LMFDB - Embedded newform 507.4.a.n.1.3

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} - x^{8} - 48x^{7} + 29x^{6} + 772x^{5} - 150x^{4} - 4745x^{3} - 966x^{2} + 9428x + 5144 $ x^9 - x^8 - 48x^7 + 29x^6 + 772x^5 - 150x^4 - 4745x^3 - 966x^2 + 9428*x + 5144
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			1.3
Root			$-2.37739$ 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)

$f(q)$	$=$	$q-3.37739 q^{2} -3.00000 q^{3} +3.40677 q^{4} +15.7127 q^{5} +10.1322 q^{6} +17.1681 q^{7} +15.5131 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-3.37739 q^{2} -3.00000 q^{3} +3.40677 q^{4} +15.7127 q^{5} +10.1322 q^{6} +17.1681 q^{7} +15.5131 q^{8} +9.00000 q^{9} -53.0679 q^{10} -52.8187 q^{11} -10.2203 q^{12} -57.9835 q^{14} -47.1380 q^{15} -79.6481 q^{16} -71.0654 q^{17} -30.3965 q^{18} -92.6916 q^{19} +53.5295 q^{20} -51.5044 q^{21} +178.390 q^{22} +190.712 q^{23} -46.5394 q^{24} +121.888 q^{25} -27.0000 q^{27} +58.4878 q^{28} -128.204 q^{29} +159.204 q^{30} -3.29674 q^{31} +144.898 q^{32} +158.456 q^{33} +240.016 q^{34} +269.757 q^{35} +30.6609 q^{36} -241.546 q^{37} +313.056 q^{38} +243.753 q^{40} +97.1824 q^{41} +173.950 q^{42} +376.151 q^{43} -179.941 q^{44} +141.414 q^{45} -644.109 q^{46} -577.354 q^{47} +238.944 q^{48} -48.2555 q^{49} -411.664 q^{50} +213.196 q^{51} -307.686 q^{53} +91.1896 q^{54} -829.924 q^{55} +266.331 q^{56} +278.075 q^{57} +432.995 q^{58} -349.914 q^{59} -160.588 q^{60} +127.467 q^{61} +11.1344 q^{62} +154.513 q^{63} +147.809 q^{64} -535.169 q^{66} +903.564 q^{67} -242.104 q^{68} -572.136 q^{69} -911.076 q^{70} -826.106 q^{71} +139.618 q^{72} +131.760 q^{73} +815.796 q^{74} -365.665 q^{75} -315.779 q^{76} -906.799 q^{77} -556.244 q^{79} -1251.48 q^{80} +81.0000 q^{81} -328.223 q^{82} +254.664 q^{83} -175.464 q^{84} -1116.63 q^{85} -1270.41 q^{86} +384.612 q^{87} -819.384 q^{88} -183.410 q^{89} -477.611 q^{90} +649.712 q^{92} +9.89023 q^{93} +1949.95 q^{94} -1456.43 q^{95} -434.693 q^{96} +780.498 q^{97} +162.978 q^{98} -475.369 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 8 q^{2} - 27 q^{3} + 32 q^{4} - 41 q^{5} + 24 q^{6} - q^{7} - 111 q^{8} + 81 q^{9}+O(q^{10}) $ 9 * q - 8 * q^2 - 27 * q^3 + 32 * q^4 - 41 * q^5 + 24 * q^6 - q^7 - 111 * q^8 + 81 * q^9	\( 9 q - 8 q^{2} - 27 q^{3} + 32 q^{4} - 41 q^{5} + 24 q^{6} - q^{7} - 111 q^{8} + 81 q^{9} + 198 q^{10} - 37 q^{11} - 96 q^{12} + 98 q^{14} + 123 q^{15} + 32 q^{16} - 134 q^{17} - 72 q^{18} + 72 q^{19} - 356 q^{20} + 3 q^{21} + 274 q^{22} + 226 q^{23} + 333 q^{24} + 612 q^{25} - 243 q^{27} - 132 q^{28} - 547 q^{29} - 594 q^{30} + 521 q^{31} - 721 q^{32} + 111 q^{33} + 100 q^{34} + 138 q^{35} + 288 q^{36} - 584 q^{37} - 416 q^{38} + 1342 q^{40} - 482 q^{41} - 294 q^{42} + 158 q^{43} - 1453 q^{44} - 369 q^{45} - 1537 q^{46} - 1500 q^{47} - 96 q^{48} + 642 q^{49} - 2777 q^{50} + 402 q^{51} + 1399 q^{53} + 216 q^{54} - 1408 q^{55} - 616 q^{56} - 216 q^{57} - 1455 q^{58} - 1541 q^{59} + 1068 q^{60} + 2092 q^{61} - 293 q^{62} - 9 q^{63} + 2481 q^{64} - 822 q^{66} - 252 q^{67} - 1579 q^{68} - 678 q^{69} - 2492 q^{70} - 2352 q^{71} - 999 q^{72} - 903 q^{73} + 1037 q^{74} - 1836 q^{75} + 485 q^{76} - 1686 q^{77} - 115 q^{79} - 5701 q^{80} + 729 q^{81} - 5147 q^{82} - 1207 q^{83} + 396 q^{84} - 4308 q^{85} - 5691 q^{86} + 1641 q^{87} - 484 q^{88} - 2336 q^{89} + 1782 q^{90} + 2087 q^{92} - 1563 q^{93} - 468 q^{94} - 222 q^{95} + 2163 q^{96} - 2155 q^{97} - 5593 q^{98} - 333 q^{99}+O(q^{100}) \) 9 * q - 8 * q^2 - 27 * q^3 + 32 * q^4 - 41 * q^5 + 24 * q^6 - q^7 - 111 * q^8 + 81 * q^9 + 198 * q^10 - 37 * q^11 - 96 * q^12 + 98 * q^14 + 123 * q^15 + 32 * q^16 - 134 * q^17 - 72 * q^18 + 72 * q^19 - 356 * q^20 + 3 * q^21 + 274 * q^22 + 226 * q^23 + 333 * q^24 + 612 * q^25 - 243 * q^27 - 132 * q^28 - 547 * q^29 - 594 * q^30 + 521 * q^31 - 721 * q^32 + 111 * q^33 + 100 * q^34 + 138 * q^35 + 288 * q^36 - 584 * q^37 - 416 * q^38 + 1342 * q^40 - 482 * q^41 - 294 * q^42 + 158 * q^43 - 1453 * q^44 - 369 * q^45 - 1537 * q^46 - 1500 * q^47 - 96 * q^48 + 642 * q^49 - 2777 * q^50 + 402 * q^51 + 1399 * q^53 + 216 * q^54 - 1408 * q^55 - 616 * q^56 - 216 * q^57 - 1455 * q^58 - 1541 * q^59 + 1068 * q^60 + 2092 * q^61 - 293 * q^62 - 9 * q^63 + 2481 * q^64 - 822 * q^66 - 252 * q^67 - 1579 * q^68 - 678 * q^69 - 2492 * q^70 - 2352 * q^71 - 999 * q^72 - 903 * q^73 + 1037 * q^74 - 1836 * q^75 + 485 * q^76 - 1686 * q^77 - 115 * q^79 - 5701 * q^80 + 729 * q^81 - 5147 * q^82 - 1207 * q^83 + 396 * q^84 - 4308 * q^85 - 5691 * q^86 + 1641 * q^87 - 484 * q^88 - 2336 * q^89 + 1782 * q^90 + 2087 * q^92 - 1563 * q^93 - 468 * q^94 - 222 * q^95 + 2163 * q^96 - 2155 * q^97 - 5593 * q^98 - 333 * 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 $

$2$	−3.37739	−1.19409	−0.597044	−	0.802208i	$-0.703658\pi$
$2$	−3.37739	−1.19409	−0.597044	+	0.802208i	$0.703658\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	3.40677	0.425846
$5$	15.7127	1.40538	0.702692	−	0.711494i	$-0.251981\pi$
$5$	15.7127	1.40538	0.702692	+	0.711494i	$0.251981\pi$
$6$	10.1322	0.689407
$7$	17.1681	0.926992	0.463496	−	0.886099i	$-0.346595\pi$
$7$	17.1681	0.926992	0.463496	+	0.886099i	$0.346595\pi$
$8$	15.5131	0.685590
$9$	9.00000	0.333333
$10$	−53.0679	−1.67815
$11$	−52.8187	−1.44777	−0.723884	−	0.689922i	$-0.757645\pi$
$11$	−52.8187	−1.44777	−0.723884	+	0.689922i	$0.757645\pi$
$12$	−10.2203	−0.245862
$13$	0	0
$13$	0	0
$14$	−57.9835	−1.10691
$15$	−47.1380	−0.811399
$16$	−79.6481	−1.24450
$17$	−71.0654	−1.01388	−0.506938	−	0.861982i	$-0.669223\pi$
$17$	−71.0654	−1.01388	−0.506938	+	0.861982i	$0.669223\pi$
$18$	−30.3965	−0.398029
$19$	−92.6916	−1.11921	−0.559603	−	0.828761i	$-0.689046\pi$
$19$	−92.6916	−1.11921	−0.559603	+	0.828761i	$0.689046\pi$
$20$	53.5295	0.598478
$21$	−51.5044	−0.535199
$22$	178.390	1.72876
$23$	190.712	1.72896	0.864482	−	0.502663i	$-0.167646\pi$
$23$	190.712	1.72896	0.864482	+	0.502663i	$0.167646\pi$
$24$	−46.5394	−0.395826
$25$	121.888	0.975106
$26$	0	0
$27$	−27.0000	−0.192450
$28$	58.4878	0.394756
$29$	−128.204	−0.820927	−0.410463	−	0.911877i	$-0.634633\pi$
$29$	−128.204	−0.820927	−0.410463	+	0.911877i	$0.634633\pi$
$30$	159.204	0.968882
$31$	−3.29674	−0.0191004	−0.00955020	−	0.999954i	$-0.503040\pi$
$31$	−3.29674	−0.0191004	−0.00955020	+	0.999954i	$0.503040\pi$
$32$	144.898	0.800454
$33$	158.456	0.835869
$34$	240.016	1.21066
$35$	269.757	1.30278
$36$	30.6609	0.141949
$37$	−241.546	−1.07324	−0.536621	−	0.843823i	$-0.680300\pi$
$37$	−241.546	−1.07324	−0.536621	+	0.843823i	$0.680300\pi$
$38$	313.056	1.33643
$39$	0	0
$40$	243.753	0.963518
$41$	97.1824	0.370179	0.185090	−	0.982722i	$-0.440742\pi$
$41$	97.1824	0.370179	0.185090	+	0.982722i	$0.440742\pi$
$42$	173.950	0.639075
$43$	376.151	1.33401	0.667006	−	0.745052i	$-0.267575\pi$
$43$	376.151	1.33401	0.667006	+	0.745052i	$0.267575\pi$
$44$	−179.941	−0.616527
$45$	141.414	0.468462
$46$	−644.109	−2.06454
$47$	−577.354	−1.79182	−0.895912	−	0.444231i	$-0.853477\pi$
$47$	−577.354	−1.79182	−0.895912	+	0.444231i	$0.853477\pi$
$48$	238.944	0.718513
$49$	−48.2555	−0.140686
$50$	−411.664	−1.16436
$51$	213.196	0.585362
$52$	0	0
$53$	−307.686	−0.797433	−0.398716	−	0.917074i	$-0.630544\pi$
$53$	−307.686	−0.797433	−0.398716	+	0.917074i	$0.630544\pi$
$54$	91.1896	0.229802
$55$	−829.924	−2.03467
$56$	266.331	0.635536
$57$	278.075	0.646174
$58$	432.995	0.980259
$59$	−349.914	−0.772117	−0.386059	−	0.922474i	$-0.626164\pi$
$59$	−349.914	−0.772117	−0.386059	+	0.922474i	$0.626164\pi$
$60$	−160.588	−0.345531
$61$	127.467	0.267548	0.133774	−	0.991012i	$-0.457290\pi$
$61$	127.467	0.267548	0.133774	+	0.991012i	$0.457290\pi$
$62$	11.1344	0.0228076
$63$	154.513	0.308997
$64$	147.809	0.288689
$65$	0	0
$66$	−535.169	−0.998102
$67$	903.564	1.64758	0.823791	−	0.566894i	$-0.191855\pi$
$67$	903.564	1.64758	0.823791	+	0.566894i	$0.191855\pi$
$68$	−242.104	−0.431755
$69$	−572.136	−0.998218
$70$	−911.076	−1.55563
$71$	−826.106	−1.38086	−0.690428	−	0.723401i	$-0.742578\pi$
$71$	−826.106	−1.38086	−0.690428	+	0.723401i	$0.742578\pi$
$72$	139.618	0.228530
$73$	131.760	0.211252	0.105626	−	0.994406i	$-0.466315\pi$
$73$	131.760	0.211252	0.105626	+	0.994406i	$0.466315\pi$
$74$	815.796	1.28155
$75$	−365.665	−0.562978
$76$	−315.779	−0.476610
$77$	−906.799	−1.34207
$78$	0	0
$79$	−556.244	−0.792182	−0.396091	−	0.918211i	$-0.629633\pi$
$79$	−556.244	−0.792182	−0.396091	+	0.918211i	$0.629633\pi$
$80$	−1251.48	−1.74900
$81$	81.0000	0.111111
$82$	−328.223	−0.442026
$83$	254.664	0.336783	0.168391	−	0.985720i	$-0.446143\pi$
$83$	254.664	0.336783	0.168391	+	0.985720i	$0.446143\pi$
$84$	−175.464	−0.227912
$85$	−1116.63	−1.42489
$86$	−1270.41	−1.59293
$87$	384.612	0.473962
$88$	−819.384	−0.992576
$89$	−183.410	−0.218443	−0.109222	−	0.994017i	$-0.534836\pi$
$89$	−183.410	−0.218443	−0.109222	+	0.994017i	$0.534836\pi$
$90$	−477.611	−0.559384
$91$	0	0
$92$	649.712	0.736273
$93$	9.89023	0.0110276
$94$	1949.95	2.13960
$95$	−1456.43	−1.57292
$96$	−434.693	−0.462142
$97$	780.498	0.816986	0.408493	−	0.912762i	$-0.366054\pi$
$97$	780.498	0.816986	0.408493	+	0.912762i	$0.366054\pi$
$98$	162.978	0.167992
$99$	−475.369	−0.482589
$100$	415.245	0.415245
$101$	−1807.67	−1.78089	−0.890444	−	0.455093i	$-0.849606\pi$
$101$	−1807.67	−1.78089	−0.890444	+	0.455093i	$0.849606\pi$
$102$	−720.047	−0.698974
$103$	1560.78	1.49308	0.746542	−	0.665338i	$-0.231712\pi$
$103$	1560.78	1.49308	0.746542	+	0.665338i	$0.231712\pi$
$104$	0	0
$105$	−809.272	−0.752160
$106$	1039.18	0.952205
$107$	322.266	0.291165	0.145582	−	0.989346i	$-0.453494\pi$
$107$	322.266	0.291165	0.145582	+	0.989346i	$0.453494\pi$
$108$	−91.9828	−0.0819541
$109$	−1607.72	−1.41277	−0.706386	−	0.707827i	$-0.749676\pi$
$109$	−1607.72	−1.41277	−0.706386	+	0.707827i	$0.749676\pi$
$110$	2802.98	2.42958
$111$	724.639	0.619637
$112$	−1367.41	−1.15364
$113$	−429.905	−0.357895	−0.178947	−	0.983859i	$-0.557269\pi$
$113$	−429.905	−0.357895	−0.178947	+	0.983859i	$0.557269\pi$
$114$	−939.167	−0.771589
$115$	2996.60	2.42986
$116$	−436.761	−0.349589
$117$	0	0
$118$	1181.80	0.921976
$119$	−1220.06	−0.939855
$120$	−731.259	−0.556287
$121$	1458.82	1.09603
$122$	−430.505	−0.319476
$123$	−291.547	−0.213723
$124$	−11.2312	−0.00813383
$125$	−48.8932	−0.0349851
$126$	−521.851	−0.368970
$127$	2371.36	1.65688	0.828442	−	0.560076i	$-0.189228\pi$
$127$	2371.36	1.65688	0.828442	+	0.560076i	$0.189228\pi$
$128$	−1658.39	−1.14517
$129$	−1128.45	−0.770193
$130$	0	0
$131$	169.200	0.112848	0.0564239	−	0.998407i	$-0.482030\pi$
$131$	169.200	0.112848	0.0564239	+	0.998407i	$0.482030\pi$
$132$	539.824	0.355952
$133$	−1591.34	−1.03749
$134$	−3051.69	−1.96736
$135$	−424.242	−0.270466
$136$	−1102.45	−0.695104
$137$	−2976.21	−1.85602	−0.928009	−	0.372558i	$-0.878481\pi$
$137$	−2976.21	−1.85602	−0.928009	+	0.372558i	$0.878481\pi$
$138$	1932.33	1.19196
$139$	−2555.15	−1.55917	−0.779585	−	0.626296i	$-0.784570\pi$
$139$	−2555.15	−1.55917	−0.779585	+	0.626296i	$0.784570\pi$
$140$	919.001	0.554784
$141$	1732.06	1.03451
$142$	2790.08	1.64886
$143$	0	0
$144$	−716.833	−0.414834
$145$	−2014.43	−1.15372
$146$	−445.006	−0.252253
$147$	144.766	0.0812254
$148$	−822.892	−0.457036
$149$	−424.529	−0.233415	−0.116707	−	0.993166i	$-0.537234\pi$
$149$	−424.529	−0.233415	−0.116707	+	0.993166i	$0.537234\pi$
$150$	1234.99	0.672245
$151$	−42.3232	−0.0228093	−0.0114047	−	0.999935i	$-0.503630\pi$
$151$	−42.3232	−0.0228093	−0.0114047	+	0.999935i	$0.503630\pi$
$152$	−1437.94	−0.767317
$153$	−639.589	−0.337959
$154$	3062.61	1.60255
$155$	−51.8007	−0.0268434
$156$	0	0
$157$	−990.187	−0.503347	−0.251674	−	0.967812i	$-0.580981\pi$
$157$	−990.187	−0.503347	−0.251674	+	0.967812i	$0.580981\pi$
$158$	1878.65	0.945935
$159$	923.058	0.460398
$160$	2276.73	1.12495
$161$	3274.17	1.60274
$162$	−273.569	−0.132676
$163$	−2645.82	−1.27139	−0.635695	−	0.771940i	$-0.719286\pi$
$163$	−2645.82	−1.27139	−0.635695	+	0.771940i	$0.719286\pi$
$164$	331.078	0.157639
$165$	2489.77	1.17472
$166$	−860.099	−0.402148
$167$	−295.428	−0.136892	−0.0684458	−	0.997655i	$-0.521804\pi$
$167$	−295.428	−0.136892	−0.0684458	+	0.997655i	$0.521804\pi$
$168$	−798.994	−0.366927
$169$	0	0
$170$	3771.29	1.70144
$171$	−834.225	−0.373069
$172$	1281.46	0.568084
$173$	1495.46	0.657213	0.328606	−	0.944467i	$-0.393421\pi$
$173$	1495.46	0.657213	0.328606	+	0.944467i	$0.393421\pi$
$174$	−1298.98	−0.565953
$175$	2092.59	0.903915
$176$	4206.91	1.80175
$177$	1049.74	0.445782
$178$	619.448	0.260840
$179$	785.097	0.327826	0.163913	−	0.986475i	$-0.447588\pi$
$179$	785.097	0.327826	0.163913	+	0.986475i	$0.447588\pi$
$180$	481.765	0.199493
$181$	−1287.01	−0.528522	−0.264261	−	0.964451i	$-0.585128\pi$
$181$	−1287.01	−0.528522	−0.264261	+	0.964451i	$0.585128\pi$
$182$	0	0
$183$	−382.400	−0.154469
$184$	2958.54	1.18536
$185$	−3795.34	−1.50832
$186$	−33.4032	−0.0131680
$187$	3753.59	1.46786
$188$	−1966.91	−0.763042
$189$	−463.539	−0.178400
$190$	4918.95	1.87820
$191$	466.452	0.176708	0.0883542	−	0.996089i	$-0.471839\pi$
$191$	466.452	0.176708	0.0883542	+	0.996089i	$0.471839\pi$
$192$	−443.426	−0.166675
$193$	−2636.99	−0.983495	−0.491747	−	0.870738i	$-0.663642\pi$
$193$	−2636.99	−0.983495	−0.491747	+	0.870738i	$0.663642\pi$
$194$	−2636.05	−0.975553
$195$	0	0
$196$	−164.395	−0.0599108
$197$	−3676.90	−1.32979	−0.664894	−	0.746937i	$-0.731523\pi$
$197$	−3676.90	−1.32979	−0.664894	+	0.746937i	$0.731523\pi$
$198$	1605.51	0.576254
$199$	220.557	0.0785670	0.0392835	−	0.999228i	$-0.487492\pi$
$199$	220.557	0.0785670	0.0392835	+	0.999228i	$0.487492\pi$
$200$	1890.87	0.668523
$201$	−2710.69	−0.951231
$202$	6105.20	2.12654
$203$	−2201.02	−0.760992
$204$	726.311	0.249274
$205$	1527.00	0.520244
$206$	−5271.35	−1.78287
$207$	1716.41	0.576322
$208$	0	0
$209$	4895.85	1.62035
$210$	2733.23	0.898146
$211$	1385.93	0.452186	0.226093	−	0.974106i	$-0.427405\pi$
$211$	1385.93	0.452186	0.226093	+	0.974106i	$0.427405\pi$
$212$	−1048.22	−0.339584
$213$	2478.32	0.797237
$214$	−1088.42	−0.347676
$215$	5910.35	1.87480
$216$	−418.855	−0.131942
$217$	−56.5989	−0.0177059
$218$	5429.91	1.68697
$219$	−395.281	−0.121966
$220$	−2827.36	−0.866457
$221$	0	0
$222$	−2447.39	−0.739901
$223$	111.847	0.0335867	0.0167933	−	0.999859i	$-0.494654\pi$
$223$	111.847	0.0335867	0.0167933	+	0.999859i	$0.494654\pi$
$224$	2487.62	0.742014
$225$	1096.99	0.325035
$226$	1451.96	0.427358
$227$	−4990.96	−1.45930	−0.729651	−	0.683819i	$-0.760318\pi$
$227$	−4990.96	−1.45930	−0.729651	+	0.683819i	$0.760318\pi$
$228$	947.337	0.275171
$229$	5787.56	1.67010	0.835050	−	0.550174i	$-0.185439\pi$
$229$	5787.56	1.67010	0.835050	+	0.550174i	$0.185439\pi$
$230$	−10120.7	−2.90147
$231$	2720.40	0.774844
$232$	−1988.85	−0.562819
$233$	−723.707	−0.203483	−0.101742	−	0.994811i	$-0.532441\pi$
$233$	−723.707	−0.203483	−0.101742	+	0.994811i	$0.532441\pi$
$234$	0	0
$235$	−9071.78	−2.51820
$236$	−1192.08	−0.328803
$237$	1668.73	0.457366
$238$	4120.62	1.12227
$239$	1239.74	0.335531	0.167766	−	0.985827i	$-0.446345\pi$
$239$	1239.74	0.335531	0.167766	+	0.985827i	$0.446345\pi$
$240$	3754.45	1.00979
$241$	755.393	0.201905	0.100953	−	0.994891i	$-0.467811\pi$
$241$	755.393	0.201905	0.100953	+	0.994891i	$0.467811\pi$
$242$	−4927.00	−1.30876
$243$	−243.000	−0.0641500
$244$	434.249	0.113934
$245$	−758.222	−0.197719
$246$	984.669	0.255204
$247$	0	0
$248$	−51.1428	−0.0130950
$249$	−763.991	−0.194442
$250$	165.131	0.0417753
$251$	2252.06	0.566330	0.283165	−	0.959071i	$-0.408616\pi$
$251$	2252.06	0.566330	0.283165	+	0.959071i	$0.408616\pi$
$252$	526.391	0.131585
$253$	−10073.2	−2.50314
$254$	−8009.01	−1.97846
$255$	3349.89	0.822659
$256$	4418.56	1.07875
$257$	3716.75	0.902118	0.451059	−	0.892494i	$-0.351046\pi$
$257$	3716.75	0.902118	0.451059	+	0.892494i	$0.351046\pi$
$258$	3811.23	0.919678
$259$	−4146.90	−0.994887
$260$	0	0
$261$	−1153.84	−0.273642
$262$	−571.454	−0.134750
$263$	−49.9521	−0.0117117	−0.00585585	−	0.999983i	$-0.501864\pi$
$263$	−49.9521	−0.0117117	−0.00585585	+	0.999983i	$0.501864\pi$
$264$	2458.15	0.573064
$265$	−4834.57	−1.12070
$266$	5374.58	1.23886
$267$	550.231	0.126118
$268$	3078.23	0.701616
$269$	3421.44	0.775497	0.387748	−	0.921765i	$-0.373253\pi$
$269$	3421.44	0.775497	0.387748	+	0.921765i	$0.373253\pi$
$270$	1432.83	0.322961
$271$	2958.89	0.663247	0.331624	−	0.943412i	$-0.392404\pi$
$271$	2958.89	0.663247	0.331624	+	0.943412i	$0.392404\pi$
$272$	5660.23	1.26177
$273$	0	0
$274$	10051.8	2.21625
$275$	−6437.99	−1.41173
$276$	−1949.13	−0.425087
$277$	−7461.08	−1.61839	−0.809193	−	0.587543i	$-0.800095\pi$
$277$	−7461.08	−1.61839	−0.809193	+	0.587543i	$0.800095\pi$
$278$	8629.73	1.86179
$279$	−29.6707	−0.00636680
$280$	4184.78	0.893173
$281$	4728.04	1.00374	0.501871	−	0.864943i	$-0.332645\pi$
$281$	4728.04	1.00374	0.501871	+	0.864943i	$0.332645\pi$
$282$	−5849.85	−1.23530
$283$	−2879.69	−0.604876	−0.302438	−	0.953169i	$-0.597801\pi$
$283$	−2879.69	−0.604876	−0.302438	+	0.953169i	$0.597801\pi$
$284$	−2814.35	−0.588032
$285$	4369.30	0.908123
$286$	0	0
$287$	1668.44	0.343153
$288$	1304.08	0.266818
$289$	137.298	0.0279458
$290$	6803.51	1.37764
$291$	−2341.50	−0.471687
$292$	448.877	0.0899607
$293$	−8200.94	−1.63517	−0.817583	−	0.575810i	$-0.804687\pi$
$293$	−8200.94	−1.63517	−0.817583	+	0.575810i	$0.804687\pi$
$294$	−488.933	−0.0969902
$295$	−5498.09	−1.08512
$296$	−3747.14	−0.735804
$297$	1426.11	0.278623
$298$	1433.80	0.278718
$299$	0	0
$300$	−1245.74	−0.239742
$301$	6457.81	1.23662
$302$	142.942	0.0272364
$303$	5423.00	1.02820
$304$	7382.71	1.39285
$305$	2002.84	0.376008
$306$	2160.14	0.403553
$307$	−2058.82	−0.382747	−0.191374	−	0.981517i	$-0.561294\pi$
$307$	−2058.82	−0.382747	−0.191374	+	0.981517i	$0.561294\pi$
$308$	−3089.25	−0.571515
$309$	−4682.33	−0.862033
$310$	174.951	0.0320534
$311$	1862.08	0.339514	0.169757	−	0.985486i	$-0.445702\pi$
$311$	1862.08	0.339514	0.169757	+	0.985486i	$0.445702\pi$
$312$	0	0
$313$	−255.896	−0.0462112	−0.0231056	−	0.999733i	$-0.507355\pi$
$313$	−255.896	−0.0462112	−0.0231056	+	0.999733i	$0.507355\pi$
$314$	3344.25	0.601041
$315$	2427.82	0.434260
$316$	−1895.00	−0.337348
$317$	−6265.68	−1.11014	−0.555072	−	0.831802i	$-0.687309\pi$
$317$	−6265.68	−1.11014	−0.555072	+	0.831802i	$0.687309\pi$
$318$	−3117.53	−0.549756
$319$	6771.57	1.18851
$320$	2322.47	0.405719
$321$	−966.798	−0.168104
$322$	−11058.1	−1.91381
$323$	6587.17	1.13474
$324$	275.948	0.0473162
$325$	0	0
$326$	8935.96	1.51815
$327$	4823.17	0.815664
$328$	1507.60	0.253791
$329$	−9912.09	−1.66101
$330$	−8408.93	−1.40272
$331$	−1071.93	−0.178002	−0.0890011	−	0.996032i	$-0.528367\pi$
$331$	−1071.93	−0.178002	−0.0890011	+	0.996032i	$0.528367\pi$
$332$	867.581	0.143418
$333$	−2173.92	−0.357747
$334$	997.775	0.163461
$335$	14197.4	2.31549
$336$	4102.22	0.666056
$337$	−5572.02	−0.900675	−0.450337	−	0.892858i	$-0.648696\pi$
$337$	−5572.02	−0.900675	−0.450337	+	0.892858i	$0.648696\pi$
$338$	0	0
$339$	1289.72	0.206631
$340$	−3804.10	−0.606783
$341$	174.130	0.0276530
$342$	2817.50	0.445477
$343$	−6717.12	−1.05741
$344$	5835.29	0.914586
$345$	−8989.79	−1.40288
$346$	−5050.76	−0.784770
$347$	−1903.44	−0.294472	−0.147236	−	0.989101i	$-0.547038\pi$
$347$	−1903.44	−0.294472	−0.147236	+	0.989101i	$0.547038\pi$
$348$	1310.28	0.201835
$349$	−1097.17	−0.168282	−0.0841409	−	0.996454i	$-0.526815\pi$
$349$	−1097.17	−0.168282	−0.0841409	+	0.996454i	$0.526815\pi$
$350$	−7067.51	−1.07935
$351$	0	0
$352$	−7653.31	−1.15887
$353$	−5420.90	−0.817352	−0.408676	−	0.912679i	$-0.634009\pi$
$353$	−5420.90	−0.817352	−0.408676	+	0.912679i	$0.634009\pi$
$354$	−3545.39	−0.532303
$355$	−12980.3	−1.94063
$356$	−624.836	−0.0930232
$357$	3660.18	0.542626
$358$	−2651.58	−0.391453
$359$	−3885.40	−0.571208	−0.285604	−	0.958348i	$-0.592194\pi$
$359$	−3885.40	−0.571208	−0.285604	+	0.958348i	$0.592194\pi$
$360$	2193.78	0.321173
$361$	1732.74	0.252622
$362$	4346.72	0.631102
$363$	−4376.46	−0.632795
$364$	0	0
$365$	2070.31	0.296890
$366$	1291.51	0.184450
$367$	−3888.26	−0.553039	−0.276520	−	0.961008i	$-0.589181\pi$
$367$	−3888.26	−0.553039	−0.276520	+	0.961008i	$0.589181\pi$
$368$	−15189.8	−2.15170
$369$	874.642	0.123393
$370$	12818.3	1.80106
$371$	−5282.39	−0.739213
$372$	33.6937	0.00469607
$373$	9960.88	1.38272	0.691361	−	0.722510i	$-0.257012\pi$
$373$	9960.88	1.38272	0.691361	+	0.722510i	$0.257012\pi$
$374$	−12677.3	−1.75275
$375$	146.680	0.0201987
$376$	−8956.57	−1.22846
$377$	0	0
$378$	1565.55	0.213025
$379$	−5288.25	−0.716726	−0.358363	−	0.933582i	$-0.616665\pi$
$379$	−5288.25	−0.716726	−0.358363	+	0.933582i	$0.616665\pi$
$380$	−4961.73	−0.669820
$381$	−7114.08	−0.956602
$382$	−1575.39	−0.211005
$383$	−688.944	−0.0919149	−0.0459575	−	0.998943i	$-0.514634\pi$
$383$	−688.944	−0.0919149	−0.0459575	+	0.998943i	$0.514634\pi$
$384$	4975.17	0.661166
$385$	−14248.2	−1.88612
$386$	8906.14	1.17438
$387$	3385.36	0.444671
$388$	2658.98	0.347910
$389$	2102.57	0.274048	0.137024	−	0.990568i	$-0.456246\pi$
$389$	2102.57	0.274048	0.137024	+	0.990568i	$0.456246\pi$
$390$	0	0
$391$	−13553.0	−1.75296
$392$	−748.593	−0.0964533
$393$	−507.599	−0.0651527
$394$	12418.3	1.58788
$395$	−8740.09	−1.11332
$396$	−1619.47	−0.205509
$397$	11254.8	1.42283	0.711413	−	0.702774i	$-0.248056\pi$
$397$	11254.8	1.42283	0.711413	+	0.702774i	$0.248056\pi$
$398$	−744.906	−0.0938159
$399$	4774.02	0.598998
$400$	−9708.17	−1.21352
$401$	3523.22	0.438756	0.219378	−	0.975640i	$-0.429597\pi$
$401$	3523.22	0.438756	0.219378	+	0.975640i	$0.429597\pi$
$402$	9155.07	1.13585
$403$	0	0
$404$	−6158.31	−0.758384
$405$	1272.73	0.156154
$406$	7433.71	0.908692
$407$	12758.2	1.55381
$408$	3307.34	0.401318
$409$	6366.65	0.769708	0.384854	−	0.922978i	$-0.374252\pi$
$409$	6366.65	0.769708	0.384854	+	0.922978i	$0.374252\pi$
$410$	−5157.26	−0.621217
$411$	8928.62	1.07157
$412$	5317.20	0.635824
$413$	−6007.37	−0.715746
$414$	−5796.98	−0.688179
$415$	4001.45	0.473310
$416$	0	0
$417$	7665.44	0.900187
$418$	−16535.2	−1.93484
$419$	8919.77	1.04000	0.519999	−	0.854167i	$-0.325932\pi$
$419$	8919.77	1.04000	0.519999	+	0.854167i	$0.325932\pi$
$420$	−2757.00	−0.320305
$421$	1090.71	0.126266	0.0631328	−	0.998005i	$-0.479891\pi$
$421$	1090.71	0.126266	0.0631328	+	0.998005i	$0.479891\pi$
$422$	−4680.82	−0.539949
$423$	−5196.19	−0.597275
$424$	−4773.17	−0.546712
$425$	−8662.05	−0.988638
$426$	−8370.25	−0.951971
$427$	2188.36	0.248015
$428$	1097.89	0.123991
$429$	0	0
$430$	−19961.5	−2.23868
$431$	3760.43	0.420263	0.210132	−	0.977673i	$-0.432611\pi$
$431$	3760.43	0.420263	0.210132	+	0.977673i	$0.432611\pi$
$432$	2150.50	0.239504
$433$	−4085.06	−0.453384	−0.226692	−	0.973967i	$-0.572791\pi$
$433$	−4085.06	−0.453384	−0.226692	+	0.973967i	$0.572791\pi$
$434$	191.157	0.0211424
$435$	6043.28	0.666099
$436$	−5477.15	−0.601623
$437$	−17677.4	−1.93507
$438$	1335.02	0.145638
$439$	9385.35	1.02036	0.510180	−	0.860068i	$-0.329579\pi$
$439$	9385.35	1.02036	0.510180	+	0.860068i	$0.329579\pi$
$440$	−12874.7	−1.39495
$441$	−434.299	−0.0468955
$442$	0	0
$443$	5755.41	0.617264	0.308632	−	0.951182i	$-0.400129\pi$
$443$	5755.41	0.617264	0.308632	+	0.951182i	$0.400129\pi$
$444$	2468.68	0.263870
$445$	−2881.87	−0.306997
$446$	−377.751	−0.0401055
$447$	1273.59	0.134762
$448$	2537.60	0.267612
$449$	−1104.67	−0.116108	−0.0580541	−	0.998313i	$-0.518490\pi$
$449$	−1104.67	−0.116108	−0.0580541	+	0.998313i	$0.518490\pi$
$450$	−3704.98	−0.388121
$451$	−5133.05	−0.535934
$452$	−1464.59	−0.152408
$453$	126.969	0.0131690
$454$	16856.4	1.74254
$455$	0	0
$456$	4313.81	0.443011
$457$	13548.5	1.38681	0.693405	−	0.720548i	$-0.256110\pi$
$457$	13548.5	1.38681	0.693405	+	0.720548i	$0.256110\pi$
$458$	−19546.9	−1.99425
$459$	1918.77	0.195121
$460$	10208.7	1.03475
$461$	−2996.03	−0.302688	−0.151344	−	0.988481i	$-0.548360\pi$
$461$	−2996.03	−0.302688	−0.151344	+	0.988481i	$0.548360\pi$
$462$	−9187.84	−0.925232
$463$	2026.10	0.203371	0.101686	−	0.994817i	$-0.467576\pi$
$463$	2026.10	0.203371	0.101686	+	0.994817i	$0.467576\pi$
$464$	10211.2	1.02164
$465$	155.402	0.0154981
$466$	2444.24	0.242977
$467$	−3284.20	−0.325428	−0.162714	−	0.986673i	$-0.552025\pi$
$467$	−3284.20	−0.325428	−0.162714	+	0.986673i	$0.552025\pi$
$468$	0	0
$469$	15512.5	1.52729
$470$	30638.9	3.00696
$471$	2970.56	0.290608
$472$	−5428.26	−0.529356
$473$	−19867.8	−1.93134
$474$	−5635.96	−0.546136
$475$	−11298.0	−1.09134
$476$	−4156.46	−0.400234
$477$	−2769.17	−0.265811
$478$	−4187.08	−0.400654
$479$	−13208.5	−1.25994	−0.629969	−	0.776621i	$-0.716932\pi$
$479$	−13208.5	−1.25994	−0.629969	+	0.776621i	$0.716932\pi$
$480$	−6830.19	−0.649488
$481$	0	0
$482$	−2551.26	−0.241092
$483$	−9822.50	−0.925340
$484$	4969.86	0.466741
$485$	12263.7	1.14818
$486$	820.706	0.0766008
$487$	−8974.09	−0.835020	−0.417510	−	0.908672i	$-0.637097\pi$
$487$	−8974.09	−0.835020	−0.417510	+	0.908672i	$0.637097\pi$
$488$	1977.41	0.183428
$489$	7937.46	0.734037
$490$	2560.81	0.236093
$491$	−12889.8	−1.18474	−0.592370	−	0.805666i	$-0.701808\pi$
$491$	−12889.8	−1.18474	−0.592370	+	0.805666i	$0.701808\pi$
$492$	−993.234	−0.0910131
$493$	9110.87	0.832319
$494$	0	0
$495$	−7469.32	−0.678224
$496$	262.579	0.0237705
$497$	−14182.7	−1.28004
$498$	2580.30	0.232181
$499$	−7371.92	−0.661348	−0.330674	−	0.943745i	$-0.607276\pi$
$499$	−7371.92	−0.661348	−0.330674	+	0.943745i	$0.607276\pi$
$500$	−166.568	−0.0148983
$501$	886.284	0.0790344
$502$	−7606.09	−0.676248
$503$	19788.2	1.75410	0.877049	−	0.480400i	$-0.159509\pi$
$503$	19788.2	1.75410	0.877049	+	0.480400i	$0.159509\pi$
$504$	2396.98	0.211845
$505$	−28403.3	−2.50283
$506$	34021.0	2.98897
$507$	0	0
$508$	8078.67	0.705577
$509$	−11682.4	−1.01732	−0.508658	−	0.860968i	$-0.669858\pi$
$509$	−11682.4	−1.01732	−0.508658	+	0.860968i	$0.669858\pi$
$510$	−11313.9	−0.982327
$511$	2262.08	0.195829
$512$	−1656.08	−0.142948
$513$	2502.67	0.215391
$514$	−12552.9	−1.07721
$515$	24524.0	2.09836
$516$	−3844.38	−0.327984
$517$	30495.1	2.59415
$518$	14005.7	1.18798
$519$	−4486.38	−0.379442
$520$	0	0
$521$	−19025.1	−1.59982	−0.799908	−	0.600122i	$-0.795119\pi$
$521$	−19025.1	−1.59982	−0.799908	+	0.600122i	$0.795119\pi$
$522$	3896.95	0.326753
$523$	5345.69	0.446942	0.223471	−	0.974711i	$-0.428261\pi$
$523$	5345.69	0.446942	0.223471	+	0.974711i	$0.428261\pi$
$524$	576.425	0.0480558
$525$	−6277.78	−0.521876
$526$	168.708	0.0139848
$527$	234.284	0.0193655
$528$	−12620.7	−1.04024
$529$	24204.0	1.98932
$530$	16328.2	1.33821
$531$	−3149.23	−0.257372
$532$	−5421.33	−0.441813
$533$	0	0
$534$	−1858.34	−0.150596
$535$	5063.66	0.409199
$536$	14017.1	1.12957
$537$	−2355.29	−0.189271
$538$	−11555.5	−0.926012
$539$	2548.79	0.203681
$540$	−1445.30	−0.115177
$541$	3058.01	0.243021	0.121510	−	0.992590i	$-0.461226\pi$
$541$	3058.01	0.243021	0.121510	+	0.992590i	$0.461226\pi$
$542$	−9993.34	−0.791975
$543$	3861.02	0.305142
$544$	−10297.2	−0.811561
$545$	−25261.7	−1.98549
$546$	0	0
$547$	17921.1	1.40082	0.700410	−	0.713740i	$-0.253000\pi$
$547$	17921.1	1.40082	0.700410	+	0.713740i	$0.253000\pi$
$548$	−10139.2	−0.790378
$549$	1147.20	0.0891827
$550$	21743.6	1.68573
$551$	11883.4	0.918786
$552$	−8875.62	−0.684369
$553$	−9549.67	−0.734346
$554$	25199.0	1.93250
$555$	11386.0	0.870828
$556$	−8704.79	−0.663967
$557$	−9106.04	−0.692702	−0.346351	−	0.938105i	$-0.612579\pi$
$557$	−9106.04	−0.692702	−0.346351	+	0.938105i	$0.612579\pi$
$558$	100.209	0.00760252
$559$	0	0
$560$	−21485.6	−1.62131
$561$	−11260.8	−0.847468
$562$	−15968.4	−1.19856
$563$	11327.9	0.847985	0.423992	−	0.905666i	$-0.360628\pi$
$563$	11327.9	0.847985	0.423992	+	0.905666i	$0.360628\pi$
$564$	5900.74	0.440542
$565$	−6754.97	−0.502980
$566$	9725.84	0.722275
$567$	1390.62	0.102999
$568$	−12815.5	−0.946701
$569$	1963.27	0.144647	0.0723237	−	0.997381i	$-0.476959\pi$
$569$	1963.27	0.144647	0.0723237	+	0.997381i	$0.476959\pi$
$570$	−14756.8	−1.08438
$571$	−2270.35	−0.166394	−0.0831971	−	0.996533i	$-0.526513\pi$
$571$	−2270.35	−0.166394	−0.0831971	+	0.996533i	$0.526513\pi$
$572$	0	0
$573$	−1399.36	−0.102023
$574$	−5634.97	−0.409755
$575$	23245.6	1.68592
$576$	1330.28	0.0962297
$577$	13949.6	1.00646	0.503231	−	0.864152i	$-0.332145\pi$
$577$	13949.6	1.00646	0.503231	+	0.864152i	$0.332145\pi$
$578$	−463.708	−0.0333697
$579$	7910.96	0.567821
$580$	−6862.69	−0.491306
$581$	4372.10	0.312195
$582$	7908.14	0.563236
$583$	16251.6	1.15450
$584$	2044.02	0.144832
$585$	0	0
$586$	27697.8	1.95253
$587$	26731.4	1.87960	0.939799	−	0.341727i	$-0.111012\pi$
$587$	26731.4	1.87960	0.939799	+	0.341727i	$0.111012\pi$
$588$	493.186	0.0345895
$589$	305.580	0.0213773
$590$	18569.2	1.29573
$591$	11030.7	0.767754
$592$	19238.7	1.33565
$593$	−10508.6	−0.727715	−0.363858	−	0.931455i	$-0.618541\pi$
$593$	−10508.6	−0.727715	−0.363858	+	0.931455i	$0.618541\pi$
$594$	−4816.52	−0.332701
$595$	−19170.4	−1.32086
$596$	−1446.27	−0.0993987
$597$	−661.670	−0.0453607
$598$	0	0
$599$	−1935.87	−0.132049	−0.0660246	−	0.997818i	$-0.521032\pi$
$599$	−1935.87	−0.132049	−0.0660246	+	0.997818i	$0.521032\pi$
$600$	−5672.61	−0.385972
$601$	−16155.5	−1.09650	−0.548249	−	0.836315i	$-0.684705\pi$
$601$	−16155.5	−1.09650	−0.548249	+	0.836315i	$0.684705\pi$
$602$	−21810.6	−1.47663
$603$	8132.08	0.549194
$604$	−144.185	−0.00971327
$605$	22922.0	1.54035
$606$	−18315.6	−1.22776
$607$	−1698.75	−0.113592	−0.0567959	−	0.998386i	$-0.518088\pi$
$607$	−1698.75	−0.113592	−0.0567959	+	0.998386i	$0.518088\pi$
$608$	−13430.8	−0.895873
$609$	6603.07	0.439359
$610$	−6764.38	−0.448987
$611$	0	0
$612$	−2178.93	−0.143918
$613$	8627.42	0.568448	0.284224	−	0.958758i	$-0.408264\pi$
$613$	8627.42	0.568448	0.284224	+	0.958758i	$0.408264\pi$
$614$	6953.46	0.457034
$615$	−4580.99	−0.300363
$616$	−14067.3	−0.920109
$617$	21415.4	1.39733	0.698664	−	0.715450i	$-0.253778\pi$
$617$	21415.4	1.39733	0.698664	+	0.715450i	$0.253778\pi$
$618$	15814.0	1.02934
$619$	17396.0	1.12957	0.564784	−	0.825238i	$-0.308959\pi$
$619$	17396.0	1.12957	0.564784	+	0.825238i	$0.308959\pi$
$620$	−176.473	−0.0114312
$621$	−5149.22	−0.332739
$622$	−6288.96	−0.405409
$623$	−3148.81	−0.202495
$624$	0	0
$625$	−16004.3	−1.02427
$626$	864.261	0.0551802
$627$	−14687.6	−0.935510
$628$	−3373.34	−0.214349
$629$	17165.6	1.08814
$630$	−8199.68	−0.518545
$631$	13059.2	0.823898	0.411949	−	0.911207i	$-0.364848\pi$
$631$	13059.2	0.823898	0.411949	+	0.911207i	$0.364848\pi$
$632$	−8629.09	−0.543112
$633$	−4157.78	−0.261070
$634$	21161.6	1.32561
$635$	37260.4	2.32856
$636$	3144.65	0.196059
$637$	0	0
$638$	−22870.2	−1.41919
$639$	−7434.95	−0.460285
$640$	−26057.7	−1.60941
$641$	22397.7	1.38012	0.690060	−	0.723752i	$-0.257584\pi$
$641$	22397.7	1.38012	0.690060	+	0.723752i	$0.257584\pi$
$642$	3265.26	0.200731
$643$	−5878.08	−0.360511	−0.180256	−	0.983620i	$-0.557693\pi$
$643$	−5878.08	−0.360511	−0.180256	+	0.983620i	$0.557693\pi$
$644$	11154.3	0.682519
$645$	−17731.0	−1.08242
$646$	−22247.5	−1.35498
$647$	−20768.3	−1.26196	−0.630979	−	0.775800i	$-0.717347\pi$
$647$	−20768.3	−1.26196	−0.630979	+	0.775800i	$0.717347\pi$
$648$	1256.56	0.0761767
$649$	18482.0	1.11785
$650$	0	0
$651$	169.797	0.0102225
$652$	−9013.69	−0.541416
$653$	15516.4	0.929865	0.464933	−	0.885346i	$-0.346079\pi$
$653$	15516.4	0.929865	0.464933	+	0.885346i	$0.346079\pi$
$654$	−16289.7	−0.973975
$655$	2658.58	0.158594
$656$	−7740.39	−0.460688
$657$	1185.84	0.0704172
$658$	33477.0	1.98339
$659$	24025.0	1.42015	0.710077	−	0.704124i	$-0.248660\pi$
$659$	24025.0	1.42015	0.710077	+	0.704124i	$0.248660\pi$
$660$	8482.08	0.500249
$661$	10778.8	0.634262	0.317131	−	0.948382i	$-0.397281\pi$
$661$	10778.8	0.634262	0.317131	+	0.948382i	$0.397281\pi$
$662$	3620.33	0.212550
$663$	0	0
$664$	3950.63	0.230895
$665$	−25004.2	−1.45808
$666$	7342.17	0.427182
$667$	−24450.0	−1.41935
$668$	−1006.45	−0.0582948
$669$	−335.541	−0.0193913
$670$	−47950.2	−2.76489
$671$	−6732.63	−0.387348
$672$	−7462.86	−0.428402
$673$	22345.9	1.27990	0.639950	−	0.768417i	$-0.278955\pi$
$673$	22345.9	1.27990	0.639950	+	0.768417i	$0.278955\pi$
$674$	18818.9	1.07549
$675$	−3290.98	−0.187659
$676$	0	0
$677$	12326.7	0.699782	0.349891	−	0.936790i	$-0.386219\pi$
$677$	12326.7	0.699782	0.349891	+	0.936790i	$0.386219\pi$
$678$	−4355.88	−0.246735
$679$	13399.7	0.757339
$680$	−17322.4	−0.976888
$681$	14972.9	0.842529
$682$	−588.104	−0.0330201
$683$	27452.5	1.53798	0.768989	−	0.639262i	$-0.220760\pi$
$683$	27452.5	1.53798	0.768989	+	0.639262i	$0.220760\pi$
$684$	−2842.01	−0.158870
$685$	−46764.2	−2.60842
$686$	22686.3	1.26264
$687$	−17362.7	−0.964232
$688$	−29959.7	−1.66018
$689$	0	0
$690$	30362.0	1.67516
$691$	18663.6	1.02749	0.513746	−	0.857942i	$-0.328257\pi$
$691$	18663.6	1.02749	0.513746	+	0.857942i	$0.328257\pi$
$692$	5094.69	0.279871
$693$	−8161.19	−0.447356
$694$	6428.65	0.351626
$695$	−40148.2	−2.19123
$696$	5966.54	0.324944
$697$	−6906.31	−0.375316
$698$	3705.59	0.200943
$699$	2171.12	0.117481
$700$	7128.98	0.384929
$701$	−4131.00	−0.222576	−0.111288	−	0.993788i	$-0.535498\pi$
$701$	−4131.00	−0.222576	−0.111288	+	0.993788i	$0.535498\pi$
$702$	0	0
$703$	22389.3	1.20118
$704$	−7807.07	−0.417955
$705$	27215.3	1.45389
$706$	18308.5	0.975990
$707$	−31034.3	−1.65087
$708$	3576.23	0.189835
$709$	−33578.1	−1.77863	−0.889317	−	0.457291i	$-0.848820\pi$
$709$	−33578.1	−1.77863	−0.889317	+	0.457291i	$0.848820\pi$
$710$	43839.7	2.31729
$711$	−5006.20	−0.264061
$712$	−2845.27	−0.149762
$713$	−628.728	−0.0330239
$714$	−12361.9	−0.647943
$715$	0	0
$716$	2674.64	0.139603
$717$	−3719.21	−0.193719
$718$	13122.5	0.682073
$719$	22527.2	1.16846	0.584229	−	0.811589i	$-0.301397\pi$
$719$	22527.2	1.16846	0.584229	+	0.811589i	$0.301397\pi$
$720$	−11263.4	−0.583001
$721$	26795.6	1.38408
$722$	−5852.12	−0.301653
$723$	−2266.18	−0.116570
$724$	−4384.53	−0.225069
$725$	−15626.6	−0.800491
$726$	14781.0	0.755613
$727$	241.718	0.0123312	0.00616562	−	0.999981i	$-0.498037\pi$
$727$	241.718	0.0123312	0.00616562	+	0.999981i	$0.498037\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	−6992.24	−0.354513
$731$	−26731.4	−1.35252
$732$	−1302.75	−0.0657800
$733$	17235.4	0.868490	0.434245	−	0.900795i	$-0.357015\pi$
$733$	17235.4	0.868490	0.434245	+	0.900795i	$0.357015\pi$
$734$	13132.2	0.660377
$735$	2274.67	0.114153
$736$	27633.7	1.38396
$737$	−47725.1	−2.38532
$738$	−2954.01	−0.147342
$739$	34304.3	1.70758	0.853792	−	0.520615i	$-0.174297\pi$
$739$	34304.3	1.70758	0.853792	+	0.520615i	$0.174297\pi$
$740$	−12929.8	−0.642312
$741$	0	0
$742$	17840.7	0.882686
$743$	−2114.59	−0.104410	−0.0522052	−	0.998636i	$-0.516625\pi$
$743$	−2114.59	−0.104410	−0.0522052	+	0.998636i	$0.516625\pi$
$744$	153.428	0.00756043
$745$	−6670.49	−0.328037
$746$	−33641.8	−1.65109
$747$	2291.97	0.112261
$748$	12787.6	0.625082
$749$	5532.70	0.269907
$750$	−495.394	−0.0241190
$751$	−40958.7	−1.99015	−0.995077	−	0.0991024i	$-0.968403\pi$
$751$	−40958.7	−1.99015	−0.995077	+	0.0991024i	$0.968403\pi$
$752$	45985.1	2.22993
$753$	−6756.18	−0.326971
$754$	0	0
$755$	−665.010	−0.0320559
$756$	−1579.17	−0.0759708
$757$	10530.8	0.505613	0.252807	−	0.967517i	$-0.418646\pi$
$757$	10530.8	0.505613	0.252807	+	0.967517i	$0.418646\pi$
$758$	17860.5	0.855834
$759$	30219.5	1.44519
$760$	−22593.9	−1.07838
$761$	787.955	0.0375340	0.0187670	−	0.999824i	$-0.494026\pi$
$761$	787.955	0.0375340	0.0187670	+	0.999824i	$0.494026\pi$
$762$	24027.0	1.14227
$763$	−27601.6	−1.30963
$764$	1589.10	0.0752506
$765$	−10049.7	−0.474962
$766$	2326.83	0.109755
$767$	0	0
$768$	−13255.7	−0.622816
$769$	−5227.62	−0.245140	−0.122570	−	0.992460i	$-0.539114\pi$
$769$	−5227.62	−0.245140	−0.122570	+	0.992460i	$0.539114\pi$
$770$	48121.9	2.25220
$771$	−11150.2	−0.520838
$772$	−8983.61	−0.418818
$773$	−4516.79	−0.210165	−0.105082	−	0.994464i	$-0.533511\pi$
$773$	−4516.79	−0.210165	−0.105082	+	0.994464i	$0.533511\pi$
$774$	−11433.7	−0.530976
$775$	−401.834	−0.0186249
$776$	12108.0	0.560117
$777$	12440.7	0.574398
$778$	−7101.21	−0.327237
$779$	−9008.00	−0.414307
$780$	0	0
$781$	43633.9	1.99916
$782$	45773.9	2.09318
$783$	3461.51	0.157987
$784$	3843.45	0.175084
$785$	−15558.5	−0.707397
$786$	1714.36	0.0777980
$787$	−6800.09	−0.308001	−0.154001	−	0.988071i	$-0.549216\pi$
$787$	−6800.09	−0.308001	−0.154001	+	0.988071i	$0.549216\pi$
$788$	−12526.4	−0.566285
$789$	149.856	0.00676176
$790$	29518.7	1.32940
$791$	−7380.67	−0.331765
$792$	−7374.46	−0.330859
$793$	0	0
$794$	−38011.8	−1.69898
$795$	14503.7	0.647036
$796$	751.385	0.0334575
$797$	−40553.3	−1.80235	−0.901174	−	0.433457i	$-0.857293\pi$
$797$	−40553.3	−1.80235	−0.901174	+	0.433457i	$0.857293\pi$
$798$	−16123.7	−0.715256
$799$	41029.9	1.81669
$800$	17661.3	0.780528
$801$	−1650.69	−0.0728144
$802$	−11899.3	−0.523913
$803$	−6959.41	−0.305844
$804$	−9234.70	−0.405078
$805$	51445.9	2.25246
$806$	0	0
$807$	−10264.3	−0.447733
$808$	−28042.6	−1.22096
$809$	−6517.83	−0.283257	−0.141628	−	0.989920i	$-0.545234\pi$
$809$	−6517.83	−0.283257	−0.141628	+	0.989920i	$0.545234\pi$
$810$	−4298.50	−0.186461
$811$	−2898.99	−0.125521	−0.0627603	−	0.998029i	$-0.519990\pi$
$811$	−2898.99	−0.125521	−0.0627603	+	0.998029i	$0.519990\pi$
$812$	−7498.37	−0.324066
$813$	−8876.68	−0.382926
$814$	−43089.3	−1.85538
$815$	−41572.9	−1.78679
$816$	−16980.7	−0.728484
$817$	−34866.1	−1.49303
$818$	−21502.7	−0.919099
$819$	0	0
$820$	5202.12	0.221544
$821$	−716.621	−0.0304632	−0.0152316	−	0.999884i	$-0.504849\pi$
$821$	−716.621	−0.0304632	−0.0152316	+	0.999884i	$0.504849\pi$
$822$	−30155.4	−1.27955
$823$	15510.0	0.656920	0.328460	−	0.944518i	$-0.393470\pi$
$823$	15510.0	0.656920	0.328460	+	0.944518i	$0.393470\pi$
$824$	24212.5	1.02364
$825$	19314.0	0.815062
$826$	20289.2	0.854664
$827$	24063.2	1.01180	0.505901	−	0.862591i	$-0.331160\pi$
$827$	24063.2	1.01180	0.505901	+	0.862591i	$0.331160\pi$
$828$	5847.40	0.245424
$829$	10037.8	0.420540	0.210270	−	0.977643i	$-0.432566\pi$
$829$	10037.8	0.420540	0.210270	+	0.977643i	$0.432566\pi$
$830$	−13514.5	−0.565173
$831$	22383.2	0.934376
$832$	0	0
$833$	3429.30	0.142639
$834$	−25889.2	−1.07490
$835$	−4641.96	−0.192385
$836$	16679.0	0.690020
$837$	89.0121	0.00367587
$838$	−30125.6	−1.24185
$839$	4005.19	0.164809	0.0824044	−	0.996599i	$-0.473740\pi$
$839$	4005.19	0.164809	0.0824044	+	0.996599i	$0.473740\pi$
$840$	−12554.3	−0.515674
$841$	−7952.74	−0.326079
$842$	−3683.74	−0.150772
$843$	−14184.1	−0.579510
$844$	4721.53	0.192562
$845$	0	0
$846$	17549.6	0.713199
$847$	25045.2	1.01601
$848$	24506.6	0.992406
$849$	8639.08	0.349225
$850$	29255.1	1.18052
$851$	−46065.8	−1.85560
$852$	8443.06	0.339500
$853$	−44062.6	−1.76867	−0.884334	−	0.466856i	$-0.845387\pi$
$853$	−44062.6	−1.76867	−0.884334	+	0.466856i	$0.845387\pi$
$854$	−7390.96	−0.296152
$855$	−13107.9	−0.524305
$856$	4999.36	0.199620
$857$	3365.44	0.134144	0.0670718	−	0.997748i	$-0.478634\pi$
$857$	3365.44	0.134144	0.0670718	+	0.997748i	$0.478634\pi$
$858$	0	0
$859$	−40969.3	−1.62731	−0.813653	−	0.581351i	$-0.802524\pi$
$859$	−40969.3	−1.62731	−0.813653	+	0.581351i	$0.802524\pi$
$860$	20135.2	0.798377
$861$	−5005.32	−0.198119
$862$	−12700.4	−0.501831
$863$	18132.8	0.715236	0.357618	−	0.933868i	$-0.383589\pi$
$863$	18132.8	0.715236	0.357618	+	0.933868i	$0.383589\pi$
$864$	−3912.24	−0.154047
$865$	23497.7	0.923637
$866$	13796.8	0.541380
$867$	−411.893	−0.0161345
$868$	−192.819	−0.00753999
$869$	29380.1	1.14690
$870$	−20410.5	−0.795381
$871$	0	0
$872$	−24940.9	−0.968582
$873$	7024.49	0.272329
$874$	59703.5	2.31064
$875$	−839.404	−0.0324309
$876$	−1346.63	−0.0519388
$877$	−33047.6	−1.27245	−0.636226	−	0.771503i	$-0.719505\pi$
$877$	−33047.6	−1.27245	−0.636226	+	0.771503i	$0.719505\pi$
$878$	−31698.0	−1.21840
$879$	24602.8	0.944064
$880$	66101.9	2.53215
$881$	1349.22	0.0515965	0.0257982	−	0.999667i	$-0.491787\pi$
$881$	1349.22	0.0515965	0.0257982	+	0.999667i	$0.491787\pi$
$882$	1466.80	0.0559973
$883$	−33934.5	−1.29330	−0.646651	−	0.762786i	$-0.723831\pi$
$883$	−33934.5	−1.29330	−0.646651	+	0.762786i	$0.723831\pi$
$884$	0	0
$885$	16494.3	0.626495
$886$	−19438.3	−0.737067
$887$	−45609.5	−1.72651	−0.863256	−	0.504766i	$-0.831579\pi$
$887$	−45609.5	−1.72651	−0.863256	+	0.504766i	$0.831579\pi$
$888$	11241.4	0.424817
$889$	40711.8	1.53592
$890$	9733.19	0.366581
$891$	−4278.32	−0.160863
$892$	381.037	0.0143028
$893$	53515.9	2.00542
$894$	−4301.40	−0.160918
$895$	12336.0	0.460722
$896$	−28471.4	−1.06157
$897$	0	0
$898$	3730.90	0.138643
$899$	422.655	0.0156800
$900$	3737.21	0.138415
$901$	21865.8	0.808498
$902$	17336.3	0.639952
$903$	−19373.4	−0.713962
$904$	−6669.18	−0.245369
$905$	−20222.3	−0.742777
$906$	−428.826	−0.0157249
$907$	8620.99	0.315607	0.157803	−	0.987471i	$-0.449559\pi$
$907$	8620.99	0.315607	0.157803	+	0.987471i	$0.449559\pi$
$908$	−17003.1	−0.621438
$909$	−16269.0	−0.593629
$910$	0	0
$911$	−43993.6	−1.59997	−0.799986	−	0.600019i	$-0.795160\pi$
$911$	−43993.6	−1.59997	−0.799986	+	0.600019i	$0.795160\pi$
$912$	−22148.1	−0.804164
$913$	−13451.0	−0.487584
$914$	−45758.6	−1.65597
$915$	−6008.53	−0.217088
$916$	19716.9	0.711205
$917$	2904.84	0.104609
$918$	−6480.43	−0.232991
$919$	15176.0	0.544733	0.272366	−	0.962194i	$-0.412194\pi$
$919$	15176.0	0.544733	0.272366	+	0.962194i	$0.412194\pi$
$920$	46486.6	1.66589
$921$	6176.47	0.220979
$922$	10118.8	0.361436
$923$	0	0
$924$	9267.76	0.329964
$925$	−29441.7	−1.04653
$926$	−6842.94	−0.242843
$927$	14047.0	0.497695
$928$	−18576.4	−0.657114
$929$	29749.7	1.05065	0.525326	−	0.850901i	$-0.323943\pi$
$929$	29749.7	1.05065	0.525326	+	0.850901i	$0.323943\pi$
$930$	−524.853	−0.0185060
$931$	4472.88	0.157457
$932$	−2465.50	−0.0866526
$933$	−5586.23	−0.196018
$934$	11092.0	0.388589
$935$	58978.9	2.06291
$936$	0	0
$937$	−46456.0	−1.61969	−0.809846	−	0.586643i	$-0.800449\pi$
$937$	−46456.0	−1.61969	−0.809846	+	0.586643i	$0.800449\pi$
$938$	−52391.8	−1.82372
$939$	767.688	0.0266800
$940$	−30905.5	−1.07237
$941$	−3243.36	−0.112360	−0.0561799	−	0.998421i	$-0.517892\pi$
$941$	−3243.36	−0.112360	−0.0561799	+	0.998421i	$0.517892\pi$
$942$	−10032.7	−0.347011
$943$	18533.9	0.640027
$944$	27870.0	0.960901
$945$	−7283.45	−0.250720
$946$	67101.5	2.30619
$947$	−53344.1	−1.83047	−0.915233	−	0.402926i	$-0.867993\pi$
$947$	−53344.1	−1.83047	−0.915233	+	0.402926i	$0.867993\pi$
$948$	5684.99	0.194768
$949$	0	0
$950$	38157.8	1.30316
$951$	18797.0	0.640942
$952$	−18927.0	−0.644356
$953$	53496.9	1.81840	0.909200	−	0.416360i	$-0.136694\pi$
$953$	53496.9	1.81840	0.909200	+	0.416360i	$0.136694\pi$
$954$	9352.58	0.317402
$955$	7329.22	0.248343
$956$	4223.50	0.142885
$957$	−20314.7	−0.686188
$958$	44610.1	1.50448
$959$	−51095.9	−1.72051
$960$	−6967.42	−0.234242
$961$	−29780.1	−0.999635
$962$	0	0
$963$	2900.39	0.0970549
$964$	2573.45	0.0859805
$965$	−41434.1	−1.38219
$966$	33174.4	1.10494
$967$	−48822.0	−1.62359	−0.811794	−	0.583944i	$-0.801509\pi$
$967$	−48822.0	−1.62359	−0.811794	+	0.583944i	$0.801509\pi$
$968$	22630.9	0.751429
$969$	−19761.5	−0.655141
$970$	−41419.4	−1.37103
$971$	36228.1	1.19734	0.598670	−	0.800996i	$-0.295696\pi$
$971$	36228.1	1.19734	0.598670	+	0.800996i	$0.295696\pi$
$972$	−827.845	−0.0273180
$973$	−43867.1	−1.44534
$974$	30309.0	0.997087
$975$	0	0
$976$	−10152.5	−0.332964
$977$	46689.8	1.52890	0.764451	−	0.644681i	$-0.223010\pi$
$977$	46689.8	1.52890	0.764451	+	0.644681i	$0.223010\pi$
$978$	−26807.9	−0.876505
$979$	9687.50	0.316255
$980$	−2583.09	−0.0841977
$981$	−14469.5	−0.470924
$982$	43533.8	1.41468
$983$	11501.9	0.373197	0.186598	−	0.982436i	$-0.440254\pi$
$983$	11501.9	0.373197	0.186598	+	0.982436i	$0.440254\pi$
$984$	−4522.81	−0.146526
$985$	−57774.0	−1.86886
$986$	−30771.0	−0.993862
$987$	29736.3	0.958983
$988$	0	0
$989$	71736.6	2.30646
$990$	25226.8	0.809859
$991$	7137.46	0.228788	0.114394	−	0.993435i	$-0.463507\pi$
$991$	7137.46	0.228788	0.114394	+	0.993435i	$0.463507\pi$
$992$	−477.690	−0.0152890
$993$	3215.80	0.102770
$994$	47900.5	1.52848
$995$	3465.53	0.110417
$996$	−2602.74	−0.0828022
$997$	−18389.5	−0.584152	−0.292076	−	0.956395i	$-0.594346\pi$
$997$	−18389.5	−0.584152	−0.292076	+	0.956395i	$0.594346\pi$
$998$	24897.9	0.789708
$999$	6521.75	0.206546

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.a.n.1.3	✓	9
3.2	odd	2		1521.4.a.bj.1.7		9
13.5	odd	4		507.4.b.j.337.15		18
13.8	odd	4		507.4.b.j.337.4		18
13.12	even	2		507.4.a.q.1.7	yes	9
39.38	odd	2		1521.4.a.be.1.3		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.n.1.3	✓	9	1.1	even	1	trivial
507.4.a.q.1.7	yes	9	13.12	even	2
507.4.b.j.337.4		18	13.8	odd	4
507.4.b.j.337.15		18	13.5	odd	4
1521.4.a.be.1.3		9	39.38	odd	2
1521.4.a.bj.1.7		9	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-3.37739 q^{2} -3.00000 q^{3} +3.40677 q^{4} +15.7127 q^{5} +10.1322 q^{6} +17.1681 q^{7} +15.5131 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-3.37739 q^{2} -3.00000 q^{3} +3.40677 q^{4} +15.7127 q^{5} +10.1322 q^{6} +17.1681 q^{7} +15.5131 q^{8} +9.00000 q^{9} -53.0679 q^{10} -52.8187 q^{11} -10.2203 q^{12} -57.9835 q^{14} -47.1380 q^{15} -79.6481 q^{16} -71.0654 q^{17} -30.3965 q^{18} -92.6916 q^{19} +53.5295 q^{20} -51.5044 q^{21} +178.390 q^{22} +190.712 q^{23} -46.5394 q^{24} +121.888 q^{25} -27.0000 q^{27} +58.4878 q^{28} -128.204 q^{29} +159.204 q^{30} -3.29674 q^{31} +144.898 q^{32} +158.456 q^{33} +240.016 q^{34} +269.757 q^{35} +30.6609 q^{36} -241.546 q^{37} +313.056 q^{38} +243.753 q^{40} +97.1824 q^{41} +173.950 q^{42} +376.151 q^{43} -179.941 q^{44} +141.414 q^{45} -644.109 q^{46} -577.354 q^{47} +238.944 q^{48} -48.2555 q^{49} -411.664 q^{50} +213.196 q^{51} -307.686 q^{53} +91.1896 q^{54} -829.924 q^{55} +266.331 q^{56} +278.075 q^{57} +432.995 q^{58} -349.914 q^{59} -160.588 q^{60} +127.467 q^{61} +11.1344 q^{62} +154.513 q^{63} +147.809 q^{64} -535.169 q^{66} +903.564 q^{67} -242.104 q^{68} -572.136 q^{69} -911.076 q^{70} -826.106 q^{71} +139.618 q^{72} +131.760 q^{73} +815.796 q^{74} -365.665 q^{75} -315.779 q^{76} -906.799 q^{77} -556.244 q^{79} -1251.48 q^{80} +81.0000 q^{81} -328.223 q^{82} +254.664 q^{83} -175.464 q^{84} -1116.63 q^{85} -1270.41 q^{86} +384.612 q^{87} -819.384 q^{88} -183.410 q^{89} -477.611 q^{90} +649.712 q^{92} +9.89023 q^{93} +1949.95 q^{94} -1456.43 q^{95} -434.693 q^{96} +780.498 q^{97} +162.978 q^{98} -475.369 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 8 q^{2} - 27 q^{3} + 32 q^{4} - 41 q^{5} + 24 q^{6} - q^{7} - 111 q^{8} + 81 q^{9}+O(q^{10}) \) 9 * q - 8 * q^2 - 27 * q^3 + 32 * q^4 - 41 * q^5 + 24 * q^6 - q^7 - 111 * q^8 + 81 * q^9	\( 9 q - 8 q^{2} - 27 q^{3} + 32 q^{4} - 41 q^{5} + 24 q^{6} - q^{7} - 111 q^{8} + 81 q^{9} + 198 q^{10} - 37 q^{11} - 96 q^{12} + 98 q^{14} + 123 q^{15} + 32 q^{16} - 134 q^{17} - 72 q^{18} + 72 q^{19} - 356 q^{20} + 3 q^{21} + 274 q^{22} + 226 q^{23} + 333 q^{24} + 612 q^{25} - 243 q^{27} - 132 q^{28} - 547 q^{29} - 594 q^{30} + 521 q^{31} - 721 q^{32} + 111 q^{33} + 100 q^{34} + 138 q^{35} + 288 q^{36} - 584 q^{37} - 416 q^{38} + 1342 q^{40} - 482 q^{41} - 294 q^{42} + 158 q^{43} - 1453 q^{44} - 369 q^{45} - 1537 q^{46} - 1500 q^{47} - 96 q^{48} + 642 q^{49} - 2777 q^{50} + 402 q^{51} + 1399 q^{53} + 216 q^{54} - 1408 q^{55} - 616 q^{56} - 216 q^{57} - 1455 q^{58} - 1541 q^{59} + 1068 q^{60} + 2092 q^{61} - 293 q^{62} - 9 q^{63} + 2481 q^{64} - 822 q^{66} - 252 q^{67} - 1579 q^{68} - 678 q^{69} - 2492 q^{70} - 2352 q^{71} - 999 q^{72} - 903 q^{73} + 1037 q^{74} - 1836 q^{75} + 485 q^{76} - 1686 q^{77} - 115 q^{79} - 5701 q^{80} + 729 q^{81} - 5147 q^{82} - 1207 q^{83} + 396 q^{84} - 4308 q^{85} - 5691 q^{86} + 1641 q^{87} - 484 q^{88} - 2336 q^{89} + 1782 q^{90} + 2087 q^{92} - 1563 q^{93} - 468 q^{94} - 222 q^{95} + 2163 q^{96} - 2155 q^{97} - 5593 q^{98} - 333 q^{99}+O(q^{100}) \) 9 * q - 8 * q^2 - 27 * q^3 + 32 * q^4 - 41 * q^5 + 24 * q^6 - q^7 - 111 * q^8 + 81 * q^9 + 198 * q^10 - 37 * q^11 - 96 * q^12 + 98 * q^14 + 123 * q^15 + 32 * q^16 - 134 * q^17 - 72 * q^18 + 72 * q^19 - 356 * q^20 + 3 * q^21 + 274 * q^22 + 226 * q^23 + 333 * q^24 + 612 * q^25 - 243 * q^27 - 132 * q^28 - 547 * q^29 - 594 * q^30 + 521 * q^31 - 721 * q^32 + 111 * q^33 + 100 * q^34 + 138 * q^35 + 288 * q^36 - 584 * q^37 - 416 * q^38 + 1342 * q^40 - 482 * q^41 - 294 * q^42 + 158 * q^43 - 1453 * q^44 - 369 * q^45 - 1537 * q^46 - 1500 * q^47 - 96 * q^48 + 642 * q^49 - 2777 * q^50 + 402 * q^51 + 1399 * q^53 + 216 * q^54 - 1408 * q^55 - 616 * q^56 - 216 * q^57 - 1455 * q^58 - 1541 * q^59 + 1068 * q^60 + 2092 * q^61 - 293 * q^62 - 9 * q^63 + 2481 * q^64 - 822 * q^66 - 252 * q^67 - 1579 * q^68 - 678 * q^69 - 2492 * q^70 - 2352 * q^71 - 999 * q^72 - 903 * q^73 + 1037 * q^74 - 1836 * q^75 + 485 * q^76 - 1686 * q^77 - 115 * q^79 - 5701 * q^80 + 729 * q^81 - 5147 * q^82 - 1207 * q^83 + 396 * q^84 - 4308 * q^85 - 5691 * q^86 + 1641 * q^87 - 484 * q^88 - 2336 * q^89 + 1782 * q^90 + 2087 * q^92 - 1563 * q^93 - 468 * q^94 - 222 * q^95 + 2163 * q^96 - 2155 * q^97 - 5593 * q^98 - 333 * q^99

Introduction

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