 q^{59} -0.433498 q^{60} +603.039 q^{61} -721.974 q^{62} -120.098 q^{63} +514.683 q^{64} -300.031 q^{66} +741.019 q^{67} -2.95077 q^{68} -378.754 q^{69} -128.396 q^{70} -572.574 q^{71} -204.182 q^{72} -26.7155 q^{73} -584.595 q^{74} -340.098 q^{75} +0.525372 q^{76} +473.090 q^{77} -207.798 q^{79} -217.135 q^{80} +81.0000 q^{81} +332.026 q^{82} +1031.37 q^{83} +1.69594 q^{84} +237.577 q^{85} +1560.60 q^{86} -537.179 q^{87} +804.318 q^{88} -1229.66 q^{89} +86.5968 q^{90} +5.34852 q^{92} -767.805 q^{93} -177.488 q^{94} -42.2996 q^{95} +5.75148 q^{96} -1795.12 q^{97} -465.264 q^{98} -319.076 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.82093	0.997349	0.498674	−	0.866789i	$-0.333820\pi$
$2$	2.82093	0.997349	0.498674	+	0.866789i	$0.333820\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−0.0423641	−0.00529552
$5$	3.41089	0.305079	0.152539	−	0.988297i	$-0.451255\pi$
$5$	3.41089	0.305079	0.152539	+	0.988297i	$0.451255\pi$
$6$	8.46278	0.575820
$7$	−13.3442	−0.720518	−0.360259	−	0.932852i	$-0.617312\pi$
$7$	−13.3442	−0.720518	−0.360259	+	0.932852i	$0.617312\pi$
$8$	−22.6869	−1.00263
$9$	9.00000	0.333333
$10$	9.62187	0.304270
$11$	−35.4529	−0.971769	−0.485885	−	0.874023i	$-0.661502\pi$
$11$	−35.4529	−0.971769	−0.485885	+	0.874023i	$0.661502\pi$
$12$	−0.127092	−0.00305737
$13$	0	0
$13$	0	0
$14$	−37.6430	−0.718607
$15$	10.2327	0.176137
$16$	−63.6593	−0.994676
$17$	69.6526	0.993720	0.496860	−	0.867831i	$-0.334486\pi$
$17$	69.6526	0.993720	0.496860	+	0.867831i	$0.334486\pi$
$18$	25.3884	0.332450
$19$	−12.4014	−0.149740	−0.0748701	−	0.997193i	$-0.523854\pi$
$19$	−12.4014	−0.149740	−0.0748701	+	0.997193i	$0.523854\pi$
$20$	−0.144499	−0.00161555
$21$	−40.0325	−0.415991
$22$	−100.010	−0.969193
$23$	−126.251	−1.14457	−0.572287	−	0.820053i	$-0.693944\pi$
$23$	−126.251	−1.14457	−0.572287	+	0.820053i	$0.693944\pi$
$24$	−68.0608	−0.578869
$25$	−113.366	−0.906927
$26$	0	0
$27$	27.0000	0.192450
$28$	0.565314	0.00381551
$29$	−179.060	−1.14657	−0.573286	−	0.819356i	$-0.694331\pi$
$29$	−179.060	−1.14657	−0.573286	+	0.819356i	$0.694331\pi$
$30$	28.8656	0.175670
$31$	−255.935	−1.48281	−0.741407	−	0.671055i	$-0.765841\pi$
$31$	−255.935	−1.48281	−0.741407	+	0.671055i	$0.765841\pi$
$32$	1.91716	0.0105909
$33$	−106.359	−0.561051
$34$	196.485	0.991085
$35$	−45.5155	−0.219815
$36$	−0.381277	−0.00176517
$37$	−207.235	−0.920790	−0.460395	−	0.887714i	$-0.652292\pi$
$37$	−207.235	−0.920790	−0.460395	+	0.887714i	$0.652292\pi$
$38$	−34.9833	−0.149343
$39$	0	0
$40$	−77.3825	−0.305881
$41$	117.701	0.448337	0.224169	−	0.974550i	$-0.428033\pi$
$41$	117.701	0.448337	0.224169	+	0.974550i	$0.428033\pi$
$42$	−112.929	−0.414888
$43$	553.224	1.96200	0.980998	−	0.194016i	$-0.0621515\pi$
$43$	553.224	1.96200	0.980998	+	0.194016i	$0.0621515\pi$
$44$	1.50193	0.00514602
$45$	30.6980	0.101693
$46$	−356.146	−1.14154
$47$	−62.9185	−0.195268	−0.0976341	−	0.995222i	$-0.531127\pi$
$47$	−62.9185	−0.195268	−0.0976341	+	0.995222i	$0.531127\pi$
$48$	−190.978	−0.574277
$49$	−164.933	−0.480854
$50$	−319.797	−0.904522
$51$	208.958	0.573724
$52$	0	0
$53$	−147.031	−0.381061	−0.190531	−	0.981681i	$-0.561021\pi$
$53$	−147.031	−0.381061	−0.190531	+	0.981681i	$0.561021\pi$
$54$	76.1651	0.191940
$55$	−120.926	−0.296466
$56$	302.738	0.722413
$57$	−37.2041	−0.0864526
$58$	−505.115	−1.14353
$59$	−274.087	−0.604798	−0.302399	−	0.953181i	$-0.597787\pi$
$59$	−274.087	−0.604798	−0.302399	+	0.953181i	$0.597787\pi$
$60$	−0.433498	−0.000932738 0
$61$	603.039	1.26576	0.632879	−	0.774251i	$-0.281873\pi$
$61$	603.039	1.26576	0.632879	+	0.774251i	$0.281873\pi$
$62$	−721.974	−1.47888
$63$	−120.098	−0.240173
$64$	514.683	1.00524
$65$	0	0
$66$	−300.031	−0.559564
$67$	741.019	1.35119	0.675596	−	0.737272i	$-0.263887\pi$
$67$	741.019	1.35119	0.675596	+	0.737272i	$0.263887\pi$
$68$	−2.95077	−0.00526226
$69$	−378.754	−0.660820
$70$	−128.396	−0.219232
$71$	−572.574	−0.957071	−0.478536	−	0.878068i	$-0.658832\pi$
$71$	−572.574	−0.957071	−0.478536	+	0.878068i	$0.658832\pi$
$72$	−204.182	−0.334210
$73$	−26.7155	−0.0428330	−0.0214165	−	0.999771i	$-0.506818\pi$
$73$	−26.7155	−0.0428330	−0.0214165	+	0.999771i	$0.506818\pi$
$74$	−584.595	−0.918349
$75$	−340.098	−0.523614
$76$	0.525372	0.000792952 0
$77$	473.090	0.700177
$78$	0	0
$79$	−207.798	−0.295938	−0.147969	−	0.988992i	$-0.547274\pi$
$79$	−207.798	−0.295938	−0.147969	+	0.988992i	$0.547274\pi$
$80$	−217.135	−0.303455
$81$	81.0000	0.111111
$82$	332.026	0.447149
$83$	1031.37	1.36395	0.681976	−	0.731374i	$-0.261121\pi$
$83$	1031.37	1.36395	0.681976	+	0.731374i	$0.261121\pi$
$84$	1.69594	0.00220289
$85$	237.577	0.303163
$86$	1560.60	1.95679
$87$	−537.179	−0.661973
$88$	804.318	0.974325
$89$	−1229.66	−1.46453	−0.732266	−	0.681019i	$-0.761537\pi$
$89$	−1229.66	−1.46453	−0.732266	+	0.681019i	$0.761537\pi$
$90$	86.5968	0.101423
$91$	0	0
$92$	5.34852	0.00606111
$93$	−767.805	−0.856103
$94$	−177.488	−0.194750
$95$	−42.2996	−0.0456826
$96$	5.75148	0.00611467
$97$	−1795.12	−1.87903	−0.939517	−	0.342502i	$-0.888726\pi$
$97$	−1795.12	−1.87903	−0.939517	+	0.342502i	$0.888726\pi$
$98$	−465.264	−0.479579
$99$	−319.076	−0.323923
$100$	4.80265	0.00480265
$101$	−769.697	−0.758295	−0.379147	−	0.925336i	$-0.623783\pi$
$101$	−769.697	−0.758295	−0.379147	+	0.925336i	$0.623783\pi$
$102$	589.455	0.572203
$103$	1543.66	1.47671	0.738357	−	0.674410i	$-0.235602\pi$
$103$	1543.66	1.47671	0.738357	+	0.674410i	$0.235602\pi$
$104$	0	0
$105$	−136.546	−0.126910
$106$	−414.764	−0.380051
$107$	2027.64	1.83196	0.915978	−	0.401228i	$-0.131417\pi$
$107$	2027.64	1.83196	0.915978	+	0.401228i	$0.131417\pi$
$108$	−1.14383	−0.00101912
$109$	−1644.97	−1.44550	−0.722751	−	0.691108i	$-0.757123\pi$
$109$	−1644.97	−1.44550	−0.722751	+	0.691108i	$0.757123\pi$
$110$	−341.123	−0.295680
$111$	−621.705	−0.531618
$112$	849.481	0.716682
$113$	−654.514	−0.544880	−0.272440	−	0.962173i	$-0.587831\pi$
$113$	−654.514	−0.544880	−0.272440	+	0.962173i	$0.587831\pi$
$114$	−104.950	−0.0862234
$115$	−430.629	−0.349186
$116$	7.58571	0.00607169
$117$	0	0
$118$	−773.179	−0.603194
$119$	−929.456	−0.715993
$120$	−232.148	−0.176601
$121$	−74.0896	−0.0556646
$122$	1701.13	1.26240
$123$	353.103	0.258848
$124$	10.8425	0.00785227
$125$	−813.039	−0.581763
$126$	−338.787	−0.239536
$127$	1325.54	0.926161	0.463080	−	0.886316i	$-0.346744\pi$
$127$	1325.54	0.926161	0.463080	+	0.886316i	$0.346744\pi$
$128$	1436.55	0.991983
$129$	1659.67	1.13276
$130$	0	0
$131$	930.114	0.620339	0.310170	−	0.950681i	$-0.399614\pi$
$131$	930.114	0.620339	0.310170	+	0.950681i	$0.399614\pi$
$132$	4.50580	0.00297106
$133$	165.486	0.107891
$134$	2090.36	1.34761
$135$	92.0939	0.0587125
$136$	−1580.20	−0.996333
$137$	2751.03	1.71559	0.857797	−	0.513989i	$-0.171833\pi$
$137$	2751.03	1.71559	0.857797	+	0.513989i	$0.171833\pi$
$138$	−1068.44	−0.659068
$139$	2436.65	1.48686	0.743432	−	0.668812i	$-0.233197\pi$
$139$	2436.65	1.48686	0.743432	+	0.668812i	$0.233197\pi$
$140$	1.92822	0.00116403
$141$	−188.755	−0.112738
$142$	−1615.19	−0.954534
$143$	0	0
$144$	−572.934	−0.331559
$145$	−610.753	−0.349795
$146$	−75.3624	−0.0427194
$147$	−494.799	−0.277621
$148$	8.77933	0.00487606
$149$	1517.58	0.834397	0.417198	−	0.908815i	$-0.363012\pi$
$149$	1517.58	0.834397	0.417198	+	0.908815i	$0.363012\pi$
$150$	−959.391	−0.522226
$151$	−583.642	−0.314544	−0.157272	−	0.987555i	$-0.550270\pi$
$151$	−583.642	−0.314544	−0.157272	+	0.987555i	$0.550270\pi$
$152$	281.349	0.150134
$153$	626.873	0.331240
$154$	1334.55	0.698321
$155$	−872.965	−0.452376
$156$	0	0
$157$	26.0932	0.0132641	0.00663206	−	0.999978i	$-0.497889\pi$
$157$	26.0932	0.0132641	0.00663206	+	0.999978i	$0.497889\pi$
$158$	−586.184	−0.295154
$159$	−441.093	−0.220006
$160$	6.53922	0.00323107
$161$	1684.72	0.824686
$162$	228.495	0.110817
$163$	−17.0905	−0.00821246	−0.00410623	−	0.999992i	$-0.501307\pi$
$163$	−17.0905	−0.00821246	−0.00410623	+	0.999992i	$0.501307\pi$
$164$	−4.98631	−0.00237418
$165$	−362.778	−0.171165
$166$	2909.43	1.36034
$167$	−3919.48	−1.81616	−0.908079	−	0.418799i	$-0.862451\pi$
$167$	−3919.48	−1.81616	−0.908079	+	0.418799i	$0.862451\pi$
$168$	908.215	0.417085
$169$	0	0
$170$	670.188	0.302359
$171$	−111.612	−0.0499134
$172$	−23.4368	−0.0103898
$173$	−1976.35	−0.868551	−0.434276	−	0.900780i	$-0.642996\pi$
$173$	−1976.35	−0.868551	−0.434276	+	0.900780i	$0.642996\pi$
$174$	−1515.34	−0.660218
$175$	1512.77	0.653457
$176$	2256.91	0.966596
$177$	−822.260	−0.349180
$178$	−3468.77	−1.46065
$179$	−2784.34	−1.16263	−0.581316	−	0.813678i	$-0.697462\pi$
$179$	−2784.34	−1.16263	−0.581316	+	0.813678i	$0.697462\pi$
$180$	−1.30049	−0.000538517 0
$181$	−1886.53	−0.774723	−0.387361	−	0.921928i	$-0.626613\pi$
$181$	−1886.53	−0.774723	−0.387361	+	0.921928i	$0.626613\pi$
$182$	0	0
$183$	1809.12	0.730786
$184$	2864.25	1.14758
$185$	−706.855	−0.280914
$186$	−2165.92	−0.853834
$187$	−2469.39	−0.965666
$188$	2.66549	0.00103405
$189$	−360.293	−0.138664
$190$	−119.324	−0.0455615
$191$	2447.94	0.927364	0.463682	−	0.886002i	$-0.346528\pi$
$191$	2447.94	0.927364	0.463682	+	0.886002i	$0.346528\pi$
$192$	1544.05	0.580375
$193$	1925.66	0.718196	0.359098	−	0.933300i	$-0.383084\pi$
$193$	1925.66	0.718196	0.359098	+	0.933300i	$0.383084\pi$
$194$	−5063.89	−1.87405
$195$	0	0
$196$	6.98724	0.00254637
$197$	1819.95	0.658203	0.329102	−	0.944294i	$-0.393254\pi$
$197$	1819.95	0.658203	0.329102	+	0.944294i	$0.393254\pi$
$198$	−900.092	−0.323064
$199$	−1273.97	−0.453816	−0.226908	−	0.973916i	$-0.572862\pi$
$199$	−1273.97	−0.453816	−0.226908	+	0.973916i	$0.572862\pi$
$200$	2571.92	0.909312
$201$	2223.06	0.780111
$202$	−2171.26	−0.756284
$203$	2389.40	0.826125
$204$	−8.85231	−0.00303817
$205$	401.465	0.136778
$206$	4354.56	1.47280
$207$	−1136.26	−0.381525
$208$	0	0
$209$	439.664	0.145513
$210$	−385.188	−0.126574
$211$	−1319.44	−0.430492	−0.215246	−	0.976560i	$-0.569055\pi$
$211$	−1319.44	−0.430492	−0.215246	+	0.976560i	$0.569055\pi$
$212$	6.22884	0.00201792
$213$	−1717.72	−0.552565
$214$	5719.83	1.82710
$215$	1886.98	0.598564
$216$	−612.547	−0.192956
$217$	3415.24	1.06839
$218$	−4640.35	−1.44167
$219$	−80.1464	−0.0247296
$220$	5.12292	0.00156994
$221$	0	0
$222$	−1753.79	−0.530209
$223$	−1203.45	−0.361384	−0.180692	−	0.983540i	$-0.557834\pi$
$223$	−1203.45	−0.361384	−0.180692	+	0.983540i	$0.557834\pi$
$224$	−25.5829	−0.00763095
$225$	−1020.29	−0.302309
$226$	−1846.34	−0.543436
$227$	4361.23	1.27518	0.637589	−	0.770377i	$-0.279932\pi$
$227$	4361.23	1.27518	0.637589	+	0.770377i	$0.279932\pi$
$228$	1.57612	0.000457811 0
$229$	−5384.29	−1.55373	−0.776864	−	0.629669i	$-0.783191\pi$
$229$	−5384.29	−1.55373	−0.776864	+	0.629669i	$0.783191\pi$
$230$	−1214.77	−0.348260
$231$	1419.27	0.404247
$232$	4062.32	1.14959
$233$	−4913.60	−1.38155	−0.690774	−	0.723070i	$-0.742730\pi$
$233$	−4913.60	−1.38155	−0.690774	+	0.723070i	$0.742730\pi$
$234$	0	0
$235$	−214.608	−0.0595722
$236$	11.6114	0.00320272
$237$	−623.394	−0.170860
$238$	−2621.93	−0.714094
$239$	−963.718	−0.260827	−0.130414	−	0.991460i	$-0.541631\pi$
$239$	−963.718	−0.260827	−0.130414	+	0.991460i	$0.541631\pi$
$240$	−651.404	−0.175200
$241$	−1544.48	−0.412817	−0.206409	−	0.978466i	$-0.566178\pi$
$241$	−1544.48	−0.412817	−0.206409	+	0.978466i	$0.566178\pi$
$242$	−209.001	−0.0555170
$243$	243.000	0.0641500
$244$	−25.5472	−0.00670284
$245$	−562.568	−0.146698
$246$	996.079	0.258161
$247$	0	0
$248$	5806.38	1.48671
$249$	3094.12	0.787478
$250$	−2293.52	−0.580221
$251$	3768.17	0.947588	0.473794	−	0.880636i	$-0.342884\pi$
$251$	3768.17	0.947588	0.473794	+	0.880636i	$0.342884\pi$
$252$	5.08783	0.00127184
$253$	4475.98	1.11226
$254$	3739.25	0.923705
$255$	712.731	0.175031
$256$	−65.0695	−0.0158861
$257$	1282.68	0.311327	0.155664	−	0.987810i	$-0.450248\pi$
$257$	1282.68	0.311327	0.155664	+	0.987810i	$0.450248\pi$
$258$	4681.81	1.12976
$259$	2765.38	0.663445
$260$	0	0
$261$	−1611.54	−0.382190
$262$	2623.78	0.618694
$263$	5029.74	1.17927	0.589633	−	0.807671i	$-0.299272\pi$
$263$	5029.74	1.17927	0.589633	+	0.807671i	$0.299272\pi$
$264$	2412.95	0.562527
$265$	−501.506	−0.116254
$266$	466.824	0.107604
$267$	−3688.97	−0.845548
$268$	−31.3926	−0.00715526
$269$	5628.46	1.27574	0.637868	−	0.770146i	$-0.279816\pi$
$269$	5628.46	1.27574	0.637868	+	0.770146i	$0.279816\pi$
$270$	259.790	0.0585568
$271$	−3368.39	−0.755037	−0.377518	−	0.926002i	$-0.623222\pi$
$271$	−3368.39	−0.755037	−0.377518	+	0.926002i	$0.623222\pi$
$272$	−4434.03	−0.988429
$273$	0	0
$274$	7760.45	1.71104
$275$	4019.15	0.881324
$276$	16.0456	0.00349938
$277$	−6507.75	−1.41160	−0.705799	−	0.708412i	$-0.749412\pi$
$277$	−6507.75	−1.41160	−0.705799	+	0.708412i	$0.749412\pi$
$278$	6873.62	1.48292
$279$	−2303.41	−0.494272
$280$	1032.61	0.220393
$281$	3625.51	0.769680	0.384840	−	0.922983i	$-0.374257\pi$
$281$	3625.51	0.769680	0.384840	+	0.922983i	$0.374257\pi$
$282$	−532.465	−0.112439
$283$	−4635.41	−0.973662	−0.486831	−	0.873496i	$-0.661847\pi$
$283$	−4635.41	−0.973662	−0.486831	+	0.873496i	$0.661847\pi$
$284$	24.2566	0.00506819
$285$	−126.899	−0.0263749
$286$	0	0
$287$	−1570.62	−0.323035
$288$	17.2545	0.00353031
$289$	−61.5178	−0.0125214
$290$	−1722.89	−0.348867
$291$	−5385.35	−1.08486
$292$	1.13178	0.000226823 0
$293$	2907.07	0.579634	0.289817	−	0.957082i	$-0.406406\pi$
$293$	2907.07	0.579634	0.289817	+	0.957082i	$0.406406\pi$
$294$	−1395.79	−0.276885
$295$	−934.879	−0.184511
$296$	4701.53	0.923212
$297$	−957.229	−0.187017
$298$	4280.99	0.832185
$299$	0	0
$300$	14.4079	0.00277281
$301$	−7382.32	−1.41365
$302$	−1646.41	−0.313710
$303$	−2309.09	−0.437802
$304$	789.461	0.148943
$305$	2056.90	0.386156
$306$	1768.36	0.330362
$307$	−933.950	−0.173627	−0.0868133	−	0.996225i	$-0.527668\pi$
$307$	−933.950	−0.173627	−0.0868133	+	0.996225i	$0.527668\pi$
$308$	−20.0421	−0.00370780
$309$	4630.98	0.852581
$310$	−2462.57	−0.451176
$311$	−3633.55	−0.662508	−0.331254	−	0.943542i	$-0.607472\pi$
$311$	−3633.55	−0.662508	−0.331254	+	0.943542i	$0.607472\pi$
$312$	0	0
$313$	−7507.26	−1.35570	−0.677852	−	0.735198i	$-0.737089\pi$
$313$	−7507.26	−1.35570	−0.677852	+	0.735198i	$0.737089\pi$
$314$	73.6071	0.0132289
$315$	−409.639	−0.0732716
$316$	8.80319	0.00156715
$317$	−1214.60	−0.215202	−0.107601	−	0.994194i	$-0.534317\pi$
$317$	−1214.60	−0.215202	−0.107601	+	0.994194i	$0.534317\pi$
$318$	−1244.29	−0.219423
$319$	6348.19	1.11420
$320$	1755.52	0.306677
$321$	6082.92	1.05768
$322$	4752.47	0.822500
$323$	−863.786	−0.148800
$324$	−3.43149	−0.000588391 0
$325$	0	0
$326$	−48.2111	−0.00819069
$327$	−4934.92	−0.834561
$328$	−2670.28	−0.449517
$329$	839.595	0.140694
$330$	−1023.37	−0.170711
$331$	−2443.20	−0.405712	−0.202856	−	0.979209i	$-0.565022\pi$
$331$	−2443.20	−0.405712	−0.202856	+	0.979209i	$0.565022\pi$
$332$	−43.6933	−0.00722283
$333$	−1865.12	−0.306930
$334$	−11056.6	−1.81134
$335$	2527.53	0.412220
$336$	2548.44	0.413777
$337$	1890.96	0.305660	0.152830	−	0.988253i	$-0.451161\pi$
$337$	1890.96	0.305660	0.152830	+	0.988253i	$0.451161\pi$
$338$	0	0
$339$	−1963.54	−0.314587
$340$	−10.0647	−0.00160540
$341$	9073.64	1.44095
$342$	−314.850	−0.0497811
$343$	6777.95	1.06698
$344$	−12551.0	−1.96716
$345$	−1291.89	−0.201602
$346$	−5575.15	−0.866248
$347$	−2791.57	−0.431872	−0.215936	−	0.976408i	$-0.569280\pi$
$347$	−2791.57	−0.431872	−0.215936	+	0.976408i	$0.569280\pi$
$348$	22.7571	0.00350549
$349$	6917.33	1.06096	0.530482	−	0.847696i	$-0.322011\pi$
$349$	6917.33	1.06096	0.530482	+	0.847696i	$0.322011\pi$
$350$	4267.43	0.651724
$351$	0	0
$352$	−67.9690	−0.0102919
$353$	−7638.37	−1.15170	−0.575849	−	0.817556i	$-0.695328\pi$
$353$	−7638.37	−1.15170	−0.575849	+	0.817556i	$0.695328\pi$
$354$	−2319.54	−0.348254
$355$	−1952.99	−0.291982
$356$	52.0933	0.00775545
$357$	−2788.37	−0.413379
$358$	−7854.42	−1.15955
$359$	5490.97	0.807249	0.403625	−	0.914925i	$-0.367750\pi$
$359$	5490.97	0.807249	0.403625	+	0.914925i	$0.367750\pi$
$360$	−696.443	−0.101960
$361$	−6705.21	−0.977578
$362$	−5321.77	−0.772669
$363$	−222.269	−0.0321380
$364$	0	0
$365$	−91.1234	−0.0130674
$366$	5103.39	0.728848
$367$	6737.43	0.958287	0.479143	−	0.877737i	$-0.340947\pi$
$367$	6737.43	0.958287	0.479143	+	0.877737i	$0.340947\pi$
$368$	8037.07	1.13848
$369$	1059.31	0.149446
$370$	−1993.99	−0.280169
$371$	1962.01	0.274561
$372$	32.5274	0.00453351
$373$	−3930.42	−0.545602	−0.272801	−	0.962070i	$-0.587950\pi$
$373$	−3930.42	−0.545602	−0.272801	+	0.962070i	$0.587950\pi$
$374$	−6965.97	−0.963106
$375$	−2439.12	−0.335881
$376$	1427.43	0.195782
$377$	0	0
$378$	−1016.36	−0.138296
$379$	−11897.7	−1.61252	−0.806258	−	0.591563i	$-0.798511\pi$
$379$	−11897.7	−1.61252	−0.806258	+	0.591563i	$0.798511\pi$
$380$	1.79199	0.000241913 0
$381$	3976.61	0.534719
$382$	6905.46	0.924906
$383$	−4748.31	−0.633492	−0.316746	−	0.948510i	$-0.602590\pi$
$383$	−4748.31	−0.633492	−0.316746	+	0.948510i	$0.602590\pi$
$384$	4309.64	0.572722
$385$	1613.66	0.213609
$386$	5432.14	0.716292
$387$	4979.02	0.653999
$388$	76.0485	0.00995046
$389$	11299.5	1.47277	0.736386	−	0.676561i	$-0.236531\pi$
$389$	11299.5	1.47277	0.736386	+	0.676561i	$0.236531\pi$
$390$	0	0
$391$	−8793.73	−1.13739
$392$	3741.82	0.482119
$393$	2790.34	0.358153
$394$	5133.95	0.656458
$395$	−708.776	−0.0902845
$396$	13.5174	0.00171534
$397$	−13853.0	−1.75129	−0.875646	−	0.482953i	$-0.839564\pi$
$397$	−13853.0	−1.75129	−0.875646	+	0.482953i	$0.839564\pi$
$398$	−3593.78	−0.452613
$399$	496.457	0.0622906
$400$	7216.79	0.902099
$401$	−5434.64	−0.676790	−0.338395	−	0.941004i	$-0.609884\pi$
$401$	−5434.64	−0.676790	−0.338395	+	0.941004i	$0.609884\pi$
$402$	6271.08	0.778042
$403$	0	0
$404$	32.6076	0.00401556
$405$	276.282	0.0338977
$406$	6740.34	0.823935
$407$	7347.09	0.894795
$408$	−4740.61	−0.575233
$409$	3472.43	0.419806	0.209903	−	0.977722i	$-0.432685\pi$
$409$	3472.43	0.419806	0.209903	+	0.977722i	$0.432685\pi$
$410$	1132.50	0.136416
$411$	8253.09	0.990498
$412$	−65.3959	−0.00781996
$413$	3657.46	0.435767
$414$	−3205.31	−0.380513
$415$	3517.90	0.416113
$416$	0	0
$417$	7309.95	0.858441
$418$	1240.26	0.145127
$419$	−4261.65	−0.496886	−0.248443	−	0.968647i	$-0.579919\pi$
$419$	−4261.65	−0.496886	−0.248443	+	0.968647i	$0.579919\pi$
$420$	5.78467	0.000672055 0
$421$	−6235.06	−0.721801	−0.360900	−	0.932604i	$-0.617531\pi$
$421$	−6235.06	−0.721801	−0.360900	+	0.932604i	$0.617531\pi$
$422$	−3722.04	−0.429351
$423$	−566.266	−0.0650894
$424$	3335.68	0.382064
$425$	−7896.22	−0.901231
$426$	−4845.57	−0.551100
$427$	−8047.06	−0.912001
$428$	−85.8992	−0.00970115
$429$	0	0
$430$	5323.05	0.596977
$431$	−2520.58	−0.281698	−0.140849	−	0.990031i	$-0.544983\pi$
$431$	−2520.58	−0.281698	−0.140849	+	0.990031i	$0.544983\pi$
$432$	−1718.80	−0.191426
$433$	−1283.89	−0.142494	−0.0712468	−	0.997459i	$-0.522698\pi$
$433$	−1283.89	−0.142494	−0.0712468	+	0.997459i	$0.522698\pi$
$434$	9634.15	1.06556
$435$	−1832.26	−0.201954
$436$	69.6878	0.00765468
$437$	1565.69	0.171389
$438$	−226.087	−0.0246641
$439$	−16942.0	−1.84190	−0.920951	−	0.389677i	$-0.872586\pi$
$439$	−16942.0	−1.84190	−0.920951	+	0.389677i	$0.872586\pi$
$440$	2743.44	0.297246
$441$	−1484.40	−0.160285
$442$	0	0
$443$	−2163.00	−0.231980	−0.115990	−	0.993250i	$-0.537004\pi$
$443$	−2163.00	−0.231980	−0.115990	+	0.993250i	$0.537004\pi$
$444$	26.3380	0.00281519
$445$	−4194.22	−0.446798
$446$	−3394.83	−0.360426
$447$	4552.75	0.481739
$448$	−6868.01	−0.724293
$449$	1673.35	0.175880	0.0879400	−	0.996126i	$-0.471972\pi$
$449$	1673.35	0.175880	0.0879400	+	0.996126i	$0.471972\pi$
$450$	−2878.17	−0.301507
$451$	−4172.85	−0.435680
$452$	27.7279	0.00288542
$453$	−1750.92	−0.181602
$454$	12302.7	1.27180
$455$	0	0
$456$	844.046	0.0866800
$457$	4652.38	0.476212	0.238106	−	0.971239i	$-0.423473\pi$
$457$	4652.38	0.476212	0.238106	+	0.971239i	$0.423473\pi$
$458$	−15188.7	−1.54961
$459$	1880.62	0.191241
$460$	18.2432	0.00184912
$461$	−3460.07	−0.349570	−0.174785	−	0.984607i	$-0.555923\pi$
$461$	−3460.07	−0.349570	−0.174785	+	0.984607i	$0.555923\pi$
$462$	4003.66	0.403176
$463$	2105.64	0.211355	0.105677	−	0.994400i	$-0.466299\pi$
$463$	2105.64	0.211355	0.105677	+	0.994400i	$0.466299\pi$
$464$	11398.8	1.14047
$465$	−2618.89	−0.261179
$466$	−13860.9	−1.37789
$467$	−1415.88	−0.140298	−0.0701489	−	0.997537i	$-0.522347\pi$
$467$	−1415.88	−0.140298	−0.0701489	+	0.997537i	$0.522347\pi$
$468$	0	0
$469$	−9888.28	−0.973557
$470$	−605.393	−0.0594143
$471$	78.2797	0.00765804
$472$	6218.19	0.606388
$473$	−19613.4	−1.90661
$474$	−1758.55	−0.170407
$475$	1405.89	0.135803
$476$	39.3756	0.00379155
$477$	−1323.28	−0.127020
$478$	−2718.58	−0.260136
$479$	985.594	0.0940145	0.0470073	−	0.998895i	$-0.485032\pi$
$479$	985.594	0.0940145	0.0470073	+	0.998895i	$0.485032\pi$
$480$	19.6177	0.00186546
$481$	0	0
$482$	−4356.88	−0.411723
$483$	5054.16	0.476133
$484$	3.13874	0.000294773 0
$485$	−6122.93	−0.573254
$486$	685.486	0.0639800
$487$	−14724.7	−1.37011	−0.685053	−	0.728494i	$-0.740221\pi$
$487$	−14724.7	−1.37011	−0.685053	+	0.728494i	$0.740221\pi$
$488$	−13681.1	−1.26909
$489$	−51.2715	−0.00474147
$490$	−1586.96	−0.146310
$491$	−16301.2	−1.49830	−0.749149	−	0.662401i	$-0.769537\pi$
$491$	−16301.2	−1.49830	−0.749149	+	0.662401i	$0.769537\pi$
$492$	−14.9589	−0.00137073
$493$	−12472.0	−1.13937
$494$	0	0
$495$	−1088.33	−0.0988221
$496$	16292.6	1.47492
$497$	7640.53	0.689587
$498$	8728.30	0.785391
$499$	−3230.42	−0.289806	−0.144903	−	0.989446i	$-0.546287\pi$
$499$	−3230.42	−0.289806	−0.144903	+	0.989446i	$0.546287\pi$
$500$	34.4437	0.00308074
$501$	−11758.4	−1.04856
$502$	10629.7	0.945076
$503$	−3577.57	−0.317129	−0.158565	−	0.987349i	$-0.550687\pi$
$503$	−3577.57	−0.317129	−0.158565	+	0.987349i	$0.550687\pi$
$504$	2724.65	0.240804
$505$	−2625.35	−0.231340
$506$	12626.4	1.10931
$507$	0	0
$508$	−56.1552	−0.00490450
$509$	13864.7	1.20735	0.603674	−	0.797231i	$-0.293703\pi$
$509$	13864.7	1.20735	0.603674	+	0.797231i	$0.293703\pi$
$510$	2010.56	0.174567
$511$	356.496	0.0308619
$512$	−11675.9	−1.00783
$513$	−334.836	−0.0288175
$514$	3618.34	0.310502
$515$	5265.25	0.450514
$516$	−70.3105	−0.00599854
$517$	2230.64	0.189756
$518$	7800.94	0.661686
$519$	−5929.06	−0.501458
$520$	0	0
$521$	−9554.66	−0.803449	−0.401725	−	0.915760i	$-0.631589\pi$
$521$	−9554.66	−0.803449	−0.401725	+	0.915760i	$0.631589\pi$
$522$	−4546.03	−0.381177
$523$	−19809.1	−1.65620	−0.828098	−	0.560583i	$-0.810577\pi$
$523$	−19809.1	−1.65620	−0.828098	+	0.560583i	$0.810577\pi$
$524$	−39.4035	−0.00328502
$525$	4538.32	0.377274
$526$	14188.5	1.17614
$527$	−17826.5	−1.47350
$528$	6770.73	0.558064
$529$	3772.38	0.310050
$530$	−1414.71	−0.115946
$531$	−2466.78	−0.201599
$532$	−7.01066	−0.000571336 0
$533$	0	0
$534$	−10406.3	−0.843306
$535$	6916.05	0.558891
$536$	−16811.4	−1.35475
$537$	−8353.02	−0.671246
$538$	15877.5	1.27235
$539$	5847.36	0.467279
$540$	−3.90148	−0.000310913 0
$541$	1011.14	0.0803551	0.0401775	−	0.999193i	$-0.487208\pi$
$541$	1011.14	0.0803551	0.0401775	+	0.999193i	$0.487208\pi$
$542$	−9501.98	−0.753035
$543$	−5659.59	−0.447286
$544$	133.535	0.0105244
$545$	−5610.81	−0.440992
$546$	0	0
$547$	15745.2	1.23074	0.615372	−	0.788237i	$-0.289006\pi$
$547$	15745.2	1.23074	0.615372	+	0.788237i	$0.289006\pi$
$548$	−116.545	−0.00908495
$549$	5427.35	0.421919
$550$	11337.7	0.878987
$551$	2220.58	0.171688
$552$	8592.76	0.662558
$553$	2772.89	0.213229
$554$	−18357.9	−1.40785
$555$	−2120.57	−0.162186
$556$	−103.227	−0.00787371
$557$	−8510.94	−0.647433	−0.323716	−	0.946154i	$-0.604932\pi$
$557$	−8510.94	−0.647433	−0.323716	+	0.946154i	$0.604932\pi$
$558$	−6497.76	−0.492961
$559$	0	0
$560$	2897.48	0.218645
$561$	−7408.16	−0.557528
$562$	10227.3	0.767639
$563$	16764.1	1.25493	0.627463	−	0.778646i	$-0.284093\pi$
$563$	16764.1	1.25493	0.627463	+	0.778646i	$0.284093\pi$
$564$	7.99646	0.000597006 0
$565$	−2232.47	−0.166232
$566$	−13076.1	−0.971080
$567$	−1080.88	−0.0800575
$568$	12990.0	0.959589
$569$	20755.4	1.52919	0.764597	−	0.644508i	$-0.222938\pi$
$569$	20755.4	1.52919	0.764597	+	0.644508i	$0.222938\pi$
$570$	−357.972	−0.0263049
$571$	−23408.6	−1.71562	−0.857810	−	0.513968i	$-0.828175\pi$
$571$	−23408.6	−1.71562	−0.857810	+	0.513968i	$0.828175\pi$
$572$	0	0
$573$	7343.81	0.535414
$574$	−4430.62	−0.322179
$575$	14312.6	1.03805
$576$	4632.14	0.335080
$577$	−10387.2	−0.749436	−0.374718	−	0.927139i	$-0.622261\pi$
$577$	−10387.2	−0.749436	−0.374718	+	0.927139i	$0.622261\pi$
$578$	−173.537	−0.0124882
$579$	5776.97	0.414650
$580$	25.8740	0.00185234
$581$	−13762.8	−0.982752
$582$	−15191.7	−1.08198
$583$	5212.68	0.370304
$584$	606.092	0.0429457
$585$	0	0
$586$	8200.63	0.578097
$587$	2898.42	0.203800	0.101900	−	0.994795i	$-0.467508\pi$
$587$	2898.42	0.203800	0.101900	+	0.994795i	$0.467508\pi$
$588$	20.9617	0.00147015
$589$	3173.94	0.222037
$590$	−2637.23	−0.184022
$591$	5459.85	0.380014
$592$	13192.4	0.915888
$593$	−8805.33	−0.609766	−0.304883	−	0.952390i	$-0.598617\pi$
$593$	−8805.33	−0.609766	−0.304883	+	0.952390i	$0.598617\pi$
$594$	−2700.27	−0.186521
$595$	−3170.27	−0.218434
$596$	−64.2910	−0.00441856
$597$	−3821.91	−0.262011
$598$	0	0
$599$	−20236.8	−1.38039	−0.690194	−	0.723624i	$-0.742475\pi$
$599$	−20236.8	−1.38039	−0.690194	+	0.723624i	$0.742475\pi$
$600$	7715.77	0.524992
$601$	−9884.92	−0.670906	−0.335453	−	0.942057i	$-0.608889\pi$
$601$	−9884.92	−0.670906	−0.335453	+	0.942057i	$0.608889\pi$
$602$	−20825.0	−1.40991
$603$	6669.17	0.450397
$604$	24.7255	0.00166567
$605$	−252.711	−0.0169821
$606$	−6513.78	−0.436641
$607$	22919.3	1.53256	0.766281	−	0.642505i	$-0.222105\pi$
$607$	22919.3	1.53256	0.766281	+	0.642505i	$0.222105\pi$
$608$	−23.7754	−0.00158589
$609$	7168.21	0.476963
$610$	5802.36	0.385132
$611$	0	0
$612$	−26.5569	−0.00175409
$613$	−916.052	−0.0603572	−0.0301786	−	0.999545i	$-0.509608\pi$
$613$	−916.052	−0.0603572	−0.0301786	+	0.999545i	$0.509608\pi$
$614$	−2634.61	−0.173166
$615$	1204.40	0.0789690
$616$	−10733.0	−0.702019
$617$	30179.4	1.96917	0.984584	−	0.174915i	$-0.0559650\pi$
$617$	30179.4	1.96917	0.984584	+	0.174915i	$0.0559650\pi$
$618$	13063.7	0.850320
$619$	−5571.12	−0.361748	−0.180874	−	0.983506i	$-0.557893\pi$
$619$	−5571.12	−0.361748	−0.180874	+	0.983506i	$0.557893\pi$
$620$	36.9824	0.00239556
$621$	−3408.78	−0.220273
$622$	−10250.0	−0.660751
$623$	16408.8	1.05522
$624$	0	0
$625$	11397.5	0.729443
$626$	−21177.5	−1.35211
$627$	1318.99	0.0840120
$628$	−1.10542	−7.02403e−5 0
$629$	−14434.5	−0.915007
$630$	−1155.56	−0.0730773
$631$	11360.1	0.716703	0.358352	−	0.933587i	$-0.383339\pi$
$631$	11360.1	0.716703	0.358352	+	0.933587i	$0.383339\pi$
$632$	4714.30	0.296717
$633$	−3958.32	−0.248545
$634$	−3426.31	−0.214631
$635$	4521.26	0.282552
$636$	18.6865	0.00116504
$637$	0	0
$638$	17907.8	1.11125
$639$	−5153.17	−0.319024
$640$	4899.89	0.302633
$641$	−24709.8	−1.52259	−0.761293	−	0.648408i	$-0.775435\pi$
$641$	−24709.8	−1.52259	−0.761293	+	0.648408i	$0.775435\pi$
$642$	17159.5	1.05488
$643$	−15226.6	−0.933869	−0.466934	−	0.884292i	$-0.654642\pi$
$643$	−15226.6	−0.933869	−0.466934	+	0.884292i	$0.654642\pi$
$644$	−71.3717	−0.00436714
$645$	5660.95	0.345581
$646$	−2436.68	−0.148405
$647$	16706.6	1.01515	0.507576	−	0.861607i	$-0.330542\pi$
$647$	16706.6	1.01515	0.507576	+	0.861607i	$0.330542\pi$
$648$	−1837.64	−0.111403
$649$	9717.18	0.587724
$650$	0	0
$651$	10245.7	0.616838
$652$	0.724024	4.34892e−5 0
$653$	−28205.8	−1.69032	−0.845158	−	0.534516i	$-0.820494\pi$
$653$	−28205.8	−1.69032	−0.845158	+	0.534516i	$0.820494\pi$
$654$	−13921.0	−0.832349
$655$	3172.51	0.189252
$656$	−7492.77	−0.445951
$657$	−240.439	−0.0142777
$658$	2368.44	0.140321
$659$	−11424.2	−0.675299	−0.337649	−	0.941272i	$-0.609632\pi$
$659$	−11424.2	−0.675299	−0.337649	+	0.941272i	$0.609632\pi$
$660$	15.3688	0.000906407 0
$661$	26518.2	1.56042	0.780211	−	0.625516i	$-0.215112\pi$
$661$	26518.2	1.56042	0.780211	+	0.625516i	$0.215112\pi$
$662$	−6892.10	−0.404636
$663$	0	0
$664$	−23398.7	−1.36754
$665$	564.453	0.0329151
$666$	−5261.36	−0.306116
$667$	22606.5	1.31234
$668$	166.045	0.00961749
$669$	−3610.34	−0.208645
$670$	7129.98	0.411127
$671$	−21379.5	−1.23002
$672$	−76.7488	−0.00440573
$673$	22816.8	1.30687	0.653434	−	0.756984i	$-0.273328\pi$
$673$	22816.8	1.30687	0.653434	+	0.756984i	$0.273328\pi$
$674$	5334.27	0.304849
$675$	−3060.88	−0.174538
$676$	0	0
$677$	17737.6	1.00696	0.503478	−	0.864008i	$-0.332053\pi$
$677$	17737.6	1.00696	0.503478	+	0.864008i	$0.332053\pi$
$678$	−5539.01	−0.313753
$679$	23954.3	1.35388
$680$	−5389.89	−0.303960
$681$	13083.7	0.736224
$682$	25596.1	1.43713
$683$	18878.4	1.05763	0.528816	−	0.848737i	$-0.322636\pi$
$683$	18878.4	1.05763	0.528816	+	0.848737i	$0.322636\pi$
$684$	4.72835	0.000264317 0
$685$	9383.45	0.523391
$686$	19120.1	1.06415
$687$	−16152.9	−0.897045
$688$	−35217.8	−1.95155
$689$	0	0
$690$	−3644.32	−0.201068
$691$	−11692.8	−0.643726	−0.321863	−	0.946786i	$-0.604309\pi$
$691$	−11692.8	−0.643726	−0.321863	+	0.946786i	$0.604309\pi$
$692$	83.7265	0.00459943
$693$	4257.81	0.233392
$694$	−7874.82	−0.430726
$695$	8311.14	0.453611
$696$	12186.9	0.663714
$697$	8198.19	0.445522
$698$	19513.3	1.05815
$699$	−14740.8	−0.797638
$700$	−64.0873	−0.00346039
$701$	20658.1	1.11304	0.556522	−	0.830833i	$-0.312135\pi$
$701$	20658.1	1.11304	0.556522	+	0.830833i	$0.312135\pi$
$702$	0	0
$703$	2569.99	0.137879
$704$	−18247.0	−0.976861
$705$	−643.823	−0.0343940
$706$	−21547.3	−1.14864
$707$	10271.0	0.546365
$708$	34.8343	0.00184909
$709$	20695.8	1.09626	0.548128	−	0.836394i	$-0.315341\pi$
$709$	20695.8	1.09626	0.548128	+	0.836394i	$0.315341\pi$
$710$	−5509.23	−0.291208
$711$	−1870.18	−0.0986461
$712$	27897.1	1.46838
$713$	32312.1	1.69719
$714$	−7865.79	−0.412283
$715$	0	0
$716$	117.956	0.00615674
$717$	−2891.16	−0.150589
$718$	15489.6	0.805109
$719$	−26299.2	−1.36411	−0.682053	−	0.731302i	$-0.738913\pi$
$719$	−26299.2	−1.36411	−0.682053	+	0.731302i	$0.738913\pi$
$720$	−1954.21	−0.101152
$721$	−20598.9	−1.06400
$722$	−18914.9	−0.974986
$723$	−4633.45	−0.238340
$724$	79.9213	0.00410256
$725$	20299.3	1.03986
$726$	−627.004	−0.0320528
$727$	−16915.6	−0.862952	−0.431476	−	0.902125i	$-0.642007\pi$
$727$	−16915.6	−0.862952	−0.431476	+	0.902125i	$0.642007\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	−257.053	−0.0130328
$731$	38533.5	1.94967
$732$	−76.6417	−0.00386989
$733$	13203.1	0.665302	0.332651	−	0.943050i	$-0.392057\pi$
$733$	13203.1	0.665302	0.332651	+	0.943050i	$0.392057\pi$
$734$	19005.8	0.955746
$735$	−1687.70	−0.0846964
$736$	−242.044	−0.0121221
$737$	−26271.3	−1.31305
$738$	2988.24	0.149050
$739$	16813.9	0.836954	0.418477	−	0.908227i	$-0.362564\pi$
$739$	16813.9	0.836954	0.418477	+	0.908227i	$0.362564\pi$
$740$	29.9453	0.00148758
$741$	0	0
$742$	5534.68	0.273834
$743$	32682.1	1.61371	0.806857	−	0.590747i	$-0.201167\pi$
$743$	32682.1	1.61371	0.806857	+	0.590747i	$0.201167\pi$
$744$	17419.1	0.858355
$745$	5176.30	0.254557
$746$	−11087.4	−0.544156
$747$	9282.37	0.454651
$748$	104.613	0.00511370
$749$	−27057.2	−1.31996
$750$	−6880.57	−0.334991
$751$	2157.76	0.104844	0.0524219	−	0.998625i	$-0.483306\pi$
$751$	2157.76	0.104844	0.0524219	+	0.998625i	$0.483306\pi$
$752$	4005.35	0.194229
$753$	11304.5	0.547090
$754$	0	0
$755$	−1990.74	−0.0959606
$756$	15.2635	0.000734296 0
$757$	13769.6	0.661117	0.330558	−	0.943786i	$-0.392763\pi$
$757$	13769.6	0.661117	0.330558	+	0.943786i	$0.392763\pi$
$758$	−33562.6	−1.60824
$759$	13427.9	0.642165
$760$	959.648	0.0458028
$761$	−16905.2	−0.805272	−0.402636	−	0.915360i	$-0.631906\pi$
$761$	−16905.2	−0.805272	−0.402636	+	0.915360i	$0.631906\pi$
$762$	11217.7	0.533302
$763$	21950.8	1.04151
$764$	−103.705	−0.00491087
$765$	2138.19	0.101054
$766$	−13394.7	−0.631813
$767$	0	0
$768$	−195.208	−0.00917185
$769$	1153.35	0.0540845	0.0270422	−	0.999634i	$-0.491391\pi$
$769$	1153.35	0.0540845	0.0270422	+	0.999634i	$0.491391\pi$
$770$	4552.01	0.213043
$771$	3848.03	0.179745
$772$	−81.5787	−0.00380322
$773$	−20713.9	−0.963811	−0.481905	−	0.876223i	$-0.660055\pi$
$773$	−20713.9	−0.963811	−0.481905	+	0.876223i	$0.660055\pi$
$774$	14045.4	0.652265
$775$	29014.3	1.34480
$776$	40725.7	1.88398
$777$	8296.14	0.383040
$778$	31875.1	1.46887
$779$	−1459.65	−0.0671341
$780$	0	0
$781$	20299.4	0.930053
$782$	−24806.5	−1.13437
$783$	−4834.61	−0.220658
$784$	10499.5	0.478294
$785$	89.0010	0.00404660
$786$	7871.35	0.357203
$787$	10870.2	0.492353	0.246177	−	0.969225i	$-0.420826\pi$
$787$	10870.2	0.492353	0.246177	+	0.969225i	$0.420826\pi$
$788$	−77.1006	−0.00348553
$789$	15089.2	0.680850
$790$	−1999.41	−0.0900451
$791$	8733.95	0.392596
$792$	7238.86	0.324775
$793$	0	0
$794$	−39078.4	−1.74665
$795$	−1504.52	−0.0671192
$796$	53.9706	0.00240319
$797$	12026.6	0.534511	0.267255	−	0.963626i	$-0.413883\pi$
$797$	12026.6	0.534511	0.267255	+	0.963626i	$0.413883\pi$
$798$	1400.47	0.0621255
$799$	−4382.43	−0.194042
$800$	−217.341	−0.00960519
$801$	−11066.9	−0.488177
$802$	−15330.7	−0.674996
$803$	947.142	0.0416238
$804$	−94.1778	−0.00413109
$805$	5746.39	0.251594
$806$	0	0
$807$	16885.4	0.736547
$808$	17462.1	0.760289
$809$	−32384.0	−1.40737	−0.703685	−	0.710512i	$-0.748463\pi$
$809$	−32384.0	−1.40737	−0.703685	+	0.710512i	$0.748463\pi$
$810$	779.371	0.0338078
$811$	42506.8	1.84046	0.920231	−	0.391375i	$-0.128000\pi$
$811$	42506.8	1.84046	0.920231	+	0.391375i	$0.128000\pi$
$812$	−101.225	−0.00437476
$813$	−10105.2	−0.435921
$814$	20725.6	0.892423
$815$	−58.2937	−0.00250545
$816$	−13302.1	−0.570670
$817$	−6860.72	−0.293790
$818$	9795.47	0.418693
$819$	0	0
$820$	−17.0077	−0.000724312 0
$821$	−20052.6	−0.852427	−0.426213	−	0.904623i	$-0.640153\pi$
$821$	−20052.6	−0.852427	−0.426213	+	0.904623i	$0.640153\pi$
$822$	23281.4	0.987872
$823$	−31625.5	−1.33948	−0.669742	−	0.742594i	$-0.733595\pi$
$823$	−31625.5	−1.33948	−0.669742	+	0.742594i	$0.733595\pi$
$824$	−35020.9	−1.48060
$825$	12057.5	0.508832
$826$	10317.4	0.434612
$827$	5440.97	0.228780	0.114390	−	0.993436i	$-0.463509\pi$
$827$	5440.97	0.228780	0.114390	+	0.993436i	$0.463509\pi$
$828$	48.1367	0.00202037
$829$	−24845.8	−1.04093	−0.520465	−	0.853883i	$-0.674241\pi$
$829$	−24845.8	−1.04093	−0.520465	+	0.853883i	$0.674241\pi$
$830$	9923.75	0.415010
$831$	−19523.2	−0.814986
$832$	0	0
$833$	−11488.0	−0.477834
$834$	20620.9	0.856165
$835$	−13368.9	−0.554071
$836$	−18.6260	−0.000770566 0
$837$	−6910.24	−0.285368
$838$	−12021.8	−0.495568
$839$	45887.8	1.88823	0.944113	−	0.329621i	$-0.106921\pi$
$839$	45887.8	1.88823	0.944113	+	0.329621i	$0.106921\pi$
$840$	3097.82	0.127244
$841$	7673.40	0.314625
$842$	−17588.7	−0.719887
$843$	10876.5	0.444375
$844$	55.8969	0.00227968
$845$	0	0
$846$	−1597.40	−0.0649168
$847$	988.665	0.0401073
$848$	9359.88	0.379033
$849$	−13906.2	−0.562144
$850$	−22274.7	−0.898842
$851$	26163.7	1.05391
$852$	72.7698	0.00292612
$853$	−33655.3	−1.35092	−0.675460	−	0.737397i	$-0.736055\pi$
$853$	−33655.3	−1.35092	−0.675460	+	0.737397i	$0.736055\pi$
$854$	−22700.2	−0.909583
$855$	−380.696	−0.0152275
$856$	−46000.9	−1.83677
$857$	41365.6	1.64880	0.824399	−	0.566009i	$-0.191513\pi$
$857$	41365.6	1.64880	0.824399	+	0.566009i	$0.191513\pi$
$858$	0	0
$859$	14866.0	0.590480	0.295240	−	0.955423i	$-0.404600\pi$
$859$	14866.0	0.590480	0.295240	+	0.955423i	$0.404600\pi$
$860$	−79.9404	−0.00316970
$861$	−4711.87	−0.186504
$862$	−7110.37	−0.280952
$863$	15469.1	0.610167	0.305083	−	0.952326i	$-0.401316\pi$
$863$	15469.1	0.610167	0.305083	+	0.952326i	$0.401316\pi$
$864$	51.7634	0.00203822
$865$	−6741.12	−0.264977
$866$	−3621.76	−0.142116
$867$	−184.553	−0.00722925
$868$	−144.684	−0.00565770
$869$	7367.05	0.287584
$870$	−5168.67	−0.201419
$871$	0	0
$872$	37319.4	1.44930
$873$	−16156.0	−0.626345
$874$	4416.69	0.170934
$875$	10849.3	0.419171
$876$	3.39533	0.000130956 0
$877$	38100.7	1.46701	0.733505	−	0.679684i	$-0.237883\pi$
$877$	38100.7	1.46701	0.733505	+	0.679684i	$0.237883\pi$
$878$	−47792.0	−1.83702
$879$	8721.21	0.334652
$880$	7698.06	0.294888
$881$	3879.84	0.148371	0.0741857	−	0.997244i	$-0.476364\pi$
$881$	3879.84	0.148371	0.0741857	+	0.997244i	$0.476364\pi$
$882$	−4187.38	−0.159860
$883$	13046.5	0.497225	0.248613	−	0.968603i	$-0.420025\pi$
$883$	13046.5	0.497225	0.248613	+	0.968603i	$0.420025\pi$
$884$	0	0
$885$	−2804.64	−0.106528
$886$	−6101.66	−0.231365
$887$	−32833.2	−1.24288	−0.621438	−	0.783463i	$-0.713451\pi$
$887$	−32833.2	−1.24288	−0.621438	+	0.783463i	$0.713451\pi$
$888$	14104.6	0.533017
$889$	−17688.2	−0.667315
$890$	−11831.6	−0.445613
$891$	−2871.69	−0.107974
$892$	50.9829	0.00191372
$893$	780.274	0.0292395
$894$	12843.0	0.480462
$895$	−9497.06	−0.354695
$896$	−19169.5	−0.714741
$897$	0	0
$898$	4720.39	0.175414
$899$	45827.6	1.70015
$900$	43.2238	0.00160088
$901$	−10241.1	−0.378668
$902$	−11771.3	−0.434525
$903$	−22147.0	−0.816173
$904$	14848.9	0.546314
$905$	−6434.74	−0.236352
$906$	−4939.23	−0.181120
$907$	21346.3	0.781468	0.390734	−	0.920504i	$-0.372221\pi$
$907$	21346.3	0.781468	0.390734	+	0.920504i	$0.372221\pi$
$908$	−184.760	−0.00675272
$909$	−6927.28	−0.252765
$910$	0	0
$911$	7792.71	0.283407	0.141704	−	0.989909i	$-0.454742\pi$
$911$	7792.71	0.283407	0.141704	+	0.989909i	$0.454742\pi$
$912$	2368.38	0.0859923
$913$	−36565.2	−1.32545
$914$	13124.0	0.474950
$915$	6170.69	0.222947
$916$	228.101	0.00822779
$917$	−12411.6	−0.446965
$918$	5305.09	0.190734
$919$	−31352.0	−1.12536	−0.562681	−	0.826674i	$-0.690230\pi$
$919$	−31352.0	−1.12536	−0.562681	+	0.826674i	$0.690230\pi$
$920$	9769.64	0.350104
$921$	−2801.85	−0.100243
$922$	−9760.61	−0.348643
$923$	0	0
$924$	−60.1262	−0.00214070
$925$	23493.4	0.835089
$926$	5939.85	0.210794
$927$	13893.0	0.492238
$928$	−343.286	−0.0121432
$929$	−12675.3	−0.447646	−0.223823	−	0.974630i	$-0.571854\pi$
$929$	−12675.3	−0.447646	−0.223823	+	0.974630i	$0.571854\pi$
$930$	−7387.71	−0.260487
$931$	2045.39	0.0720032
$932$	208.161	0.00731601
$933$	−10900.7	−0.382499
$934$	−3994.09	−0.139926
$935$	−8422.80	−0.294604
$936$	0	0
$937$	−12308.5	−0.429138	−0.214569	−	0.976709i	$-0.568835\pi$
$937$	−12308.5	−0.429138	−0.214569	+	0.976709i	$0.568835\pi$
$938$	−27894.1	−0.970976
$939$	−22521.8	−0.782717
$940$	9.09167	0.000315466 0
$941$	−20465.2	−0.708978	−0.354489	−	0.935060i	$-0.615345\pi$
$941$	−20465.2	−0.708978	−0.354489	+	0.935060i	$0.615345\pi$
$942$	220.821	0.00763774
$943$	−14859.9	−0.513155
$944$	17448.2	0.601578
$945$	−1228.92	−0.0423034
$946$	−55328.0	−1.90155
$947$	−11170.8	−0.383318	−0.191659	−	0.981462i	$-0.561387\pi$
$947$	−11170.8	−0.383318	−0.191659	+	0.981462i	$0.561387\pi$
$948$	26.4096	0.000904792 0
$949$	0	0
$950$	3965.91	0.135443
$951$	−3643.81	−0.124247
$952$	21086.5	0.717876
$953$	14109.4	0.479587	0.239794	−	0.970824i	$-0.422920\pi$
$953$	14109.4	0.479587	0.239794	+	0.970824i	$0.422920\pi$
$954$	−3732.87	−0.126684
$955$	8349.64	0.282919
$956$	40.8271	0.00138122
$957$	19044.6	0.643285
$958$	2780.29	0.0937653
$959$	−36710.2	−1.23612
$960$	5266.57	0.177060
$961$	35711.6	1.19874
$962$	0	0
$963$	18248.8	0.610652
$964$	65.4307	0.00218608
$965$	6568.19	0.219106
$966$	14257.4	0.474870
$967$	40785.8	1.35634	0.678171	−	0.734904i	$-0.262773\pi$
$967$	40785.8	1.35634	0.678171	+	0.734904i	$0.262773\pi$
$968$	1680.87	0.0558110
$969$	−2591.36	−0.0859096
$970$	−17272.4	−0.571734
$971$	4230.29	0.139811	0.0699055	−	0.997554i	$-0.477730\pi$
$971$	4230.29	0.139811	0.0699055	+	0.997554i	$0.477730\pi$
$972$	−10.2945	−0.000339708 0
$973$	−32515.1	−1.07131
$974$	−41537.4	−1.36647
$975$	0	0
$976$	−38389.0	−1.25902
$977$	59925.0	1.96230	0.981151	−	0.193242i	$-0.0619004\pi$
$977$	59925.0	1.96230	0.981151	+	0.193242i	$0.0619004\pi$
$978$	−144.633	−0.00472890
$979$	43594.9	1.42319
$980$	23.8327	0.000776844 0
$981$	−14804.8	−0.481834
$982$	−45984.6	−1.49433
$983$	23333.3	0.757087	0.378543	−	0.925584i	$-0.376425\pi$
$983$	23333.3	0.757087	0.378543	+	0.925584i	$0.376425\pi$
$984$	−8010.83	−0.259528
$985$	6207.64	0.200804
$986$	−35182.5	−1.13635
$987$	2518.79	0.0812298
$988$	0	0
$989$	−69845.2	−2.24565
$990$	−3070.11	−0.0985601
$991$	−32523.5	−1.04253	−0.521263	−	0.853396i	$-0.674539\pi$
$991$	−32523.5	−1.04253	−0.521263	+	0.853396i	$0.674539\pi$
$992$	−490.668	−0.0157044
$993$	−7329.61	−0.234238
$994$	21553.4	0.687759
$995$	−4345.37	−0.138450
$996$	−131.080	−0.00417010
$997$	−7114.35	−0.225992	−0.112996	−	0.993595i	$-0.536045\pi$
$997$	−7114.35	−0.225992	−0.112996	+	0.993595i	$0.536045\pi$
$998$	−9112.77	−0.289038
$999$	−5595.35	−0.177206

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.a.o.1.7	✓	9
3.2	odd	2		1521.4.a.bi.1.3		9
13.5	odd	4		507.4.b.k.337.6		18
13.8	odd	4		507.4.b.k.337.13		18
13.12	even	2		507.4.a.p.1.3	yes	9
39.38	odd	2		1521.4.a.bf.1.7		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.o.1.7	✓	9	1.1	even	1	trivial
507.4.a.p.1.3	yes	9	13.12	even	2
507.4.b.k.337.6		18	13.5	odd	4
507.4.b.k.337.13		18	13.8	odd	4
1521.4.a.bf.1.7		9	39.38	odd	2
1521.4.a.bi.1.3		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)

\(f(q)\)	\(=\)	\(q+2.82093 q^{2} +3.00000 q^{3} -0.0423641 q^{4} +3.41089 q^{5} +8.46278 q^{6} -13.3442 q^{7} -22.6869 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.82093 q^{2} +3.00000 q^{3} -0.0423641 q^{4} +3.41089 q^{5} +8.46278 q^{6} -13.3442 q^{7} -22.6869 q^{8} +9.00000 q^{9} +9.62187 q^{10} -35.4529 q^{11} -0.127092 q^{12} -37.6430 q^{14} +10.2327 q^{15} -63.6593 q^{16} +69.6526 q^{17} +25.3884 q^{18} -12.4014 q^{19} -0.144499 q^{20} -40.0325 q^{21} -100.010 q^{22} -126.251 q^{23} -68.0608 q^{24} -113.366 q^{25} +27.0000 q^{27} +0.565314 q^{28} -179.060 q^{29} +28.8656 q^{30} -255.935 q^{31} +1.91716 q^{32} -106.359 q^{33} +196.485 q^{34} -45.5155 q^{35} -0.381277 q^{36} -207.235 q^{37} -34.9833 q^{38} -77.3825 q^{40} +117.701 q^{41} -112.929 q^{42} +553.224 q^{43} +1.50193 q^{44} +30.6980 q^{45} -356.146 q^{46} -62.9185 q^{47} -190.978 q^{48} -164.933 q^{49} -319.797 q^{50} +208.958 q^{51} -147.031 q^{53} +76.1651 q^{54} -120.926 q^{55} +302.738 q^{56} -37.2041 q^{57} -505.115 q^{58} -274.087 q^{59} -0.433498 q^{60} +603.039 q^{61} -721.974 q^{62} -120.098 q^{63} +514.683 q^{64} -300.031 q^{66} +741.019 q^{67} -2.95077 q^{68} -378.754 q^{69} -128.396 q^{70} -572.574 q^{71} -204.182 q^{72} -26.7155 q^{73} -584.595 q^{74} -340.098 q^{75} +0.525372 q^{76} +473.090 q^{77} -207.798 q^{79} -217.135 q^{80} +81.0000 q^{81} +332.026 q^{82} +1031.37 q^{83} +1.69594 q^{84} +237.577 q^{85} +1560.60 q^{86} -537.179 q^{87} +804.318 q^{88} -1229.66 q^{89} +86.5968 q^{90} +5.34852 q^{92} -767.805 q^{93} -177.488 q^{94} -42.2996 q^{95} +5.75148 q^{96} -1795.12 q^{97} -465.264 q^{98} -319.076 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 6 q^{2} + 27 q^{3} + 44 q^{4} - 33 q^{5} - 18 q^{6} - 83 q^{7} - 87 q^{8} + 81 q^{9}+O(q^{10}) \) 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

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